3.6.34 \(\int \frac {x^3 (d+e x)^{3/2}}{a+b x+c x^2} \, dx\) [534]

Optimal. Leaf size=581 \[ \frac {2 \left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (b^2-a c\right ) (d+e x)^{3/2}}{3 c^3}-\frac {2 (c d+b e) (d+e x)^{5/2}}{5 c^2 e^2}+\frac {2 (d+e x)^{7/2}}{7 c e^2}+\frac {\sqrt {2} \left (2 b^3 c d e-4 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-3 a e^2\right )+a c^2 \left (c d^2-a e^2\right )-\frac {2 b^4 c d e-8 a b^2 c^2 d e+4 a^2 c^3 d e-b^5 e^2-b^3 c \left (c d^2-5 a e^2\right )+a b c^2 \left (3 c d^2-5 a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{9/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left (2 b^3 c d e-4 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-3 a e^2\right )+a c^2 \left (c d^2-a e^2\right )+\frac {2 b^4 c d e-8 a b^2 c^2 d e+4 a^2 c^3 d e-b^5 e^2-b^3 c \left (c d^2-5 a e^2\right )+a b c^2 \left (3 c d^2-5 a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{9/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

2/3*(-a*c+b^2)*(e*x+d)^(3/2)/c^3-2/5*(b*e+c*d)*(e*x+d)^(5/2)/c^2/e^2+2/7*(e*x+d)^(7/2)/c/e^2+2*(2*a*b*c*e-a*c^
2*d-b^3*e+b^2*c*d)*(e*x+d)^(1/2)/c^4+arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1
/2))*2^(1/2)*(2*b^3*c*d*e-4*a*b*c^2*d*e-b^4*e^2-b^2*c*(-3*a*e^2+c*d^2)+a*c^2*(-a*e^2+c*d^2)+(-2*b^4*c*d*e+8*a*
b^2*c^2*d*e-4*a^2*c^3*d*e+b^5*e^2+b^3*c*(-5*a*e^2+c*d^2)-a*b*c^2*(-5*a*e^2+3*c*d^2))/(-4*a*c+b^2)^(1/2))/c^(9/
2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)+arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)
))^(1/2))*2^(1/2)*(2*b^3*c*d*e-4*a*b*c^2*d*e-b^4*e^2-b^2*c*(-3*a*e^2+c*d^2)+a*c^2*(-a*e^2+c*d^2)+(2*b^4*c*d*e-
8*a*b^2*c^2*d*e+4*a^2*c^3*d*e-b^5*e^2-b^3*c*(-5*a*e^2+c*d^2)+a*b*c^2*(-5*a*e^2+3*c*d^2))/(-4*a*c+b^2)^(1/2))/c
^(9/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

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Rubi [A]
time = 13.80, antiderivative size = 581, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {911, 1301, 1180, 214} \begin {gather*} \frac {\sqrt {2} \left (-\frac {4 a^2 c^3 d e-b^3 c \left (c d^2-5 a e^2\right )-8 a b^2 c^2 d e+a b c^2 \left (3 c d^2-5 a e^2\right )+b^5 \left (-e^2\right )+2 b^4 c d e}{\sqrt {b^2-4 a c}}-b^2 c \left (c d^2-3 a e^2\right )-4 a b c^2 d e+a c^2 \left (c d^2-a e^2\right )+b^4 \left (-e^2\right )+2 b^3 c d e\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{c^{9/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {2} \left (\frac {4 a^2 c^3 d e-b^3 c \left (c d^2-5 a e^2\right )-8 a b^2 c^2 d e+a b c^2 \left (3 c d^2-5 a e^2\right )+b^5 \left (-e^2\right )+2 b^4 c d e}{\sqrt {b^2-4 a c}}-b^2 c \left (c d^2-3 a e^2\right )-4 a b c^2 d e+a c^2 \left (c d^2-a e^2\right )+b^4 \left (-e^2\right )+2 b^3 c d e\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c^{9/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {2 \left (b^2-a c\right ) (d+e x)^{3/2}}{3 c^3}+\frac {2 \sqrt {d+e x} \left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right )}{c^4}-\frac {2 (d+e x)^{5/2} (b e+c d)}{5 c^2 e^2}+\frac {2 (d+e x)^{7/2}}{7 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*(d + e*x)^(3/2))/(a + b*x + c*x^2),x]

[Out]

(2*(b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e)*Sqrt[d + e*x])/c^4 + (2*(b^2 - a*c)*(d + e*x)^(3/2))/(3*c^3) - (2*(
c*d + b*e)*(d + e*x)^(5/2))/(5*c^2*e^2) + (2*(d + e*x)^(7/2))/(7*c*e^2) + (Sqrt[2]*(2*b^3*c*d*e - 4*a*b*c^2*d*
e - b^4*e^2 - b^2*c*(c*d^2 - 3*a*e^2) + a*c^2*(c*d^2 - a*e^2) - (2*b^4*c*d*e - 8*a*b^2*c^2*d*e + 4*a^2*c^3*d*e
- b^5*e^2 - b^3*c*(c*d^2 - 5*a*e^2) + a*b*c^2*(3*c*d^2 - 5*a*e^2))/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c
]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(c^(9/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) +
(Sqrt[2]*(2*b^3*c*d*e - 4*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 3*a*e^2) + a*c^2*(c*d^2 - a*e^2) + (2*b^4*c*
d*e - 8*a*b^2*c^2*d*e + 4*a^2*c^3*d*e - b^5*e^2 - b^3*c*(c*d^2 - 5*a*e^2) + a*b*c^2*(3*c*d^2 - 5*a*e^2))/Sqrt[
b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(c^(9/2)*Sqrt[
2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 911

