3.6.35 \(\int \frac {x^2 (d+e x)^{3/2}}{a+b x+c x^2} \, dx\) [535]
Optimal. Leaf size=441 \[ -\frac {2 \left (b c d-b^2 e+a c e\right ) \sqrt {d+e x}}{c^3}-\frac {2 b (d+e x)^{3/2}}{3 c^2}+\frac {2 (d+e x)^{5/2}}{5 c e}+\frac {\sqrt {2} \left ((c d-b e) \left (b c d-b^2 e+2 a c e\right )+\frac {2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{7/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left ((c d-b e) \left (b c d-b^2 e+2 a c e\right )-\frac {2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{7/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]
[Out]
-2/3*b*(e*x+d)^(3/2)/c^2+2/5*(e*x+d)^(5/2)/c/e-2*(a*c*e-b^2*e+b*c*d)*(e*x+d)^(1/2)/c^3+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*e+c*d)*(2*a*c*e-b^2*e+b*c*d)+(2*b^3*c*d*e-
6*a*b*c^2*d*e-b^4*e^2-b^2*c*(-4*a*e^2+c*d^2)+2*a*c^2*(-a*e^2+c*d^2))/(-4*a*c+b^2)^(1/2))/c^(7/2)/(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*e+c*d)*(2*a*c*e-b^2*e+b*c*d)+(-2*b^3*c*d*e+6*a*b*c^2*d*e+b^4*e^2+b^2*c*(-4*a*e^2+c*d^2)-2*a*c^2*(-a*e^2
+c*d^2))/(-4*a*c+b^2)^(1/2))/c^(7/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
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Rubi [A]
time = 1.52, antiderivative size = 441, 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 {2 \sqrt {d+e x} \left (a c e+b^2 (-e)+b c d\right )}{c^3}+\frac {\sqrt {2} \left ((c d-b e) \left (2 a c e+b^2 (-e)+b c d\right )+\frac {-b^2 c \left (c d^2-4 a e^2\right )-6 a b c^2 d e+2 a c^2 \left (c d^2-a e^2\right )+b^4 \left (-e^2\right )+2 b^3 c d e}{\sqrt {b^2-4 a c}}\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^{7/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {2} \left ((c d-b e) \left (2 a c e+b^2 (-e)+b c d\right )-\frac {-b^2 c \left (c d^2-4 a e^2\right )-6 a b c^2 d e+2 a c^2 \left (c d^2-a e^2\right )+b^4 \left (-e^2\right )+2 b^3 c d e}{\sqrt {b^2-4 a c}}\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^{7/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {2 b (d+e x)^{3/2}}{3 c^2}+\frac {2 (d+e x)^{5/2}}{5 c e} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x^2*(d + e*x)^(3/2))/(a + b*x + c*x^2),x]
[Out]
(-2*(b*c*d - b^2*e + a*c*e)*Sqrt[d + e*x])/c^3 - (2*b*(d + e*x)^(3/2))/(3*c^2) + (2*(d + e*x)^(5/2))/(5*c*e) +
(Sqrt[2]*((c*d - b*e)*(b*c*d - b^2*e + 2*a*c*e) + (2*b^3*c*d*e - 6*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 4*a
*e^2) + 2*a*c^2*(c*d^2 - 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^(7/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*((c*d - b*e)*(b*c*d - b^2
*e + 2*a*c*e) - (2*b^3*c*d*e - 6*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 4*a*e^2) + 2*a*c^2*(c*d^2 - a*e^2))/Sq
rt[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^(7/2)*Sq
