3.6.25 \(\int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx\) [525]

Optimal. Leaf size=490 \[ -\frac {2 b \left (b^2-2 a c\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (c^2 d^2+b^2 e^2+c e (b d-a e)\right ) (d+e x)^{3/2}}{3 c^3 e^3}-\frac {2 (2 c d+b e) (d+e x)^{5/2}}{5 c^2 e^3}+\frac {2 (d+e x)^{7/2}}{7 c e^3}+\frac {\sqrt {2} \left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e-\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{9/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e+\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{9/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

2/3*(c^2*d^2+b^2*e^2+c*e*(-a*e+b*d))*(e*x+d)^(3/2)/c^3/e^3-2/5*(b*e+2*c*d)*(e*x+d)^(5/2)/c^2/e^3+2/7*(e*x+d)^(
7/2)/c/e^3-2*b*(-2*a*c+b^2)*(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)*(b^3*c*d-2*a*b*c^2*d-b^4*e+3*a*b^2*c*e-a^2*c^2*e+(5*a^2*b*c^2*e-2*a^2*c^3*d-5*a*b^3*c*e+
4*a*b^2*c^2*d+b^5*e-b^4*c*d)/(-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)*(b^3*c*d-2*a*b*c^2*d-b^4*e+3*a*b^2*c*e
-a^2*c^2*e+(-5*a^2*b*c^2*e+2*a^2*c^3*d+5*a*b^3*c*e-4*a*b^2*c^2*d-b^5*e+b^4*c*d)/(-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 = 12.35, antiderivative size = 490, 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 {-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d}{\sqrt {b^2-4 a c}}-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\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 {-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d}{\sqrt {b^2-4 a c}}-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\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 b \left (b^2-2 a c\right ) \sqrt {d+e x}}{c^4}+\frac {2 (d+e x)^{3/2} \left (c e (b d-a e)+b^2 e^2+c^2 d^2\right )}{3 c^3 e^3}-\frac {2 (d+e x)^{5/2} (b e+2 c d)}{5 c^2 e^3}+\frac {2 (d+e x)^{7/2}}{7 c e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^4*Sqrt[d + e*x])/(a + b*x + c*x^2),x]

[Out]