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1301

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*((d + e*x^2)^q/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {x^3 (d+e x)^{3/2}}{a+b x+c x^2} \, dx &=\frac {2 \text {Subst}\left (\int \frac {x^4 \left (-\frac {d}{e}+\frac {x^2}{e}\right )^3}{\frac {c d^2-b d e+a e^2}{e^2}-\frac {(2 c d-b e) x^2}{e^2}+\frac {c x^4}{e^2}} \, dx,x,\sqrt {d+e x}\right )}{e}\\ &=\frac {2 \text {Subst}\left (\int \left (\frac {e \left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right )}{c^4}+\frac {\left (b^2-a c\right ) e x^2}{c^3}-\frac {(c d+b e) x^4}{c^2 e}+\frac {x^6}{c e}-\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \left (c d^2-b d e+a e^2\right )+\left (2 b^3 c d e-4 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-3 a e^2\right )+a c^2 \left (c d^2-a e^2\right )\right ) x^2}{c^4 e \left (\frac {c d^2-b d e+a e^2}{e^2}-\frac {(2 c d-b e) x^2}{e^2}+\frac {c x^4}{e^2}\right )}\right ) \, dx,x,\sqrt {d+e x}\right )}{e}\\ &=\frac {2 \left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (b^2-a c\right ) (d+e x)^{3/2}}{3 c^3}-\frac {2 (c d+b e) (d+e x)^{5/2}}{5 c^2 e^2}+\frac {2 (d+e x)^{7/2}}{7 c e^2}-\frac {2 \text {Subst}\left (\int \frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \left (c d^2-b d e+a e^2\right )+\left (2 b^3 c d e-4 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-3 a e^2\right )+a c^2 \left (c d^2-a e^2\right )\right ) x^2}{\frac {c d^2-b d e+a e^2}{e^2}-\frac {(2 c d-b e) x^2}{e^2}+\frac {c x^4}{e^2}} \, dx,x,\sqrt {d+e x}\right )}{c^4 e^2}\\ &=\frac {2 \left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (b^2-a c\right ) (d+e x)^{3/2}}{3 c^3}-\frac {2 (c d+b e) (d+e x)^{5/2}}{5 c^2 e^2}+\frac {2 (d+e x)^{7/2}}{7 c e^2}-\frac {\left (2 b^3 c d e-4 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-3 a e^2\right )+a c^2 \left (c d^2-a e^2\right )-\frac {2 b^4 c d e-8 a b^2 c^2 d e+4 a^2 c^3 d e-b^5 e^2-b^3 c \left (c d^2-5 a e^2\right )+a b c^2 \left (3 c d^2-5 a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{-\frac {\sqrt {b^2-4 a c}}{2 e}-\frac {2 c d-b e}{2 e^2}+\frac {c x^2}{e^2}} \, dx,x,\sqrt {d+e x}\right )}{c^4 e^2}-\frac {\left (2 b^3 c d e-4 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-3 a e^2\right )+a c^2 \left (c d^2-a e^2\right )+\frac {2 b^4 c d e-8 a b^2 c^2 d e+4 a^2 c^3 d e-b^5 e^2-b^3 c \left (c d^2-5 a e^2\right )+a b c^2 \left (3 c d^2-5 a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b^2-4 a c}}{2 e}-\frac {2 c d-b e}{2 e^2}+\frac {c x^2}{e^2}} \, dx,x,\sqrt {d+e x}\right )}{c^4 e^2}\\ &=\frac {2 \left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (b^2-a c\right ) (d+e x)^{3/2}}{3 c^3}-\frac {2 (c d+b e) (d+e x)^{5/2}}{5 c^2 e^2}+\frac {2 (d+e x)^{7/2}}{7 c e^2}+\frac {\sqrt {2} \left (2 b^3 c d e-4 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-3 a e^2\right )+a c^2 \left (c d^2-a e^2\right )-\frac {2 b^4 c d e-8 a b^2 c^2 d e+4 a^2 c^3 d e-b^5 e^2-b^3 c \left (c d^2-5 a e^2\right )+a b c^2 \left (3 c d^2-5 a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{9/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left (2 b^3 c d e-4 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-3 a e^2\right )+a c^2 \left (c d^2-a e^2\right )+\frac {2 b^4 c d e-8 a b^2 c^2 d e+4 a^2 c^3 d e-b^5 e^2-b^3 c \left (c d^2-5 a e^2\right )+a b c^2 \left (3 c d^2-5 a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{9/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.46, size = 755, normalized size = 1.30 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (105 b^3 e^3+3 c^3 (2 d-5 e x) (d+e x)^2-35 b c e^2 (4 b d+6 a e+b e x)+7 c^2 e \left (3 b (d+e x)^2+5 a e (4 d+e x)\right )\right )}{105 c^4 e^2}+\frac {\left (i b^5 e^2+b^4 e \left (-2 i c d+\sqrt {-b^2+4 a c} e\right )+i b^3 c \left (c d^2+e \left (2 i \sqrt {-b^2+4 a c} d-5 a e\right )\right )+a c^2 \left (a \sqrt {-b^2+4 a c} e^2-c d \left (\sqrt {-b^2+4 a c} d+4 i a e\right )\right )+a b c^2 \left (-3 i c d^2+e \left (4 \sqrt {-b^2+4 a c} d+5 i a e\right )\right )+b^2 c \left (-3 a \sqrt {-b^2+4 a c} e^2+c d \left (\sqrt {-b^2+4 a c} d+8 i a e\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{c^{9/2} \sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\left (-i b^5 e^2+b^4 e \left (2 i c d+\sqrt {-b^2+4 a c} e\right )+a c^2 \left (a \sqrt {-b^2+4 a c} e^2+c d \left (-\sqrt {-b^2+4 a c} d+4 i a e\right )\right )+a b c^2 \left (3 i c d^2+e \left (4 \sqrt {-b^2+4 a c} d-5 i a e\right )\right )+b^3 c \left (-i c d^2+e \left (-2 \sqrt {-b^2+4 a c} d+5 i a e\right )\right )+b^2 c \left (-3 a \sqrt {-b^2+4 a c} e^2+c d \left (\sqrt {-b^2+4 a c} d-8 i a e\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{c^{9/2} \sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(d + e*x)^(3/2))/(a + b*x + c*x^2),x]

[Out]