rt[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^2 (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 )^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 )}{e}\\ &=\frac {2 \text {Subst}\left (\int \left (-\frac {e \left (b c d-b^2 e+a c e\right )}{c^3}-\frac {b e x^2}{c^2}+\frac {x^4}{c}+\frac {\left (b c d-b^2 e+a c e\right ) \left (c d^2-b d e+a e^2\right )-(c d-b e) \left (b c d-b^2 e+2 a c e\right ) x^2}{c^3 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 c d-b^2 e+a c e\right ) \sqrt {d+e x}}{c^3}-\frac {2 b (d+e x)^{3/2}}{3 c^2}+\frac {2 (d+e x)^{5/2}}{5 c e}+\frac {2 \text {Subst}\left (\int \frac {\left (b c d-b^2 e+a c e\right ) \left (c d^2-b d e+a e^2\right )-(c d-b e) \left (b c d-b^2 e+2 a c 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^3 e^2}\\ &=-\frac {2 \left (b c d-b^2 e+a c e\right ) \sqrt {d+e x}}{c^3}-\frac {2 b (d+e x)^{3/2}}{3 c^2}+\frac {2 (d+e x)^{5/2}}{5 c e}-\frac {\left ((c d-b e) \left (b c d-b^2 e+2 a c e\right )-\frac {2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-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^3 e^2}-\frac {\left ((c d-b e) \left (b c d-b^2 e+2 a c e\right )+\frac {2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-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^3 e^2}\\ &=-\frac {2 \left (b c d-b^2 e+a c e\right ) \sqrt {d+e x}}{c^3}-\frac {2 b (d+e x)^{3/2}}{3 c^2}+\frac {2 (d+e x)^{5/2}}{5 c e}+\frac {\sqrt {2} \left ((c d-b e) \left (b c d-b^2 e+2 a c e\right )+\frac {2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-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^{7/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left ((c d-b e) \left (b c d-b^2 e+2 a c e\right )-\frac {2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-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^{7/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}
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Mathematica [A]
time = 1.84, size = 537, normalized size = 1.22 \begin {gather*} \frac {\frac {2 \sqrt {c} \sqrt {d+e x} \left (15 b^2 e^2+3 c^2 (d+e x)^2-5 c e (4 b d+3 a e+b e x)\right )}{e}-\frac {15 \sqrt {2} \left (-b^4 e^2+b^3 e \left (2 c d+\sqrt {b^2-4 a c} e\right )+b c \left (-2 a \sqrt {b^2-4 a c} e^2+c d \left (\sqrt {b^2-4 a c} d-6 a e\right )\right )-b^2 c \left (c d^2+2 e \left (\sqrt {b^2-4 a c} d-2 a e\right )\right )+2 a c^2 \left (c d^2+e \left (\sqrt {b^2-4 a c} d-a e\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {15 \sqrt {2} \left (b^4 e^2+b^3 e \left (-2 c d+\sqrt {b^2-4 a c} e\right )+2 a c^2 \left (-c d^2+e \left (\sqrt {b^2-4 a c} d+a e\right )\right )+b^2 c \left (c d^2-2 e \left (\sqrt {b^2-4 a c} d+2 a e\right )\right )+b c \left (-2 a \sqrt {b^2-4 a c} e^2+c d \left (\sqrt {b^2-4 a c} d+6 a e\right )\right )\right ) \tan ^{-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 )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}}{15 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(x^2*(d + e*x)^(3/2))/(a + b*x + c*x^2),x]
[Out]
((2*Sqrt[c]*Sqrt[d + e*x]*(15*b^2*e^2 + 3*c^2*(d + e*x)^2 - 5*c*e*(4*b*d + 3*a*e + b*e*x)))/e - (15*Sqrt[2]*(-
(b^4*e^2) + b^3*e*(2*c*d + Sqrt[b^2 - 4*a*c]*e) + b*c*(-2*a*Sqrt[b^2 - 4*a*c]*e^2 + c*d*(Sqrt[b^2 - 4*a*c]*d -
6*a*e)) - b^2*c*(c*d^2 + 2*e*(Sqrt[b^2 - 4*a*c]*d - 2*a*e)) + 2*a*c^2*(c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d - a*e))
)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2
*c*d + (b - Sqrt[b^2 - 4*a*c])*e]) - (15*Sqrt[2]*(b^4*e^2 + b^3*e*(-2*c*d + Sqrt[b^2 - 4*a*c]*e) + 2*a*c^2*(-(