(-2*b*(b^2 - 2*a*c)*Sqrt[d + e*x])/c^4 + (2*(c^2*d^2 + b^2*e^2 + c*e*(b*d - a*e))*(d + e*x)^(3/2))/(3*c^3*e^3)
- (2*(2*c*d + b*e)*(d + e*x)^(5/2))/(5*c^2*e^3) + (2*(d + e*x)^(7/2))/(7*c*e^3) + (Sqrt[2]*(b^3*c*d - 2*a*b*c
^2*d - b^4*e + 3*a*b^2*c*e - a^2*c^2*e - (b^4*c*d - 4*a*b^2*c^2*d + 2*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*
b*c^2*e)/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]*(b^3*c*d - 2*a*b*c^2*d - b^4*e + 3*a*b^2*c*e - a^
2*c^2*e + (b^4*c*d - 4*a*b^2*c^2*d + 2*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*b*c^2*e)/Sqrt[b^2 - 4*a*c])*Arc
Tanh[(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^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx &=\frac {2 \text {Subst}\left (\int \frac {x^2 \left (-\frac {d}{e}+\frac {x^2}{e}\right )^4}{\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 {\left (b^3-2 a b c\right ) e}{c^4}+\frac {\left (c^2 d^2+b^2 e^2+c e (b d-a e)\right ) x^2}{c^3 e^2}-\frac {(2 c d+b e) x^4}{c^2 e^2}+\frac {x^6}{c e^2}+\frac {b \left (b^2-2 a c\right ) \left (c d^2-b d e+a e^2\right )-\left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e\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 b \left (b^2-2 a c\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (c^2 d^2+b^2 e^2+c e (b d-a e)\right ) (d+e x)^{3/2}}{3 c^3 e^3}-\frac {2 (2 c d+b e) (d+e x)^{5/2}}{5 c^2 e^3}+\frac {2 (d+e x)^{7/2}}{7 c e^3}+\frac {2 \text {Subst}\left (\int \frac {b \left (b^2-2 a c\right ) \left (c d^2-b d e+a e^2\right )+\left (-b^3 c d+2 a b c^2 d+b^4 e-3 a b^2 c e+a^2 c^2 e\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 b \left (b^2-2 a c\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (c^2 d^2+b^2 e^2+c e (b d-a e)\right ) (d+e x)^{3/2}}{3 c^3 e^3}-\frac {2 (2 c d+b e) (d+e x)^{5/2}}{5 c^2 e^3}+\frac {2 (d+e x)^{7/2}}{7 c e^3}-\frac {\left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e-\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\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 (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e+\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\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 b \left (b^2-2 a c\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (c^2 d^2+b^2 e^2+c e (b d-a e)\right ) (d+e x)^{3/2}}{3 c^3 e^3}-\frac {2 (2 c d+b e) (d+e x)^{5/2}}{5 c^2 e^3}+\frac {2 (d+e x)^{7/2}}{7 c e^3}+\frac {\sqrt {2} \left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e-\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\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 (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e+\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\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 = 1.89, size = 627, normalized size = 1.28 \begin {gather*} \frac {2 \sqrt {d+e x} \left (-105 b^3 e^3-7 c^2 e (d+e x) (-2 b d+5 a e+3 b e x)+c^3 \left (8 d^3-4 d^2 e x+3 d e^2 x^2+15 e^3 x^3\right )+35 b c e^2 (6 a e+b (d+e x))\right )}{105 c^4 e^3}+\frac {\left (i b^5 e-b^3 c \left (\sqrt {-b^2+4 a c} d+5 i a e\right )+a b c^2 \left (2 \sqrt {-b^2+4 a c} d+5 i a e\right )+a b^2 c \left (4 i c d-3 \sqrt {-b^2+4 a c} e\right )+b^4 \left (-i c d+\sqrt {-b^2+4 a c} e\right )+a^2 c^2 \left (-2 i c d+\sqrt {-b^2+4 a c} e\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+a b c^2 \left (2 \sqrt {-b^2+4 a c} d-5 i a e\right )+b^3 c \left (-\sqrt {-b^2+4 a c} d+5 i a e\right )+a b^2 c \left (-4 i c d-3 \sqrt {-b^2+4 a c} e\right )+b^4 \left (i c d+\sqrt {-b^2+4 a c} e\right )+a^2 c^2 \left (2 i c d+\sqrt {-b^2+4 a c} e\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^4*Sqrt[d + e*x])/(a + b*x + c*x^2),x]

[Out]

(2*Sqrt[d + e*x]*(-105*b^3*e^3 - 7*c^2*e*(d + e*x)*(-2*b*d + 5*a*e + 3*b*e*x) + c^3*(8*d^3 - 4*d^2*e*x + 3*d*e
^2*x^2 + 15*e^3*x^3) + 35*b*c*e^2*(6*a*e + b*(d + e*x))))/(105*c^4*e^3) + ((I*b^5*e - b^3*c*(Sqrt[-b^2 + 4*a*c
]*d + (5*I)*a*e) + a*b*c^2*(2*Sqrt[-b^2 + 4*a*c]*d + (5*I)*a*e) + a*b^2*c*((4*I)*c*d - 3*Sqrt[-b^2 + 4*a*c]*e)
+ b^4*((-I)*c*d + Sqrt[-b^2 + 4*a*c]*e) + a^2*c^2*((-2*I)*c*d + Sqrt[-b^2 + 4*a*c]*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 + a*b*c^2*(2*Sqrt[-b^2 + 4*a*c]*d - (5*I)*a*e) + b^3*c*(-(Sqrt[-b
^2 + 4*a*c]*d) + (5*I)*a*e) + a*b^2*c*((-4*I)*c*d - 3*Sqrt[-b^2 + 4*a*c]*e) + b^4*(I*c*d + Sqrt[-b^2 + 4*a*c]*
e) + a^2*c^2*((2*I)*c*d + Sqrt[-b^2 + 4*a*c]*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 = 0.25, size = 708, normalized size = 1.44