(-2*Sqrt[d + e*x]*(105*b^3*e^3 + 3*c^3*(2*d - 5*e*x)*(d + e*x)^2 - 35*b*c*e^2*(4*b*d + 6*a*e + b*e*x) + 7*c^2*
e*(3*b*(d + e*x)^2 + 5*a*e*(4*d + e*x))))/(105*c^4*e^2) + ((I*b^5*e^2 + b^4*e*((-2*I)*c*d + Sqrt[-b^2 + 4*a*c]
*e) + I*b^3*c*(c*d^2 + e*((2*I)*Sqrt[-b^2 + 4*a*c]*d - 5*a*e)) + a*c^2*(a*Sqrt[-b^2 + 4*a*c]*e^2 - c*d*(Sqrt[-
b^2 + 4*a*c]*d + (4*I)*a*e)) + a*b*c^2*((-3*I)*c*d^2 + e*(4*Sqrt[-b^2 + 4*a*c]*d + (5*I)*a*e)) + b^2*c*(-3*a*S
qrt[-b^2 + 4*a*c]*e^2 + c*d*(Sqrt[-b^2 + 4*a*c]*d + (8*I)*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-
2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(c^(9/2)*Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[-2*c*d + (b - I*Sqrt[-b^2 + 4*a*c
])*e]) + (((-I)*b^5*e^2 + b^4*e*((2*I)*c*d + Sqrt[-b^2 + 4*a*c]*e) + a*c^2*(a*Sqrt[-b^2 + 4*a*c]*e^2 + c*d*(-(
Sqrt[-b^2 + 4*a*c]*d) + (4*I)*a*e)) + a*b*c^2*((3*I)*c*d^2 + e*(4*Sqrt[-b^2 + 4*a*c]*d - (5*I)*a*e)) + b^3*c*(
(-I)*c*d^2 + e*(-2*Sqrt[-b^2 + 4*a*c]*d + (5*I)*a*e)) + b^2*c*(-3*a*Sqrt[-b^2 + 4*a*c]*e^2 + c*d*(Sqrt[-b^2 +
4*a*c]*d - (8*I)*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]])/(
c^(9/2)*Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[-2*c*d + (b + I*Sqrt[-b^2 + 4*a*c])*e])

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Maple [A]
time = 0.16, size = 893, normalized size = 1.54

method	result	size
derivativedivides	\(\frac {\frac {2 \left (\frac {\left (e x +d \right )^{\frac {7}{2}} c^{3}}{7}-\frac {b \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {a \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {b^{2} c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a b c \,e^{3} \sqrt {e x +d}-a \,c^{2} d \,e^{2} \sqrt {e x +d}-b^{3} e^{3} \sqrt {e x +d}+b^{2} c d \,e^{2} \sqrt {e x +d}\right )}{c^{4}}-\frac {8 e^{2} \left (\frac {\left (-5 a^{2} b \,c^{2} e^{3}+4 a^{2} c^{3} d \,e^{2}+5 a \,b^{3} e^{3} c -8 a \,b^{2} c^{2} d \,e^{2}+3 a b \,c^{3} d^{2} e -b^{5} e^{3}+2 b^{4} d \,e^{2} c -b^{3} c^{2} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c \,e^{2}-4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{3} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (5 a^{2} b \,c^{2} e^{3}-4 a^{2} c^{3} d \,e^{2}-5 a \,b^{3} e^{3} c +8 a \,b^{2} c^{2} d \,e^{2}-3 a b \,c^{3} d^{2} e +b^{5} e^{3}-2 b^{4} d \,e^{2} c +b^{3} c^{2} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c \,e^{2}-4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{3} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c^{2} d^{2}\right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c^{3}}}{e^{2}}\)	\(893\)
default	\(\frac {\frac {2 \left (\frac {\left (e x +d \right )^{\frac {7}{2}} c^{3}}{7}-\frac {b \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {a \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {b^{2} c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a b c \,e^{3} \sqrt {e x +d}-a \,c^{2} d \,e^{2} \sqrt {e x +d}-b^{3} e^{3} \sqrt {e x +d}+b^{2} c d \,e^{2} \sqrt {e x +d}\right )}{c^{4}}-\frac {8 e^{2} \left (\frac {\left (-5 a^{2} b \,c^{2} e^{3}+4 a^{2} c^{3} d \,e^{2}+5 a \,b^{3} e^{3} c -8 a \,b^{2} c^{2} d \,e^{2}+3 a b \,c^{3} d^{2} e -b^{5} e^{3}+2 b^{4} d \,e^{2} c -b^{3} c^{2} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c \,e^{2}-4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{3} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (5 a^{2} b \,c^{2} e^{3}-4 a^{2} c^{3} d \,e^{2}-5 a \,b^{3} e^{3} c +8 a \,b^{2} c^{2} d \,e^{2}-3 a b \,c^{3} d^{2} e +b^{5} e^{3}-2 b^{4} d \,e^{2} c +b^{3} c^{2} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c \,e^{2}-4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{3} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c^{2} d^{2}\right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c^{3}}}{e^{2}}\)	\(893\)
risch	\(\text {Expression too large to display}\)	\(2998\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(e*x+d)^(3/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