c*d^2) + e*(Sqrt[b^2 - 4*a*c]*d + a*e)) + b^2*c*(c*d^2 - 2*e*(Sqrt[b^2 - 4*a*c]*d + 2*a*e)) + b*c*(-2*a*Sqrt[b
^2 - 4*a*c]*e^2 + c*d*(Sqrt[b^2 - 4*a*c]*d + 6*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + (b
+ Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]))/(15*c^(7/2))
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Maple [A]
time = 0.14, size = 674, normalized size = 1.53
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method |
result |
size |
| | |
derivativedivides |
\(\frac {-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {5}{2}} c^{2}}{5}+\frac {b c e \left (e x +d \right )^{\frac {3}{2}}}{3}+a c \,e^{2} \sqrt {e x +d}-b^{2} e^{2} \sqrt {e x +d}+b d e c \sqrt {e x +d}\right )}{c^{3}}+\frac {8 e \left (\frac {\left (-2 a^{2} c^{2} e^{3}+4 a \,b^{2} e^{3} c -6 a b \,c^{2} d \,e^{2}+2 a \,c^{3} d^{2} e -b^{4} e^{3}+2 b^{3} d \,e^{2} c -b^{2} c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (2 a^{2} c^{2} e^{3}-4 a \,b^{2} e^{3} c +6 a b \,c^{2} d \,e^{2}-2 a \,c^{3} d^{2} e +b^{4} e^{3}-2 b^{3} d \,e^{2} c +b^{2} c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2}\right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c^{2}}}{e}\) |
\(674\) |
default |
\(\frac {-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {5}{2}} c^{2}}{5}+\frac {b c e \left (e x +d \right )^{\frac {3}{2}}}{3}+a c \,e^{2} \sqrt {e x +d}-b^{2} e^{2} \sqrt {e x +d}+b d e c \sqrt {e x +d}\right )}{c^{3}}+\frac {8 e \left (\frac {\left (-2 a^{2} c^{2} e^{3}+4 a \,b^{2} e^{3} c -6 a b \,c^{2} d \,e^{2}+2 a \,c^{3} d^{2} e -b^{4} e^{3}+2 b^{3} d \,e^{2} c -b^{2} c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (2 a^{2} c^{2} e^{3}-4 a \,b^{2} e^{3} c +6 a b \,c^{2} d \,e^{2}-2 a \,c^{3} d^{2} e +b^{4} e^{3}-2 b^{3} d \,e^{2} c +b^{2} c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2}\right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c^{2}}}{e}\) |
\(674\) |
risch |
\(\text {Expression too large to display}\) |
\(2358\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*(e*x+d)^(3/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
[Out]
2/e*(-1/c^3*(-1/5*(e*x+d)^(5/2)*c^2+1/3*b*c*e*(e*x+d)^(3/2)+a*c*e^2*(e*x+d)^(1/2)-b^2*e^2*(e*x+d)^(1/2)+b*d*e*
c*(e*x+d)^(1/2))+4*e/c^2*(1/8*(-2*a^2*c^2*e^3+4*a*b^2*e^3*c-6*a*b*c^2*d*e^2+2*a*c^3*d^2*e-b^4*e^3+2*b^3*d*e^2*
c-b^2*c^2*d^2*e+2*(-e^2*(4*a*c-b^2))^(1/2)*a*b*c*e^2-2*(-e^2*(4*a*c-b^2))^(1/2)*a*c^2*d*e-(-e^2*(4*a*c-b^2))^(
1/2)*b^3*e^2+2*(-e^2*(4*a*c-b^2))^(1/2)*b^2*c*d*e-(-e^2*(4*a*c-b^2))^(1/2)*b*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*(2*a^2*c^2*e^3-4*a*b^2*e^3*c+6*a*b*c^2*d*e^2-2*a*c^3*d^2*e+b^4*e^3-2*b^3*d*e^2*
c+b^2*c^2*d^2*e+2*(-e^2*(4*a*c-b^2))^(1/2)*a*b*c*e^2-2*(-e^2*(4*a*c-b^2))^(1/2)*a*c^2*d*e-(-e^2*(4*a*c-b^2))^(
1/2)*b^3*e^2+2*(-e^2*(4*a*c-b^2))^(1/2)*b^2*c*d*e-(-e^2*(4*a*c-b^2))^(1/2)*b*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^2*(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="maxima")
[Out]
integrate((x*e + d)^(3/2)*x^2/(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. 8462 vs.
