3.19.76 \(\int (a+b x) (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=264 \[ \frac {4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{3 e^5 (a+b x)}-\frac {8 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{5 e^5 (a+b x)}+\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^5 (a+b x)}-\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^5 (a+b x)} \]

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Rubi [A]  time = 0.10, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {770, 21, 43} \begin {gather*} \frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^5 (a+b x)}-\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^5 (a+b x)}+\frac {4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{3 e^5 (a+b x)}-\frac {8 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{5 e^5 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)) - (8*b*(b*d - a*e)^3*(d + e*
x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)) + (4*b^2*(b*d - a*e)^2*(d + e*x)^(9/2)*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(3*e^5*(a + b*x)) - (8*b^3*(b*d - a*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e
^5*(a + b*x)) + (2*b^4*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^5*(a + b*x))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^3 (d+e x)^{3/2} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^4 (d+e x)^{3/2} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^4 (d+e x)^{3/2}}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^{5/2}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{7/2}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{9/2}}{e^4}+\frac {b^4 (d+e x)^{11/2}}{e^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}-\frac {8 b (b d-a e)^3 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}+\frac {4 b^2 (b d-a e)^2 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}-\frac {8 b^3 (b d-a e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}+\frac {2 b^4 (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^5 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 172, normalized size = 0.65 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{5/2} \left (3003 a^4 e^4+1716 a^3 b e^3 (5 e x-2 d)+286 a^2 b^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+52 a b^3 e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+b^4 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{15015 e^5 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(5/2)*(3003*a^4*e^4 + 1716*a^3*b*e^3*(-2*d + 5*e*x) + 286*a^2*b^2*e^2*(8*d^2 -
20*d*e*x + 35*e^2*x^2) + 52*a*b^3*e*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + b^4*(128*d^4 - 320*d
^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4)))/(15015*e^5*(a + b*x))

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IntegrateAlgebraic [A]  time = 49.35, size = 241, normalized size = 0.91 \begin {gather*} \frac {2 (d+e x)^{5/2} \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (3003 a^4 e^4+8580 a^3 b e^3 (d+e x)-12012 a^3 b d e^3+18018 a^2 b^2 d^2 e^2+10010 a^2 b^2 e^2 (d+e x)^2-25740 a^2 b^2 d e^2 (d+e x)-12012 a b^3 d^3 e+25740 a b^3 d^2 e (d+e x)+5460 a b^3 e (d+e x)^3-20020 a b^3 d e (d+e x)^2+3003 b^4 d^4-8580 b^4 d^3 (d+e x)+10010 b^4 d^2 (d+e x)^2+1155 b^4 (d+e x)^4-5460 b^4 d (d+e x)^3\right )}{15015 e^4 (a e+b e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*Sqrt[(a*e + b*e*x)^2/e^2]*(3003*b^4*d^4 - 12012*a*b^3*d^3*e + 18018*a^2*b^2*d^2*e^2 - 12012
*a^3*b*d*e^3 + 3003*a^4*e^4 - 8580*b^4*d^3*(d + e*x) + 25740*a*b^3*d^2*e*(d + e*x) - 25740*a^2*b^2*d*e^2*(d +
e*x) + 8580*a^3*b*e^3*(d + e*x) + 10010*b^4*d^2*(d + e*x)^2 - 20020*a*b^3*d*e*(d + e*x)^2 + 10010*a^2*b^2*e^2*
(d + e*x)^2 - 5460*b^4*d*(d + e*x)^3 + 5460*a*b^3*e*(d + e*x)^3 + 1155*b^4*(d + e*x)^4))/(15015*e^4*(a*e + b*e
*x))