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 {2 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}+\frac {b \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a b c \,e^{3} \sqrt {e x +d}-b^{3} e^{3} \sqrt {e x +d}\right )}{c^{4}}-\frac {8 e^{3} \left (\frac {\left (-5 a^{2} b \,c^{2} e^{2}+2 a^{2} c^{3} d e +5 a \,b^{3} e^{2} c -4 a \,b^{2} c^{2} d e -b^{5} e^{2}+b^{4} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d \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^{2}-2 a^{2} c^{3} d e -5 a \,b^{3} e^{2} c +4 a \,b^{2} c^{2} d e +b^{5} e^{2}-b^{4} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d \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^{3}}\)	\(708\)
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 {2 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}+\frac {b \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a b c \,e^{3} \sqrt {e x +d}-b^{3} e^{3} \sqrt {e x +d}\right )}{c^{4}}-\frac {8 e^{3} \left (\frac {\left (-5 a^{2} b \,c^{2} e^{2}+2 a^{2} c^{3} d e +5 a \,b^{3} e^{2} c -4 a \,b^{2} c^{2} d e -b^{5} e^{2}+b^{4} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d \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^{2}-2 a^{2} c^{3} d e -5 a \,b^{3} e^{2} c +4 a \,b^{2} c^{2} d e +b^{5} e^{2}-b^{4} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d \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^{3}}\)	\(708\)
risch	\(\text {Expression too large to display}\)	\(2219\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x+d)^(1/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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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^4*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