\(2 (405) = 810\).
time = 7.60, size = 8462, normalized size = 19.19 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="fricas")
[Out]
-1/30*(15*sqrt(2)*c^3*sqrt(((b^4*c^3 - 4*a*b^2*c^4 + 2*a^2*c^5)*d^3 - 3*(b^5*c^2 - 5*a*b^3*c^3 + 5*a^2*b*c^4)*
d^2*e + 3*(b^6*c - 6*a*b^4*c^2 + 9*a^2*b^2*c^3 - 2*a^3*c^4)*d*e^2 - (b^7 - 7*a*b^5*c + 14*a^2*b^3*c^2 - 7*a^3*
b*c^3)*e^3 + (b^2*c^7 - 4*a*c^8)*sqrt(((b^6*c^6 - 4*a*b^4*c^7 + 4*a^2*b^2*c^8)*d^6 - 6*(b^7*c^5 - 5*a*b^5*c^6
+ 7*a^2*b^3*c^7 - 2*a^3*b*c^8)*d^5*e + 3*(5*b^8*c^4 - 30*a*b^6*c^5 + 55*a^2*b^4*c^6 - 30*a^3*b^2*c^7 + 3*a^4*c
^8)*d^4*e^2 - 2*(10*b^9*c^3 - 70*a*b^7*c^4 + 160*a^2*b^5*c^5 - 130*a^3*b^3*c^6 + 29*a^4*b*c^7)*d^3*e^3 + 3*(5*
b^10*c^2 - 40*a*b^8*c^3 + 110*a^2*b^6*c^4 - 120*a^3*b^4*c^5 + 45*a^4*b^2*c^6 - 2*a^5*c^7)*d^2*e^4 - 6*(b^11*c
- 9*a*b^9*c^2 + 29*a^2*b^7*c^3 - 40*a^3*b^5*c^4 + 22*a^4*b^3*c^5 - 3*a^5*b*c^6)*d*e^5 + (b^12 - 10*a*b^10*c +
37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12*a^5*b^2*c^5 + a^6*c^6)*e^6)/(b^2*c^14 - 4*a*c^15)))/(b^2
*c^7 - 4*a*c^8))*e*log(sqrt(2)*((b^6*c^4 - 6*a*b^4*c^5 + 8*a^2*b^2*c^6)*d^4 - (4*b^7*c^3 - 28*a*b^5*c^4 + 53*a
^2*b^3*c^5 - 20*a^3*b*c^6)*d^3*e + 3*(2*b^8*c^2 - 16*a*b^6*c^3 + 39*a^2*b^4*c^4 - 29*a^3*b^2*c^5 + 4*a^4*c^6)*
d^2*e^2 - (4*b^9*c - 36*a*b^7*c^2 + 107*a^2*b^5*c^3 - 118*a^3*b^3*c^4 + 40*a^4*b*c^5)*d*e^3 + (b^10 - 10*a*b^8
*c + 35*a^2*b^6*c^2 - 51*a^3*b^4*c^3 + 29*a^4*b^2*c^4 - 4*a^5*c^5)*e^4 - ((b^4*c^8 - 6*a*b^2*c^9 + 8*a^2*c^10)
*d - (b^5*c^7 - 7*a*b^3*c^8 + 12*a^2*b*c^9)*e)*sqrt(((b^6*c^6 - 4*a*b^4*c^7 + 4*a^2*b^2*c^8)*d^6 - 6*(b^7*c^5
- 5*a*b^5*c^6 + 7*a^2*b^3*c^7 - 2*a^3*b*c^8)*d^5*e + 3*(5*b^8*c^4 - 30*a*b^6*c^5 + 55*a^2*b^4*c^6 - 30*a^3*b^2
*c^7 + 3*a^4*c^8)*d^4*e^2 - 2*(10*b^9*c^3 - 70*a*b^7*c^4 + 160*a^2*b^5*c^5 - 130*a^3*b^3*c^6 + 29*a^4*b*c^7)*d
^3*e^3 + 3*(5*b^10*c^2 - 40*a*b^8*c^3 + 110*a^2*b^6*c^4 - 120*a^3*b^4*c^5 + 45*a^4*b^2*c^6 - 2*a^5*c^7)*d^2*e^
4 - 6*(b^11*c - 9*a*b^9*c^2 + 29*a^2*b^7*c^3 - 40*a^3*b^5*c^4 + 22*a^4*b^3*c^5 - 3*a^5*b*c^6)*d*e^5 + (b^12 -
10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12*a^5*b^2*c^5 + a^6*c^6)*e^6)/(b^2*c^14 - 4*
a*c^15)))*sqrt(((b^4*c^3 - 4*a*b^2*c^4 + 2*a^2*c^5)*d^3 - 3*(b^5*c^2 - 5*a*b^3*c^3 + 5*a^2*b*c^4)*d^2*e + 3*(b
^6*c - 6*a*b^4*c^2 + 9*a^2*b^2*c^3 - 2*a^3*c^4)*d*e^2 - (b^7 - 7*a*b^5*c + 14*a^2*b^3*c^2 - 7*a^3*b*c^3)*e^3 +
(b^2*c^7 - 4*a*c^8)*sqrt(((b^6*c^6 - 4*a*b^4*c^7 + 4*a^2*b^2*c^8)*d^6 - 6*(b^7*c^5 - 5*a*b^5*c^6 + 7*a^2*b^3*
c^7 - 2*a^3*b*c^8)*d^5*e + 3*(5*b^8*c^4 - 30*a*b^6*c^5 + 55*a^2*b^4*c^6 - 30*a^3*b^2*c^7 + 3*a^4*c^8)*d^4*e^2
- 2*(10*b^9*c^3 - 70*a*b^7*c^4 + 160*a^2*b^5*c^5 - 130*a^3*b^3*c^6 + 29*a^4*b*c^7)*d^3*e^3 + 3*(5*b^10*c^2 - 4
0*a*b^8*c^3 + 110*a^2*b^6*c^4 - 120*a^3*b^4*c^5 + 45*a^4*b^2*c^6 - 2*a^5*c^7)*d^2*e^4 - 6*(b^11*c - 9*a*b^9*c^
2 + 29*a^2*b^7*c^3 - 40*a^3*b^5*c^4 + 22*a^4*b^3*c^5 - 3*a^5*b*c^6)*d*e^5 + (b^12 - 10*a*b^10*c + 37*a^2*b^8*c