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fricas [A]  time = 0.44, size = 311, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (1155 \, b^{4} e^{6} x^{6} + 128 \, b^{4} d^{6} - 832 \, a b^{3} d^{5} e + 2288 \, a^{2} b^{2} d^{4} e^{2} - 3432 \, a^{3} b d^{3} e^{3} + 3003 \, a^{4} d^{2} e^{4} + 210 \, {\left (7 \, b^{4} d e^{5} + 26 \, a b^{3} e^{6}\right )} x^{5} + 35 \, {\left (b^{4} d^{2} e^{4} + 208 \, a b^{3} d e^{5} + 286 \, a^{2} b^{2} e^{6}\right )} x^{4} - 20 \, {\left (2 \, b^{4} d^{3} e^{3} - 13 \, a b^{3} d^{2} e^{4} - 715 \, a^{2} b^{2} d e^{5} - 429 \, a^{3} b e^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{4} d^{4} e^{2} - 104 \, a b^{3} d^{3} e^{3} + 286 \, a^{2} b^{2} d^{2} e^{4} + 4576 \, a^{3} b d e^{5} + 1001 \, a^{4} e^{6}\right )} x^{2} - 2 \, {\left (32 \, b^{4} d^{5} e - 208 \, a b^{3} d^{4} e^{2} + 572 \, a^{2} b^{2} d^{3} e^{3} - 858 \, a^{3} b d^{2} e^{4} - 3003 \, a^{4} d e^{5}\right )} x\right )} \sqrt {e x + d}}{15015 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(1155*b^4*e^6*x^6 + 128*b^4*d^6 - 832*a*b^3*d^5*e + 2288*a^2*b^2*d^4*e^2 - 3432*a^3*b*d^3*e^3 + 3003*a
^4*d^2*e^4 + 210*(7*b^4*d*e^5 + 26*a*b^3*e^6)*x^5 + 35*(b^4*d^2*e^4 + 208*a*b^3*d*e^5 + 286*a^2*b^2*e^6)*x^4 -
20*(2*b^4*d^3*e^3 - 13*a*b^3*d^2*e^4 - 715*a^2*b^2*d*e^5 - 429*a^3*b*e^6)*x^3 + 3*(16*b^4*d^4*e^2 - 104*a*b^3
*d^3*e^3 + 286*a^2*b^2*d^2*e^4 + 4576*a^3*b*d*e^5 + 1001*a^4*e^6)*x^2 - 2*(32*b^4*d^5*e - 208*a*b^3*d^4*e^2 +
572*a^2*b^2*d^3*e^3 - 858*a^3*b*d^2*e^4 - 3003*a^4*d*e^5)*x)*sqrt(e*x + d)/e^5

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giac [B]  time = 0.27, size = 944, normalized size = 3.58

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(60060*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3*b*d^2*e^(-1)*sgn(b*x + a) + 18018*(3*(x*e + d)^(5/2)
- 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*b^2*d^2*e^(-2)*sgn(b*x + a) + 5148*(5*(x*e + d)^(7/2) - 21*
(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a*b^3*d^2*e^(-3)*sgn(b*x + a) + 143*(35*(x*
e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d
^4)*b^4*d^2*e^(-4)*sgn(b*x + a) + 24024*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*
b*d*e^(-1)*sgn(b*x + a) + 15444*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x
*e + d)*d^3)*a^2*b^2*d*e^(-2)*sgn(b*x + a) + 1144*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^
(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a*b^3*d*e^(-3)*sgn(b*x + a) + 130*(63*(x*e + d)^(
11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4
- 693*sqrt(x*e + d)*d^5)*b^4*d*e^(-4)*sgn(b*x + a) + 45045*sqrt(x*e + d)*a^4*d^2*sgn(b*x + a) + 30030*((x*e +
d)^(3/2) - 3*sqrt(x*e + d)*d)*a^4*d*sgn(b*x + a) + 5148*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e +
d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^3*b*e^(-1)*sgn(b*x + a) + 858*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)
*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^2*b^2*e^(-2)*sgn(b*x + a) +
260*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(
x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a*b^3*e^(-3)*sgn(b*x + a) + 15*(231*(x*e + d)^(13/2) - 1638*(x*e +
d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)
^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b^4*e^(-4)*sgn(b*x + a) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d
+ 15*sqrt(x*e + d)*d^2)*a^4*sgn(b*x + a))*e^(-1)

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maple [A]  time = 0.05, size = 202, normalized size = 0.77 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (1155 b^{4} e^{4} x^{4}+5460 a \,b^{3} e^{4} x^{3}-840 b^{4} d \,e^{3} x^{3}+10010 a^{2} b^{2} e^{4} x^{2}-3640 a \,b^{3} d \,e^{3} x^{2}+560 b^{4} d^{2} e^{2} x^{2}+8580 a^{3} b \,e^{4} x -5720 a^{2} b^{2} d \,e^{3} x +2080 a \,b^{3} d^{2} e^{2} x -320 b^{4} d^{3} e x +3003 a^{4} e^{4}-3432 a^{3} b d \,e^{3}+2288 a^{2} b^{2} d^{2} e^{2}-832 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{15015 \left (b x +a \right )^{3} e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(e*x+d)^(5/2)*(1155*b^4*e^4*x^4+5460*a*b^3*e^4*x^3-840*b^4*d*e^3*x^3+10010*a^2*b^2*e^4*x^2-3640*a*b^3*
d*e^3*x^2+560*b^4*d^2*e^2*x^2+8580*a^3*b*e^4*x-5720*a^2*b^2*d*e^3*x+2080*a*b^3*d^2*e^2*x-320*b^4*d^3*e*x+3003*
a^4*e^4-3432*a^3*b*d*e^3+2288*a^2*b^2*d^2*e^2-832*a*b^3*d^3*e+128*b^4*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)^3