integrate(sqrt(x*e + d)*x^4/(c*x^2 + b*x + a), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5512 vs. \(2 (457) = 914\).
time = 2.60, size = 5512, normalized size = 11.25 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/210*(105*sqrt(2)*c^4*sqrt(((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 - (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 + (b^2*c^9 - 4*a*c^10)*sqrt(((b^14*c^2 - 12*a*b^12*
c^3 + 56*a^2*b^10*c^4 - 128*a^3*b^8*c^5 + 148*a^4*b^6*c^6 - 80*a^5*b^4*c^7 + 16*a^6*b^2*c^8)*d^2 - 2*(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 + (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^2)/(b^2*c^18 - 4*a*c^19)))/(b^2*c^9 - 4*a*c^10))*e^3*log(sqrt(2)
*((b^12*c - 12*a*b^10*c^2 + 54*a^2*b^8*c^3 - 112*a^3*b^6*c^4 + 104*a^4*b^4*c^5 - 32*a^5*b^2*c^6)*d - (b^13 - 1
3*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 - (b^6*c^9 -
8*a*b^4*c^10 + 18*a^2*b^2*c^11 - 8*a^3*c^12)*sqrt(((b^14*c^2 - 12*a*b^12*c^3 + 56*a^2*b^10*c^4 - 128*a^3*b^8*
c^5 + 148*a^4*b^6*c^6 - 80*a^5*b^4*c^7 + 16*a^6*b^2*c^8)*d^2 - 2*(b^15*c - 13*a*b^13*c^2 + 67*a^2*b^11*c^3 - 1
74*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 + (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^2)/(b^2*c^18 - 4*a*c^19)))*sqrt(((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 - (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 + (b^2*c^9 - 4*a*c^10)*sqrt(((b^14*c^
2 - 12*a*b^12*c^3 + 56*a^2*b^10*c^4 - 128*a^3*b^8*c^5 + 148*a^4*b^6*c^6 - 80*a^5*b^4*c^7 + 16*a^6*b^2*c^8)*d^2
- 2*(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 + (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^2)/(b^2*c^18 - 4*a*c^19)))/(b^2*c^9 - 4*a*c^10)) -
4*((a^4*b^7*c - 6*a^5*b^5*c^2 + 10*a^6*b^3*c^3 - 4*a^7*b*c^4)*d - (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)*sqrt(x*e + d)) - 105*sqrt(2)*c^4*sqrt(((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 - (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 + (b^2*c^9 -
4*a*c^10)*sqrt(((b^14*c^2 - 12*a*b^12*c^3 + 56*a^2*b^10*c^4 - 128*a^3*b^8*c^5 + 148*a^4*b^6*c^6 - 80*a^5*b^4*c
^7 + 16*a^6*b^2*c^8)*d^2 - 2*(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 - 1
66*a^5*b^5*c^6 + 50*a^6*b^3*c^7 - 4*a^7*b*c^8)*d*e + (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^2)/(b^2*c^18 - 4*a*c^19)))
/(b^2*c^9 - 4*a*c^10))*e^3*log(-sqrt(2)*((b^12*c - 12*a*b^10*c^2 + 54*a^2*b^8*c^3 - 112*a^3*b^6*c^4 + 104*a^4*
b^4*c^5 - 32*a^5*b^2*c^6)*d - (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 - (b^6*c^9 - 8*a*b^4*c^10 + 18*a^2*b^2*c^11 - 8*a^3*c^12)*sqrt(((b^14*c^2 - 12*a*b^
12*c^3 + 56*a^2*b^10*c^4 - 128*a^3*b^8*c^5 + 148*a^4*b^6*c^6 - 80*a^5*b^4*c^7 + 16*a^6*b^2*c^8)*d^2 - 2*(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 + (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^2)/(b^2*c^18 - 4*a*c^19)))*sqrt(((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 - (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 + (b^2*c^9 - 4*a*c^10)*sqrt(((b^14*c^2 - 12*a*b^12*c^3 + 56*a^2*b^10*c^4 - 128*a^3*b^8*c^5 + 148*a^4*b^6*c^6
- 80*a^5*b^4*c^7 + 16*a^6*b^2*c^8)*d^2 - 2*(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 + (b^16 - 14*a*b^14*c + 79*a^2*b^12*c^2 - 23
0*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^2)/(b^2*c^1
8 - 4*a*c^19)))/(b^2*c^9 - 4*a*c^10)) - 4*((a^4*b^7*c - 6*a^5*b^5*c^2 + 10*a^6*b^3*c^3 - 4*a^7*b*c^4)*d - (a^4
*b^8 - 7*a^5*b^6*c + 15*a^6*b^4*c^2 - 10*a^7*b^2*c^3 + a^8*c^4)*e)*sqrt(x*e + d)) + 105*sqrt(2)*c^4*sqrt(((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 - (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 - (b^2*c^9 - 4*a*c^10)*sqrt(((b^14*c^2 - 12*a*b^12*c^3 + 56*a^2*b^10*c^4 - 128*a^3*b
^8*c^5 + 148*a^4*b^6*c^6 - 80*a^5*b^4*c^7 + 16*a^6*b^2*c^8)*d^2 - 2*(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 + (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^2)/(b^2*c^18 - 4*a*c^19)))/(b^2*c^9 - 4*a*c^10))*e^3*log(sqrt(2)*((b^12*c - 12*a*b^10*c^2 + 54*a^
2*b^8*c^3 - 112*a^3*b^6*c^4 + 104*a^4*b^4*c^5 -...