^2 - 62*a^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12*a^5*b^2*c^5 + a^6*c^6)*e^6)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c
^8)) + 4*((a^2*b^3*c^4 - 2*a^3*b*c^5)*d^5 - (4*a^2*b^4*c^3 - 11*a^3*b^2*c^4 + 3*a^4*c^5)*d^4*e + 2*(3*a^2*b^5*
c^2 - 10*a^3*b^3*c^3 + 5*a^4*b*c^4)*d^3*e^2 - 2*(2*a^2*b^6*c - 7*a^3*b^4*c^2 + 3*a^4*b^2*c^3 + a^5*c^4)*d^2*e^
3 + (a^2*b^7 - 2*a^3*b^5*c - 6*a^4*b^3*c^2 + 8*a^5*b*c^3)*d*e^4 - (a^3*b^6 - 5*a^4*b^4*c + 6*a^5*b^2*c^2 - a^6
*c^3)*e^5)*sqrt(x*e + d)) - 15*sqrt(2)*c^3*sqrt(((b^4*c^3 - 4*a*b^2*c^4 + 2*a^2*c^5)*d^3 - 3*(b^5*c^2 - 5*a*b^
3*c^3 + 5*a^2*b*c^4)*d^2*e + 3*(b^6*c - 6*a*b^4*c^2 + 9*a^2*b^2*c^3 - 2*a^3*c^4)*d*e^2 - (b^7 - 7*a*b^5*c + 14
*a^2*b^3*c^2 - 7*a^3*b*c^3)*e^3 + (b^2*c^7 - 4*a*c^8)*sqrt(((b^6*c^6 - 4*a*b^4*c^7 + 4*a^2*b^2*c^8)*d^6 - 6*(b
^7*c^5 - 5*a*b^5*c^6 + 7*a^2*b^3*c^7 - 2*a^3*b*c^8)*d^5*e + 3*(5*b^8*c^4 - 30*a*b^6*c^5 + 55*a^2*b^4*c^6 - 30*
a^3*b^2*c^7 + 3*a^4*c^8)*d^4*e^2 - 2*(10*b^9*c^3 - 70*a*b^7*c^4 + 160*a^2*b^5*c^5 - 130*a^3*b^3*c^6 + 29*a^4*b
*c^7)*d^3*e^3 + 3*(5*b^10*c^2 - 40*a*b^8*c^3 + 110*a^2*b^6*c^4 - 120*a^3*b^4*c^5 + 45*a^4*b^2*c^6 - 2*a^5*c^7)
*d^2*e^4 - 6*(b^11*c - 9*a*b^9*c^2 + 29*a^2*b^7*c^3 - 40*a^3*b^5*c^4 + 22*a^4*b^3*c^5 - 3*a^5*b*c^6)*d*e^5 + (
b^12 - 10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12*a^5*b^2*c^5 + a^6*c^6)*e^6)/(b^2*c^
14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c^8))*e*log(-sqrt(2)*((b^6*c^4 - 6*a*b^4*c^5 + 8*a^2*b^2*c^6)*d^4 - (4*b^7*c^3
- 28*a*b^5*c^4 + 53*a^2*b^3*c^5 - 20*a^3*b*c^6)*d^3*e + 3*(2*b^8*c^2 - 16*a*b^6*c^3 + 39*a^2*b^4*c^4 - 29*a^3
*b^2*c^5 + 4*a^4*c^6)*d^2*e^2 - (4*b^9*c - 36*a*b^7*c^2 + 107*a^2*b^5*c^3 - 118*a^3*b^3*c^4 + 40*a^4*b*c^5)*d*
e^3 + (b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 51*a^3*b^4*c^3 + 29*a^4*b^2*c^4 - 4*a^5*c^5)*e^4 - ((b^4*c^8 - 6*a
*b^2*c^9 + 8*a^2*c^10)*d - (b^5*c^7 - 7*a*b^3*c^8 + 12*a^2*b*c^9)*e)*sqrt(((b^6*c^6 - 4*a*b^4*c^7 + 4*a^2*b^2*
c^8)*d^6 - 6*(b^7*c^5 - 5*a*b^5*c^6 + 7*a^2*b^3*c^7 - 2*a^3*b*c^8)*d^5*e + 3*(5*b^8*c^4 - 30*a*b^6*c^5 + 55*a^
2*b^4*c^6 - 30*a^3*b^2*c^7 + 3*a^4*c^8)*d^4*e^2 - 2*(10*b^9*c^3 - 70*a*b^7*c^4 + 160*a^2*b^5*c^5 - 130*a^3*b^3
*c^6 + 29*a^4*b*c^7)*d^3*e^3 + 3*(5*b^10*c^2 - 40*a*b^8*c^3 + 110*a^2*b^6*c^4 - 120*a^3*b^4*c^5 + 45*a^4*b^2*c
^6 - 2*a^5*c^7)*d^2*e^4 - 6*(b^11*c - 9*a*b^9*c...
________________________________________________________________________________________
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**2*(e*x+d)**(3/2)/(c*x**2+b*x+a),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1160 vs.
\(2 (405) = 810\).