2/e^2*(1/c^4*(1/7*(e*x+d)^(7/2)*c^3-1/5*b*c^2*e*(e*x+d)^(5/2)-1/5*c^3*d*(e*x+d)^(5/2)-1/3*a*c^2*e^2*(e*x+d)^(3
/2)+1/3*b^2*c*e^2*(e*x+d)^(3/2)+2*a*b*c*e^3*(e*x+d)^(1/2)-a*c^2*d*e^2*(e*x+d)^(1/2)-b^3*e^3*(e*x+d)^(1/2)+b^2*
c*d*e^2*(e*x+d)^(1/2))-4*e^2/c^3*(1/8*(-5*a^2*b*c^2*e^3+4*a^2*c^3*d*e^2+5*a*b^3*e^3*c-8*a*b^2*c^2*d*e^2+3*a*b*
c^3*d^2*e-b^5*e^3+2*b^4*d*e^2*c-b^3*c^2*d^2*e-(-e^2*(4*a*c-b^2))^(1/2)*a^2*c^2*e^2+3*(-e^2*(4*a*c-b^2))^(1/2)*
a*b^2*c*e^2-4*(-e^2*(4*a*c-b^2))^(1/2)*a*b*c^2*d*e+(-e^2*(4*a*c-b^2))^(1/2)*a*c^3*d^2-(-e^2*(4*a*c-b^2))^(1/2)
*b^4*e^2+2*(-e^2*(4*a*c-b^2))^(1/2)*b^3*c*d*e-(-e^2*(4*a*c-b^2))^(1/2)*b^2*c^2*d^2)/c/(-e^2*(4*a*c-b^2))^(1/2)
*2^(1/2)/((e*b-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((e*b-2*c*d+(-e^2*(4*a*
c-b^2))^(1/2))*c)^(1/2))-1/8*(5*a^2*b*c^2*e^3-4*a^2*c^3*d*e^2-5*a*b^3*e^3*c+8*a*b^2*c^2*d*e^2-3*a*b*c^3*d^2*e+
b^5*e^3-2*b^4*d*e^2*c+b^3*c^2*d^2*e-(-e^2*(4*a*c-b^2))^(1/2)*a^2*c^2*e^2+3*(-e^2*(4*a*c-b^2))^(1/2)*a*b^2*c*e^
2-4*(-e^2*(4*a*c-b^2))^(1/2)*a*b*c^2*d*e+(-e^2*(4*a*c-b^2))^(1/2)*a*c^3*d^2-(-e^2*(4*a*c-b^2))^(1/2)*b^4*e^2+2
*(-e^2*(4*a*c-b^2))^(1/2)*b^3*c*d*e-(-e^2*(4*a*c-b^2))^(1/2)*b^2*c^2*d^2)/c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/(
(-e*b+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-e*b+2*c*d+(-e^2*(4*a*c-b^2))
^(1/2))*c)^(1/2))))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

integrate((x*e + d)^(3/2)*x^3/(c*x^2 + b*x + a), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 11377 vs. \(2 (536) = 1072\).
time = 21.02, size = 11377, normalized size = 19.58 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