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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**4*(e*x+d)**(1/2)/(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. 1171 vs. \(2 (457) = 914\).
time = 1.70, size = 1171, normalized size = 2.39 \begin {gather*} -\frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} d e - {\left (b^{6} - 7 \, a b^{4} c + 13 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} e^{2}\right )} c^{2} - 2 \, {\left ({\left (b^{3} c^{3} - 2 \, a b c^{4}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (b^{4} c^{2} - 2 \, a b^{2} c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a b^{3} c^{2} - 2 \, a^{2} b c^{3}\right )} \sqrt {b^{2} - 4 \, a c} e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c \right |} + \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, {\left (b^{4} c^{4} - 4 \, a b^{2} c^{5} + 2 \, a^{2} c^{6}\right )} d^{2} - {\left (3 \, b^{5} c^{3} - 14 \, a b^{3} c^{4} + 12 \, a^{2} b c^{5}\right )} d e + {\left (b^{6} c^{2} - 5 \, a b^{4} c^{3} + 5 \, a^{2} b^{2} c^{4}\right )} e^{2}\right )}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {{\left (2 \, c^{8} d e^{24} - b c^{7} e^{25} + \sqrt {-4 \, {\left (c^{8} d^{2} e^{24} - b c^{7} d e^{25} + a c^{7} e^{26}\right )} c^{8} e^{24} + {\left (2 \, c^{8} d e^{24} - b c^{7} e^{25}\right )}^{2}}\right )} e^{\left (-24\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 (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} d e - {\left (b^{6} - 7 \, a b^{4} c + 13 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} e^{2}\right )} c^{2} + 2 \, {\left ({\left (b^{3} c^{3} - 2 \, a b c^{4}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (b^{4} c^{2} - 2 \, a b^{2} c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a b^{3} c^{2} - 2 \, a^{2} b c^{3}\right )} \sqrt {b^{2} - 4 \, a c} e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c \right |} + \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, {\left (b^{4} c^{4} - 4 \, a b^{2} c^{5} + 2 \, a^{2} c^{6}\right )} d^{2} - {\left (3 \, b^{5} c^{3} - 14 \, a b^{3} c^{4} + 12 \, a^{2} b c^{5}\right )} d e + {\left (b^{6} c^{2} - 5 \, a b^{4} c^{3} + 5 \, a^{2} b^{2} c^{4}\right )} e^{2}\right )}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {{\left (2 \, c^{8} d e^{24} - b c^{7} e^{25} - \sqrt {-4 \, {\left (c^{8} d^{2} e^{24} - b c^{7} d e^{25} + a c^{7} e^{26}\right )} c^{8} e^{24} + {\left (2 \, c^{8} d e^{24} - b c^{7} e^{25}\right )}^{2}}\right )} e^{\left (-24\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^{18} - 42 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{6} d e^{18} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{6} d^{2} e^{18} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{5} e^{19} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{5} d e^{19} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{4} e^{20} - 35 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{5} e^{20} - 105 \, \sqrt {x e + d} b^{3} c^{3} e^{21} + 210 \, \sqrt {x e + d} a b c^{4} e^{21}\right )} e^{\left (-21\right )}}{105 \, c^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

-1/4*(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*((b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d*e - (b^6 - 7*a*b
^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e^2)*c^2 - 2*((b^3*c^3 - 2*a*b*c^4)*sqrt(b^2 - 4*a*c)*d^2 - (b^4*c^2 - 2*a*
b^2*c^3)*sqrt(b^2 - 4*a*c)*d*e + (a*b^3*c^2 - 2*a^2*b*c^3)*sqrt(b^2 - 4*a*c)*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqr
t(b^2 - 4*a*c)*c)*e)*abs(c) + sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(2*(b^4*c^4 - 4*a*b^2*c^5 + 2*a
^2*c^6)*d^2 - (3*b^5*c^3 - 14*a*b^3*c^4 + 12*a^2*b*c^5)*d*e + (b^6*c^2 - 5*a*b^4*c^3 + 5*a^2*b^2*c^4)*e^2))*ar
ctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*c^8*d*e^24 - b*c^7*e^25 + sqrt(-4*(c^8*d^2*e^24 - b*c^7*d*e^25 + a*c^7
*e^26)*c^8*e^24 + (2*c^8*d*e^24 - b*c^7*e^25)^2))*e^(-24)/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*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*((b^5*
c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d*e - (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e^2)*c^2 + 2*((b^3*c^3 - 2
*a*b*c^4)*sqrt(b^2 - 4*a*c)*d^2 - (b^4*c^2 - 2*a*b^2*c^3)*sqrt(b^2 - 4*a*c)*d*e + (a*b^3*c^2 - 2*a^2*b*c^3)*sq
rt(b^2 - 4*a*c)*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(c) + sqrt(-4*c^2*d + 2*(b*c - sqrt(b
^2 - 4*a*c)*c)*e)*(2*(b^4*c^4 - 4*a*b^2*c^5 + 2*a^2*c^6)*d^2 - (3*b^5*c^3 - 14*a*b^3*c^4 + 12*a^2*b*c^5)*d*e +
(b^6*c^2 - 5*a*b^4*c^3 + 5*a^2*b^2*c^4)*e^2))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*c^8*d*e^24 - b*c^7*e^
25 - sqrt(-4*(c^8*d^2*e^24 - b*c^7*d*e^25 + a*c^7*e^26)*c^8*e^24 + (2*c^8*d*e^24 - b*c^7*e^25)^2))*e^(-24)/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^18 - 42*(x*e + d)^(5/2)*c^6*d*e^18 + 35*(x*e + d)^(3/2)*c^6*d^2*e^18 - 21*(x*e + d)^(5/2)
*b*c^5*e^19 + 35*(x*e + d)^(3/2)*b*c^5*d*e^19 + 35*(x*e + d)^(3/2)*b^2*c^4*e^20 - 35*(x*e + d)^(3/2)*a*c^5*e^2
0 - 105*sqrt(x*e + d)*b^3*c^3*e^21 + 210*sqrt(x*e + d)*a*b*c^4*e^21)*e^(-21)/c^7