time = 1.09, size = 1160, normalized size = 2.63 \begin {gather*} -\frac {{\left ({\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{2} e - 2 \, {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d e^{2} + {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\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 (\sqrt {b^{2} - 4 \, a c} b c^{4} d^{3} + \sqrt {b^{2} - 4 \, a c} b^{3} c^{2} d e^{2} - {\left (2 \, b^{2} c^{3} - a c^{4}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} e - {\left (a b^{2} c^{2} - a^{2} 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^{2} c^{5} - 2 \, a c^{6}\right )} d^{3} - {\left (5 \, b^{3} c^{4} - 14 \, a b c^{5}\right )} d^{2} e + 2 \, {\left (2 \, b^{4} c^{3} - 7 \, a b^{2} c^{4} + 2 \, a^{2} c^{5}\right )} d e^{2} - {\left (b^{5} c^{2} - 4 \, a b^{3} c^{3} + 2 \, a^{2} b 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^{6} d e^{6} - b c^{5} e^{7} + \sqrt {-4 \, {\left (c^{6} d^{2} e^{6} - b c^{5} d e^{7} + a c^{5} e^{8}\right )} c^{6} e^{6} + {\left (2 \, c^{6} d e^{6} - b c^{5} e^{7}\right )}^{2}}\right )} e^{\left (-6\right )}}{c^{6}}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{6} d^{2} - \sqrt {b^{2} - 4 \, a c} b c^{5} d e + \sqrt {b^{2} - 4 \, a c} a c^{5} e^{2}\right )} c^{2}} + \frac {{\left ({\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{2} e - 2 \, {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d e^{2} + {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\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 (\sqrt {b^{2} - 4 \, a c} b c^{4} d^{3} + \sqrt {b^{2} - 4 \, a c} b^{3} c^{2} d e^{2} - {\left (2 \, b^{2} c^{3} - a c^{4}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} e - {\left (a b^{2} c^{2} - a^{2} 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^{2} c^{5} - 2 \, a c^{6}\right )} d^{3} - {\left (5 \, b^{3} c^{4} - 14 \, a b c^{5}\right )} d^{2} e + 2 \, {\left (2 \, b^{4} c^{3} - 7 \, a b^{2} c^{4} + 2 \, a^{2} c^{5}\right )} d e^{2} - {\left (b^{5} c^{2} - 4 \, a b^{3} c^{3} + 2 \, a^{2} b 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^{6} d e^{6} - b c^{5} e^{7} - \sqrt {-4 \, {\left (c^{6} d^{2} e^{6} - b c^{5} d e^{7} + a c^{5} e^{8}\right )} c^{6} e^{6} + {\left (2 \, c^{6} d e^{6} - b c^{5} e^{7}\right )}^{2}}\right )} e^{\left (-6\right )}}{c^{6}}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{6} d^{2} - \sqrt {b^{2} - 4 \, a c} b c^{5} d e + \sqrt {b^{2} - 4 \, a c} a c^{5} e^{2}\right )} c^{2}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{4} e^{4} - 5 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{3} e^{5} - 15 \, \sqrt {x e + d} b c^{3} d e^{5} + 15 \, \sqrt {x e + d} b^{2} c^{2} e^{6} - 15 \, \sqrt {x e + d} a c^{3} e^{6}\right )} e^{\left (-5\right )}}{15 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="giac")
[Out]
-1/4*(((b^3*c^2 - 4*a*b*c^3)*d^2*e - 2*(b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d*e^2 + (b^5 - 6*a*b^3*c + 8*a^2*b*c^
2)*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*c^2 - 2*(sqrt(b^2 - 4*a*c)*b*c^4*d^3 + sqrt(b^2 - 4*a
*c)*b^3*c^2*d*e^2 - (2*b^2*c^3 - a*c^4)*sqrt(b^2 - 4*a*c)*d^2*e - (a*b^2*c^2 - a^2*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^2*c^5 - 2*a*c^6)*d^3 - (5*b^3*c^4 - 14*a*b*c^