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maxima [B]  time = 0.70, size = 488, normalized size = 1.85 \begin {gather*} \frac {2 \, {\left (105 \, b^{3} e^{5} x^{5} - 16 \, b^{3} d^{5} + 88 \, a b^{2} d^{4} e - 198 \, a^{2} b d^{3} e^{2} + 231 \, a^{3} d^{2} e^{3} + 35 \, {\left (4 \, b^{3} d e^{4} + 11 \, a b^{2} e^{5}\right )} x^{4} + 5 \, {\left (b^{3} d^{2} e^{3} + 110 \, a b^{2} d e^{4} + 99 \, a^{2} b e^{5}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{3} e^{2} - 11 \, a b^{2} d^{2} e^{3} - 264 \, a^{2} b d e^{4} - 77 \, a^{3} e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{4} e - 44 \, a b^{2} d^{3} e^{2} + 99 \, a^{2} b d^{2} e^{3} + 462 \, a^{3} d e^{4}\right )} x\right )} \sqrt {e x + d} a}{1155 \, e^{4}} + \frac {2 \, {\left (1155 \, b^{3} e^{6} x^{6} + 128 \, b^{3} d^{6} - 624 \, a b^{2} d^{5} e + 1144 \, a^{2} b d^{4} e^{2} - 858 \, a^{3} d^{3} e^{3} + 105 \, {\left (14 \, b^{3} d e^{5} + 39 \, a b^{2} e^{6}\right )} x^{5} + 35 \, {\left (b^{3} d^{2} e^{4} + 156 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} - 5 \, {\left (8 \, b^{3} d^{3} e^{3} - 39 \, a b^{2} d^{2} e^{4} - 1430 \, a^{2} b d e^{5} - 429 \, a^{3} e^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{3} d^{4} e^{2} - 78 \, a b^{2} d^{3} e^{3} + 143 \, a^{2} b d^{2} e^{4} + 1144 \, a^{3} d e^{5}\right )} x^{2} - {\left (64 \, b^{3} d^{5} e - 312 \, a b^{2} d^{4} e^{2} + 572 \, a^{2} b d^{3} e^{3} - 429 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d} b}{15015 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(105*b^3*e^5*x^5 - 16*b^3*d^5 + 88*a*b^2*d^4*e - 198*a^2*b*d^3*e^2 + 231*a^3*d^2*e^3 + 35*(4*b^3*d*e^4
+ 11*a*b^2*e^5)*x^4 + 5*(b^3*d^2*e^3 + 110*a*b^2*d*e^4 + 99*a^2*b*e^5)*x^3 - 3*(2*b^3*d^3*e^2 - 11*a*b^2*d^2*e
^3 - 264*a^2*b*d*e^4 - 77*a^3*e^5)*x^2 + (8*b^3*d^4*e - 44*a*b^2*d^3*e^2 + 99*a^2*b*d^2*e^3 + 462*a^3*d*e^4)*x
)*sqrt(e*x + d)*a/e^4 + 2/15015*(1155*b^3*e^6*x^6 + 128*b^3*d^6 - 624*a*b^2*d^5*e + 1144*a^2*b*d^4*e^2 - 858*a
^3*d^3*e^3 + 105*(14*b^3*d*e^5 + 39*a*b^2*e^6)*x^5 + 35*(b^3*d^2*e^4 + 156*a*b^2*d*e^5 + 143*a^2*b*e^6)*x^4 -
5*(8*b^3*d^3*e^3 - 39*a*b^2*d^2*e^4 - 1430*a^2*b*d*e^5 - 429*a^3*e^6)*x^3 + 3*(16*b^3*d^4*e^2 - 78*a*b^2*d^3*e
^3 + 143*a^2*b*d^2*e^4 + 1144*a^3*d*e^5)*x^2 - (64*b^3*d^5*e - 312*a*b^2*d^4*e^2 + 572*a^2*b*d^3*e^3 - 429*a^3
*d^2*e^4)*x)*sqrt(e*x + d)*b/e^5

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  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((b*x+a)*(e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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