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Mupad [B]
time = 4.86, size = 2500, normalized size = 5.10 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(d + e*x)^(1/2))/(a + b*x + c*x^2),x)

[Out]

(d + e*x)^(3/2)*((4*d^2)/(c*e^3) - (2*(a*e^5 + c*d^2*e^3 - b*d*e^4))/(3*c^2*e^6) + (((8*d)/(c*e^3) + (2*(b*e^4
- 2*c*d*e^3))/(c^2*e^6))*(b*e^4 - 2*c*d*e^3))/(3*c*e^3)) - atan(((((8*(a*b^5*c^5*e^4 + 8*a^3*b*c^7*e^4 - b^6*
c^5*d*e^3 - 6*a^2*b^3*c^6*e^4 + b^5*c^6*d^2*e^2 + 6*a*b^4*c^6*d*e^3 - 6*a*b^3*c^7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^
2 - 8*a^2*b^2*c^7*d*e^3))/c^7 - (8*(d + e*x)^(1/2)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) -
b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129
*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d
*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*
a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c -
b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(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 + 8*a^5*c^6*d + b^8*e*(-(4*
a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 1
38*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36
*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*
c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^
2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(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^4 - 2*a^5*c^5*e^4 + 35*a^2*b^6*c^2*e^4 - 50*a^3*b^4*c^3*e^4
+ 25*a^4*b^2*c^4*e^4 + 2*a^4*c^6*d^2*e^2 + b^8*c^2*d^2*e^2 - 10*a*b^8*c*e^4 - 2*b^9*c*d*e^3 + 20*a^2*b^4*c^4*
d^2*e^2 - 16*a^3*b^2*c^5*d^2*e^2 + 18*a*b^7*c^2*d*e^3 - 18*a^4*b*c^5*d*e^3 - 8*a*b^6*c^3*d^2*e^2 - 54*a^2*b^5*
c^3*d*e^3 + 60*a^3*b^3*c^4*d*e^3))/c^7)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d -
52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^
4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c -
b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d
*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1
/2) - 10*a^3*b^2*c^3*e*(-(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*(
a*b^5*c^5*e^4 + 8*a^3*b*c^7*e^4 - b^6*c^5*d*e^3 - 6*a^2*b^3*c^6*e^4 + b^5*c^6*d^2*e^2 + 6*a*b^4*c^6*d*e^3 - 6*
a*b^3*c^7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^2 - 8*a^2*b^2*c^7*d*e^3))/c^7 + (8*(d + e*x)^(1/2)*(-(b^11*e + 8*a^5*c^6
*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a
^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*
a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b
^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^
3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(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 + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 6
6*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2
) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c -
b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*
c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(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^4 - 2*a^5*c^5*e^4 + 3
5*a^2*b^6*c^2*e^4 - 50*a^3*b^4*c^3*e^4 + 25*a^4*b^2*c^4*e^4 + 2*a^4*c^6*d^2*e^2 + b^8*c^2*d^2*e^2 - 10*a*b^8*c
*e^4 - 2*b^9*c*d*e^3 + 20*a^2*b^4*c^4*d^2*e^2 - 16*a^3*b^2*c^5*d^2*e^2 + 18*a*b^7*c^2*d*e^3 - 18*a^4*b*c^5*d*e
^3 - 8*a*b^6*c^3*d^2*e^2 - 54*a^2*b^5*c^3*d*e^3 + 60*a^3*b^3*c^4*d*e^3))/c^7)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*
(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*
e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d
- 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-
(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) +
15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*...

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