5)*d^2*e + 2*(2*b^4*c^3 - 7*a*b^2*c^4 + 2*a^2*c^5)*d*e^2 - (b^5*c^2 - 4*a*b^3*c^3 + 2*a^2*b*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^6*d*e^6 - b*c^5*e^7 + sq
rt(-4*(c^6*d^2*e^6 - b*c^5*d*e^7 + a*c^5*e^8)*c^6*e^6 + (2*c^6*d*e^6 - b*c^5*e^7)^2))*e^(-6)/c^6))/((sqrt(b^2
- 4*a*c)*c^6*d^2 - sqrt(b^2 - 4*a*c)*b*c^5*d*e + sqrt(b^2 - 4*a*c)*a*c^5*e^2)*c^2) + 1/4*(((b^3*c^2 - 4*a*b*c^
3)*d^2*e - 2*(b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d*e^2 + (b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*e^3)*sqrt(-4*c^2*d + 2*
(b*c - sqrt(b^2 - 4*a*c)*c)*e)*c^2 + 2*(sqrt(b^2 - 4*a*c)*b*c^4*d^3 + sqrt(b^2 - 4*a*c)*b^3*c^2*d*e^2 - (2*b^2
*c^3 - a*c^4)*sqrt(b^2 - 4*a*c)*d^2*e - (a*b^2*c^2 - a^2*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^2*c^5 - 2*a*c^6)*d^3 - (5*b^3*c^4 - 14*a*b*c^5)*d^2*e + 2*(2*b^4*c^3 -
7*a*b^2*c^4 + 2*a^2*c^5)*d*e^2 - (b^5*c^2 - 4*a*b^3*c^3 + 2*a^2*b*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^6*d*e^6 - b*c^5*e^7 - sqrt(-4*(c^6*d^2*e^6 - b*c^5
*d*e^7 + a*c^5*e^8)*c^6*e^6 + (2*c^6*d*e^6 - b*c^5*e^7)^2))*e^(-6)/c^6))/((sqrt(b^2 - 4*a*c)*c^6*d^2 - sqrt(b^
2 - 4*a*c)*b*c^5*d*e + sqrt(b^2 - 4*a*c)*a*c^5*e^2)*c^2) + 2/15*(3*(x*e + d)^(5/2)*c^4*e^4 - 5*(x*e + d)^(3/2)
*b*c^3*e^5 - 15*sqrt(x*e + d)*b*c^3*d*e^5 + 15*sqrt(x*e + d)*b^2*c^2*e^6 - 15*sqrt(x*e + d)*a*c^3*e^6)*e^(-5)/
c^5
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Mupad [B]
time = 5.72, size = 2500, normalized size = 5.67 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2*(d + e*x)^(3/2))/(a + b*x + c*x^2),x)
[Out]
atan(((((8*(4*a^3*c^6*e^5 + a*b^4*c^4*e^5 - b^5*c^4*d*e^4 - 5*a^2*b^2*c^5*e^5 + 4*a^2*c^7*d^2*e^3 - b^3*c^6*d^
3*e^2 + 2*b^4*c^5*d^2*e^3 + 4*a*b*c^7*d^3*e^2 + 4*a*b^3*c^5*d*e^4 - 9*a*b^2*c^6*d^2*e^3))/c^5 - (8*(d + e*x)^(
1/2)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 - b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*
c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 +
a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 -
6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e^3*(-(4*a*c
- b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e + 3*b^5*c*d*e^2*(-(4*a*c - b^2)
^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 - 3*a^2*c^4*d^2*e*(-(4*a*c - b
^2)^3)^(1/2) - 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^
3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 -
8*a*b^2*c^8)))^(1/2)*(b^3*c^7*e^3 - 2*b^2*c^8*d*e^2 - 4*a*b*c^8*e^3 + 8*a*c^9*d*e^2))/c^5)*(-(b^9*e^3 + 8*a^3*
c^6*d^3 - b^6*c^3*d^3 - b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e
^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 + a^3*c^3*e^3*(-(4*a*c - b
^2)^3)^(1/2) + b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 - 6*a^2*b^2*c^2*e^3*(-(4*
a*c - b^2)^3)^(1/2) - 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b
^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e + 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c