-1/210*(105*sqrt(2)*c^4*sqrt(((b^6*c^3 - 6*a*b^4*c^4 + 9*a^2*b^2*c^5 - 2*a^3*c^6)*d^3 - 3*(b^7*c^2 - 7*a*b^5*c
^3 + 14*a^2*b^3*c^4 - 7*a^3*b*c^5)*d^2*e + 3*(b^8*c - 8*a*b^6*c^2 + 20*a^2*b^4*c^3 - 16*a^3*b^2*c^4 + 2*a^4*c^
5)*d*e^2 - (b^9 - 9*a*b^7*c + 27*a^2*b^5*c^2 - 30*a^3*b^3*c^3 + 9*a^4*b*c^4)*e^3 + (b^2*c^9 - 4*a*c^10)*sqrt((
(b^10*c^6 - 8*a*b^8*c^7 + 22*a^2*b^6*c^8 - 24*a^3*b^4*c^9 + 9*a^4*b^2*c^10)*d^6 - 6*(b^11*c^5 - 9*a*b^9*c^6 +
29*a^2*b^7*c^7 - 40*a^3*b^5*c^8 + 22*a^4*b^3*c^9 - 3*a^5*b*c^10)*d^5*e + 3*(5*b^12*c^4 - 50*a*b^10*c^5 + 185*a
^2*b^8*c^6 - 310*a^3*b^6*c^7 + 230*a^4*b^4*c^8 - 60*a^5*b^2*c^9 + 3*a^6*c^10)*d^4*e^2 - 2*(10*b^13*c^3 - 110*a
*b^11*c^4 + 460*a^2*b^9*c^5 - 910*a^3*b^7*c^6 + 860*a^4*b^5*c^7 - 340*a^5*b^3*c^8 + 39*a^6*b*c^9)*d^3*e^3 + 3*
(5*b^14*c^2 - 60*a*b^12*c^3 + 280*a^2*b^10*c^4 - 640*a^3*b^8*c^5 + 740*a^4*b^6*c^6 - 400*a^5*b^4*c^7 + 80*a^6*
b^2*c^8 - 2*a^7*c^9)*d^2*e^4 - 6*(b^15*c - 13*a*b^13*c^2 + 67*a^2*b^11*c^3 - 174*a^3*b^9*c^4 + 239*a^4*b^7*c^5
- 166*a^5*b^5*c^6 + 50*a^6*b^3*c^7 - 4*a^7*b*c^8)*d*e^5 + (b^16 - 14*a*b^14*c + 79*a^2*b^12*c^2 - 230*a^3*b^1
0*c^3 + 367*a^4*b^8*c^4 - 314*a^5*b^6*c^5 + 130*a^6*b^4*c^6 - 20*a^7*b^2*c^7 + a^8*c^8)*e^6)/(b^2*c^18 - 4*a*c
^19)))/(b^2*c^9 - 4*a*c^10))*e^2*log(sqrt(2)*((b^9*c^4 - 9*a*b^7*c^5 + 27*a^2*b^5*c^6 - 31*a^3*b^3*c^7 + 12*a^
4*b*c^8)*d^4 - (4*b^10*c^3 - 40*a*b^8*c^4 + 140*a^2*b^6*c^5 - 203*a^3*b^4*c^6 + 111*a^4*b^2*c^7 - 12*a^5*c^8)*
d^3*e + 3*(2*b^11*c^2 - 22*a*b^9*c^3 + 88*a^2*b^7*c^4 - 155*a^3*b^5*c^5 + 114*a^4*b^3*c^6 - 24*a^5*b*c^7)*d^2*
e^2 - (4*b^12*c - 48*a*b^10*c^2 + 216*a^2*b^8*c^3 - 449*a^3*b^6*c^4 + 423*a^4*b^4*c^5 - 141*a^5*b^2*c^6 + 4*a^
6*c^7)*d*e^3 + (b^13 - 13*a*b^11*c + 65*a^2*b^9*c^2 - 156*a^3*b^7*c^3 + 181*a^4*b^5*c^4 - 86*a^5*b^3*c^5 + 8*a
^6*b*c^6)*e^4 - ((b^5*c^10 - 7*a*b^3*c^11 + 12*a^2*b*c^12)*d - (b^6*c^9 - 8*a*b^4*c^10 + 18*a^2*b^2*c^11 - 8*a
^3*c^12)*e)*sqrt(((b^10*c^6 - 8*a*b^8*c^7 + 22*a^2*b^6*c^8 - 24*a^3*b^4*c^9 + 9*a^4*b^2*c^10)*d^6 - 6*(b^11*c^
5 - 9*a*b^9*c^6 + 29*a^2*b^7*c^7 - 40*a^3*b^5*c^8 + 22*a^4*b^3*c^9 - 3*a^5*b*c^10)*d^5*e + 3*(5*b^12*c^4 - 50*
a*b^10*c^5 + 185*a^2*b^8*c^6 - 310*a^3*b^6*c^7 + 230*a^4*b^4*c^8 - 60*a^5*b^2*c^9 + 3*a^6*c^10)*d^4*e^2 - 2*(1
0*b^13*c^3 - 110*a*b^11*c^4 + 460*a^2*b^9*c^5 - 910*a^3*b^7*c^6 + 860*a^4*b^5*c^7 - 340*a^5*b^3*c^8 + 39*a^6*b
*c^9)*d^3*e^3 + 3*(5*b^14*c^2 - 60*a*b^12*c^3 + 280*a^2*b^10*c^4 - 640*a^3*b^8*c^5 + 740*a^4*b^6*c^6 - 400*a^5
*b^4*c^7 + 80*a^6*b^2*c^8 - 2*a^7*c^9)*d^2*e^4 - 6*(b^15*c - 13*a*b^13*c^2 + 67*a^2*b^11*c^3 - 174*a^3*b^9*c^4
+ 239*a^4*b^7*c^5 - 166*a^5*b^5*c^6 + 50*a^6*b^3*c^7 - 4*a^7*b*c^8)*d*e^5 + (b^16 - 14*a*b^14*c + 79*a^2*b^12
*c^2 - 230*a^3*b^10*c^3 + 367*a^4*b^8*c^4 - 314*a^5*b^6*c^5 + 130*a^6*b^4*c^6 - 20*a^7*b^2*c^7 + a^8*c^8)*e^6)
/(b^2*c^18 - 4*a*c^19)))*sqrt(((b^6*c^3 - 6*a*b^4*c^4 + 9*a^2*b^2*c^5 - 2*a^3*c^6)*d^3 - 3*(b^7*c^2 - 7*a*b^5*
c^3 + 14*a^2*b^3*c^4 - 7*a^3*b*c^5)*d^2*e + 3*(b^8*c - 8*a*b^6*c^2 + 20*a^2*b^4*c^3 - 16*a^3*b^2*c^4 + 2*a^4*c
^5)*d*e^2 - (b^9 - 9*a*b^7*c + 27*a^2*b^5*c^2 - 30*a^3*b^3*c^3 + 9*a^4*b*c^4)*e^3 + (b^2*c^9 - 4*a*c^10)*sqrt(
((b^10*c^6 - 8*a*b^8*c^7 + 22*a^2*b^6*c^8 - 24*a^3*b^4*c^9 + 9*a^4*b^2*c^10)*d^6 - 6*(b^11*c^5 - 9*a*b^9*c^6 +
29*a^2*b^7*c^7 - 40*a^3*b^5*c^8 + 22*a^4*b^3*c^9 - 3*a^5*b*c^10)*d^5*e + 3*(5*b^12*c^4 - 50*a*b^10*c^5 + 185*
a^2*b^8*c^6 - 310*a^3*b^6*c^7 + 230*a^4*b^4*c^8 - 60*a^5*b^2*c^9 + 3*a^6*c^10)*d^4*e^2 - 2*(10*b^13*c^3 - 110*
a*b^11*c^4 + 460*a^2*b^9*c^5 - 910*a^3*b^7*c^6 + 860*a^4*b^5*c^7 - 340*a^5*b^3*c^8 + 39*a^6*b*c^9)*d^3*e^3 + 3
*(5*b^14*c^2 - 60*a*b^12*c^3 + 280*a^2*b^10*c^4 - 640*a^3*b^8*c^5 + 740*a^4*b^6*c^6 - 400*a^5*b^4*c^7 + 80*a^6
*b^2*c^8 - 2*a^7*c^9)*d^2*e^4 - 6*(b^15*c - 13*a*b^13*c^2 + 67*a^2*b^11*c^3 - 174*a^3*b^9*c^4 + 239*a^4*b^7*c^
5 - 166*a^5*b^5*c^6 + 50*a^6*b^3*c^7 - 4*a^7*b*c^8)*d*e^5 + (b^16 - 14*a*b^14*c + 79*a^2*b^12*c^2 - 230*a^3*b^
10*c^3 + 367*a^4*b^8*c^4 - 314*a^5*b^6*c^5 + 130*a^6*b^4*c^6 - 20*a^7*b^2*c^7 + a^8*c^8)*e^6)/(b^2*c^18 - 4*a*
c^19)))/(b^2*c^9 - 4*a*c^10)) - 4*((a^3*b^5*c^4 - 4*a^4*b^3*c^5 + 3*a^5*b*c^6)*d^5 - (4*a^3*b^6*c^3 - 19*a^4*b
^4*c^4 + 21*a^5*b^2*c^5 - 3*a^6*c^6)*d^4*e + 2*(3*a^3*b^7*c^2 - 16*a^4*b^5*c^3 + 22*a^5*b^3*c^4 - 6*a^6*b*c^5)
*d^3*e^2 - 2*(2*a^3*b^8*c - 11*a^4*b^6*c^2 + 15*a^5*b^4*c^3 - 2*a^6*b^2*c^4 - a^7*c^5)*d^2*e^3 + (a^3*b^9 - 4*
a^4*b^7*c - 3*a^5*b^5*c^2 + 20*a^6*b^3*c^3 - 11*a^7*b*c^4)*d*e^4 - (a^4*b^8 - 7*a^5*b^6*c + 15*a^6*b^4*c^2 - 1
0*a^7*b^2*c^3 + a^8*c^4)*e^5)*sqrt(x*e + d)) - 105*sqrt(2)*c^4*sqrt(((b^6*c^3 - 6*a*b^4*c^4 + 9*a^2*b^2*c^5 -
2*a^3*c^6)*d^3 - 3*(b^7*c^2 - 7*a*b^5*c^3 + 14*a^2*b^3*c^4 - 7*a^3*b*c^5)*d^2*e + 3*(b^8*c - 8*a*b^6*c^2 + 20*
a^2*b^4*c^3 - 16*a^3*b^2*c^4 + 2*a^4*c^5)*d*e^2 - (b^9 - 9*a*b^7*c + 27*a^2*b^5*c^2 - 30*a^3*b^3*c^3 + 9*a^4*b
*c^4)*e^3 + (b^2*c^9 - 4*a*c^10)*sqrt(((b^10*c^6 - 8*a*b^8*c^7 + 22*a^2*b^6*c^8 - 24*a^3*b^4*c^9 + 9*a^4*b^2*c
^10)*d^6 - 6*(b^11*c^5 - 9*a*b^9*c^6 + 29*a^2*b^7*c^7 - 40*a^3*b^5*c^8 + 22*a^4*b^3*c^9 - 3*a^5*b*c^10)*d^5*e
+ 3*(5*b^12*c^4 - 50*a*b^10*c^5 + 185*a^2*b^8*c...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x+d)**(3/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1362 vs. \(2 (536) = 1072\).
time = 1.09, size = 1362, normalized size = 2.34 \begin {gather*} \frac {{\left ({\left ({\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} d^{2} e - 2 \, {\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} d e^{2} + {\left (b^{6} - 7 \, a b^{4} c + 13 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2} - 2 \, {\left ({\left (b^{2} c^{4} - a c^{5}\right )} \sqrt {b^{2} - 4 \, a c} d^{3} - {\left (2 \, b^{3} c^{3} - 3 \, a b c^{4}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} e + {\left (b^{4} c^{2} - a b^{2} c^{3} - a^{2} c^{4}\right )} \sqrt {b^{2} - 4 \, a c} d e^{2} - {\left (a b^{3} c^{2} - 2 \, a^{2} b c^{3}\right )} \sqrt {b^{2} - 4 \, a c} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c \right |} + {\left (2 \, {\left (b^{3} c^{5} - 3 \, a b c^{6}\right )} d^{3} - {\left (5 \, b^{4} c^{4} - 19 \, a b^{2} c^{5} + 8 \, a^{2} c^{6}\right )} d^{2} e + 2 \, {\left (2 \, b^{5} c^{3} - 9 \, a b^{3} c^{4} + 7 \, a^{2} b c^{5}\right )} d e^{2} - {\left (b^{6} c^{2} - 5 \, a b^{4} c^{3} + 5 \, a^{2} b^{2} c^{4}\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {{\left (2 \, c^{8} d e^{16} - b c^{7} e^{17} + \sqrt {-4 \, {\left (c^{8} d^{2} e^{16} - b c^{7} d e^{17} + a c^{7} e^{18}\right )} c^{8} e^{16} + {\left (2 \, c^{8} d e^{16} - b c^{7} e^{17}\right )}^{2}}\right )} e^{\left (-16\right )}}{c^{8}}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{7} d^{2} - \sqrt {b^{2} - 4 \, a c} b c^{6} d e + \sqrt {b^{2} - 4 \, a c} a c^{6} e^{2}\right )} c^{2}} - \frac {{\left ({\left ({\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} d^{2} e - 2 \, {\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} d e^{2} + {\left (b^{6} - 7 \, a b^{4} c + 13 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2} + 2 \, {\left ({\left (b^{2} c^{4} - a c^{5}\right )} \sqrt {b^{2} - 4 \, a c} d^{3} - {\left (2 \, b^{3} c^{3} - 3 \, a b c^{4}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} e + {\left (b^{4} c^{2} - a b^{2} c^{3} - a^{2} c^{4}\right )} \sqrt {b^{2} - 4 \, a c} d e^{2} - {\left (a b^{3} c^{2} - 2 \, a^{2} b c^{3}\right )} \sqrt {b^{2} - 4 \, a c} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c \right |} + {\left (2 \, {\left (b^{3} c^{5} - 3 \, a b c^{6}\right )} d^{3} - {\left (5 \, b^{4} c^{4} - 19 \, a b^{2} c^{5} + 8 \, a^{2} c^{6}\right )} d^{2} e + 2 \, {\left (2 \, b^{5} c^{3} - 9 \, a b^{3} c^{4} + 7 \, a^{2} b c^{5}\right )} d e^{2} - {\left (b^{6} c^{2} - 5 \, a b^{4} c^{3} + 5 \, a^{2} b^{2} c^{4}\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {{\left (2 \, c^{8} d e^{16} - b c^{7} e^{17} - \sqrt {-4 \, {\left (c^{8} d^{2} e^{16} - b c^{7} d e^{17} + a c^{7} e^{18}\right )} c^{8} e^{16} + {\left (2 \, c^{8} d e^{16} - b c^{7} e^{17}\right )}^{2}}\right )} e^{\left (-16\right )}}{c^{8}}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{7} d^{2} - \sqrt {b^{2} - 4 \, a c} b c^{6} d e + \sqrt {b^{2} - 4 \, a c} a c^{6} e^{2}\right )} c^{2}} + \frac {2 \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{6} e^{12} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{6} d e^{12} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{5} e^{13} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{4} e^{14} - 35 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{5} e^{14} + 105 \, \sqrt {x e + d} b^{2} c^{4} d e^{14} - 105 \, \sqrt {x e + d} a c^{5} d e^{14} - 105 \, \sqrt {x e + d} b^{3} c^{3} e^{15} + 210 \, \sqrt {x e + d} a b c^{4} e^{15}\right )} e^{\left (-14\right )}}{105 \, c^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