^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 - 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c^2
*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^3*c^2*d*e^2*(-(4*a*c - b
^2)^3)^(1/2) + 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) - (
8*(d + e*x)^(1/2)*(b^8*e^6 + 2*a^4*c^4*e^6 + 20*a^2*b^4*c^2*e^6 - 16*a^3*b^2*c^3*e^6 + 2*a^2*c^6*d^4*e^2 - 12*
a^3*c^5*d^2*e^4 + b^4*c^4*d^4*e^2 - 4*b^5*c^3*d^3*e^3 + 6*b^6*c^2*d^2*e^4 - 8*a*b^6*c*e^6 - 4*b^7*c*d*e^5 + 54
*a^2*b^2*c^4*d^2*e^4 + 28*a*b^5*c^2*d*e^5 + 28*a^3*b*c^4*d*e^5 - 4*a*b^2*c^5*d^4*e^2 + 20*a*b^3*c^4*d^3*e^3 -
36*a*b^4*c^3*d^2*e^4 - 20*a^2*b*c^5*d^3*e^3 - 56*a^2*b^3*c^3*d*e^5))/c^5)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3
*d^3 - b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^
2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 + a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^
3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 - 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/
2) - 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30
*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e + 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2
*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 - 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c^2*d^2*e*(-(4*a*c -
b^2)^3)^(1/2) + 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 9*
a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*1i - (((8*(4*a^3*c^6
*e^5 + a*b^4*c^4*e^5 - b^5*c^4*d*e^4 - 5*a^2*b^2*c^5*e^5 + 4*a^2*c^7*d^2*e^3 - b^3*c^6*d^3*e^2 + 2*b^4*c^5*d^2
*e^3 + 4*a*b*c^7*d^3*e^2 + 4*a*b^3*c^5*d*e^4 - 9*a*b^2*c^6*d^2*e^3))/c^5 + (8*(d + e*x)^(1/2)*(-(b^9*e^3 + 8*a
^3*c^6*d^3 - b^6*c^3*d^3 - b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*
d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 + a^3*c^3*e^3*(-(4*a*c
- b^2)^3)^(1/2) + b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 - 6*a^2*b^2*c^2*e^3*(-
(4*a*c - b^2)^3)^(1/2) - 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*
a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e + 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^
3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 - 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*
c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^3*c^2*d*e^2*(-(4*a*c
- b^2)^3)^(1/2) + 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*
(b^3*c^7*e^3 - 2*b^2*c^8*d*e^2 - 4*a*b*c^8*e^3 + 8*a*c^9*d*e^2))/c^5)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3
- b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e
- 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 + a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*c^
3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 - 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) -
2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b...
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