1/4*(((b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d^2*e - 2*(b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d*e^2 + (b^6 - 7*a*b^4
*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*c^2 - 2*((b^2*c^4 - a*c
^5)*sqrt(b^2 - 4*a*c)*d^3 - (2*b^3*c^3 - 3*a*b*c^4)*sqrt(b^2 - 4*a*c)*d^2*e + (b^4*c^2 - a*b^2*c^3 - a^2*c^4)*
sqrt(b^2 - 4*a*c)*d*e^2 - (a*b^3*c^2 - 2*a^2*b*c^3)*sqrt(b^2 - 4*a*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 -
4*a*c)*c)*e)*abs(c) + (2*(b^3*c^5 - 3*a*b*c^6)*d^3 - (5*b^4*c^4 - 19*a*b^2*c^5 + 8*a^2*c^6)*d^2*e + 2*(2*b^5*
c^3 - 9*a*b^3*c^4 + 7*a^2*b*c^5)*d*e^2 - (b^6*c^2 - 5*a*b^4*c^3 + 5*a^2*b^2*c^4)*e^3)*sqrt(-4*c^2*d + 2*(b*c +
sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*c^8*d*e^16 - b*c^7*e^17 + sqrt(-4*(c^8*d^2
*e^16 - b*c^7*d*e^17 + a*c^7*e^18)*c^8*e^16 + (2*c^8*d*e^16 - b*c^7*e^17)^2))*e^(-16)/c^8))/((sqrt(b^2 - 4*a*c
)*c^7*d^2 - sqrt(b^2 - 4*a*c)*b*c^6*d*e + sqrt(b^2 - 4*a*c)*a*c^6*e^2)*c^2) - 1/4*(((b^4*c^2 - 5*a*b^2*c^3 + 4
*a^2*c^4)*d^2*e - 2*(b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d*e^2 + (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)
*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*c^2 + 2*((b^2*c^4 - a*c^5)*sqrt(b^2 - 4*a*c)*d^3 - (2*b
^3*c^3 - 3*a*b*c^4)*sqrt(b^2 - 4*a*c)*d^2*e + (b^4*c^2 - a*b^2*c^3 - a^2*c^4)*sqrt(b^2 - 4*a*c)*d*e^2 - (a*b^3
*c^2 - 2*a^2*b*c^3)*sqrt(b^2 - 4*a*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(c) + (2*(b^3*c
^5 - 3*a*b*c^6)*d^3 - (5*b^4*c^4 - 19*a*b^2*c^5 + 8*a^2*c^6)*d^2*e + 2*(2*b^5*c^3 - 9*a*b^3*c^4 + 7*a^2*b*c^5)
*d*e^2 - (b^6*c^2 - 5*a*b^4*c^3 + 5*a^2*b^2*c^4)*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arctan
(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*c^8*d*e^16 - b*c^7*e^17 - sqrt(-4*(c^8*d^2*e^16 - b*c^7*d*e^17 + a*c^7*e^1
8)*c^8*e^16 + (2*c^8*d*e^16 - b*c^7*e^17)^2))*e^(-16)/c^8))/((sqrt(b^2 - 4*a*c)*c^7*d^2 - sqrt(b^2 - 4*a*c)*b*
c^6*d*e + sqrt(b^2 - 4*a*c)*a*c^6*e^2)*c^2) + 2/105*(15*(x*e + d)^(7/2)*c^6*e^12 - 21*(x*e + d)^(5/2)*c^6*d*e^
12 - 21*(x*e + d)^(5/2)*b*c^5*e^13 + 35*(x*e + d)^(3/2)*b^2*c^4*e^14 - 35*(x*e + d)^(3/2)*a*c^5*e^14 + 105*sqr
t(x*e + d)*b^2*c^4*d*e^14 - 105*sqrt(x*e + d)*a*c^5*d*e^14 - 105*sqrt(x*e + d)*b^3*c^3*e^15 + 210*sqrt(x*e + d
)*a*b*c^4*e^15)*e^(-14)/c^7

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Mupad [B]
time = 7.14, size = 2500, normalized size = 4.30 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(d + e*x)^(3/2))/(a + b*x + c*x^2),x)

[Out]

atan(((((8*(4*a^3*c^8*d*e^4 - 8*a^3*b*c^7*e^5 - a*b^5*c^5*e^5 + b^6*c^5*d*e^4 + 6*a^2*b^3*c^6*e^5 + 4*a^2*c^9*
d^3*e^2 + b^4*c^7*d^3*e^2 - 2*b^5*c^6*d^2*e^3 - 5*a*b^4*c^6*d*e^4 - 5*a*b^2*c^8*d^3*e^2 + 11*a*b^3*c^7*d^2*e^3
- 12*a^2*b*c^8*d^2*e^3 + 3*a^2*b^2*c^7*d*e^4))/c^7 - (8*(d + e*x)^(1/2)*(-(b^11*e^3 - 8*a^4*c^7*d^3 - b^8*c^3
*d^3 + b^8*e^3*(-(4*a*c - b^2)^3)^(1/2) + 10*a*b^6*c^4*d^3 - 36*a^5*b*c^5*e^3 + 24*a^5*c^6*d*e^2 + 3*b^9*c^2*d
^2*e - 33*a^2*b^4*c^5*d^3 + 38*a^3*b^2*c^6*d^3 + 63*a^2*b^7*c^2*e^3 - 138*a^3*b^5*c^3*e^3 + 129*a^4*b^3*c^4*e^
3 + a^4*c^4*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^5*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e^3 - 3*b^10*c*d*
e^2 + 15*a^2*b^4*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e^
3*(-(4*a*c - b^2)^3)^(1/2) - 33*a*b^7*c^3*d^2*e + 36*a*b^8*c^2*d*e^2 + 84*a^4*b*c^6*d^2*e - 3*b^7*c*d*e^2*(-(4
*a*c - b^2)^3)^(1/2) + 4*a*b^3*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^5*d^3*(-(4*a*c - b^2)^3)^(1/2) + 1
26*a^2*b^5*c^4*d^2*e - 156*a^2*b^6*c^3*d*e^2 - 189*a^3*b^3*c^5*d^2*e + 288*a^3*b^4*c^4*d*e^2 - 198*a^4*b^2*c^5
*d*e^2 - 3*a^3*c^5*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*b^6*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 15*a*b^4*c^3*d^
2*e*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^5*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^3*b*c^4*d*e^2*(-(4*a*c - b^2
)^3)^(1/2) + 18*a^2*b^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 30*a^2*b^3*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2
*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2 - 4*a*b*c^10*e^3 + 8*a*c^11*d*
e^2))/c^7)*(-(b^11*e^3 - 8*a^4*c^7*d^3 - b^8*c^3*d^3 + b^8*e^3*(-(4*a*c - b^2)^3)^(1/2) + 10*a*b^6*c^4*d^3 - 3
6*a^5*b*c^5*e^3 + 24*a^5*c^6*d*e^2 + 3*b^9*c^2*d^2*e - 33*a^2*b^4*c^5*d^3 + 38*a^3*b^2*c^6*d^3 + 63*a^2*b^7*c^
2*e^3 - 138*a^3*b^5*c^3*e^3 + 129*a^4*b^3*c^4*e^3 + a^4*c^4*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^5*c^3*d^3*(-(4*a*
c - b^2)^3)^(1/2) - 13*a*b^9*c*e^3 - 3*b^10*c*d*e^2 + 15*a^2*b^4*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2
*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 33*a*b^7*c^3*d^2*e + 36*a*b^8*c^2
*d*e^2 + 84*a^4*b*c^6*d^2*e - 3*b^7*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b^3*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2
) - 3*a^2*b*c^5*d^3*(-(4*a*c - b^2)^3)^(1/2) + 126*a^2*b^5*c^4*d^2*e - 156*a^2*b^6*c^3*d*e^2 - 189*a^3*b^3*c^5
*d^2*e + 288*a^3*b^4*c^4*d*e^2 - 198*a^4*b^2*c^5*d*e^2 - 3*a^3*c^5*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*b^6*c^2*
d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 15*a*b^4*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^5*c^2*d*e^2*(-(4*a*c - b
^2)^3)^(1/2) + 12*a^3*b*c^4*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a^2*b^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3
0*a^2*b^3*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2) - (8*(d + e*x)
^(1/2)*(b^10*e^6 - 2*a^5*c^5*e^6 + 35*a^2*b^6*c^2*e^6 - 50*a^3*b^4*c^3*e^6 + 25*a^4*b^2*c^4*e^6 - 2*a^3*c^7*d^
4*e^2 + 12*a^4*c^6*d^2*e^4 + b^6*c^4*d^4*e^2 - 4*b^7*c^3*d^3*e^3 + 6*b^8*c^2*d^2*e^4 - 10*a*b^8*c*e^6 - 4*b^9*
c*d*e^5 + 9*a^2*b^2*c^6*d^4*e^2 - 56*a^2*b^3*c^5*d^3*e^3 + 120*a^2*b^4*c^4*d^2*e^4 - 96*a^3*b^2*c^5*d^2*e^4 +
36*a*b^7*c^2*d*e^5 - 36*a^4*b*c^5*d*e^5 - 6*a*b^4*c^5*d^4*e^2 + 28*a*b^5*c^4*d^3*e^3 - 48*a*b^6*c^3*d^2*e^4 -
108*a^2*b^5*c^3*d*e^5 + 28*a^3*b*c^6*d^3*e^3 + 120*a^3*b^3*c^4*d*e^5))/c^7)*(-(b^11*e^3 - 8*a^4*c^7*d^3 - b^8*
c^3*d^3 + b^8*e^3*(-(4*a*c - b^2)^3)^(1/2) + 10*a*b^6*c^4*d^3 - 36*a^5*b*c^5*e^3 + 24*a^5*c^6*d*e^2 + 3*b^9*c^
2*d^2*e - 33*a^2*b^4*c^5*d^3 + 38*a^3*b^2*c^6*d^3 + 63*a^2*b^7*c^2*e^3 - 138*a^3*b^5*c^3*e^3 + 129*a^4*b^3*c^4
*e^3 + a^4*c^4*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^5*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e^3 - 3*b^10*c
*d*e^2 + 15*a^2*b^4*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c
*e^3*(-(4*a*c - b^2)^3)^(1/2) - 33*a*b^7*c^3*d^2*e + 36*a*b^8*c^2*d*e^2 + 84*a^4*b*c^6*d^2*e - 3*b^7*c*d*e^2*(
-(4*a*c - b^2)^3)^(1/2) + 4*a*b^3*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^5*d^3*(-(4*a*c - b^2)^3)^(1/2)
+ 126*a^2*b^5*c^4*d^2*e - 156*a^2*b^6*c^3*d*e^2 - 189*a^3*b^3*c^5*d^2*e + 288*a^3*b^4*c^4*d*e^2 - 198*a^4*b^2*
c^5*d*e^2 - 3*a^3*c^5*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*b^6*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 15*a*b^4*c^3
*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^5*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^3*b*c^4*d*e^2*(-(4*a*c -
b^2)^3)^(1/2) + 18*a^2*b^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 30*a^2*b^3*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))
/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*1i - (((8*(4*a^3*c^8*d*e^4 - 8*a^3*b*c^7*e^5 - a*b^5*c^5*e^
5 + b^6*c^5*d*e^4 + 6*a^2*b^3*c^6*e^5 + 4*a^2*c^9*d^3*e^2 + b^4*c^7*d^3*e^2 - 2*b^5*c^6*d^2*e^3 - 5*a*b^4*c^6*
d*e^4 - 5*a*b^2*c^8*d^3*e^2 + 11*a*b^3*c^7*d^2*e^3 - 12*a^2*b*c^8*d^2*e^3 + 3*a^2*b^2*c^7*d*e^4))/c^7 + (8*(d
+ e*x)^(1/2)*(-(b^11*e^3 - 8*a^4*c^7*d^3 - b^8*c^3*d^3 + b^8*e^3*(-(4*a*c - b^2)^3)^(1/2) + 10*a*b^6*c^4*d^3 -
36*a^5*b*c^5*e^3 + 24*a^5*c^6*d*e^2 + 3*b^9*c^2*d^2*e - 33*a^2*b^4*c^5*d^3 + 38*a^3*b^2*c^6*d^3 + 63*a^2*b^7*
c^2*e^3 - 138*a^3*b^5*c^3*e^3 + 129*a^4*b^3*c^4...

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