3.19.75 \(\int (a+b x) (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=264 \[ \frac {12 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^5 (a+b x)}-\frac {8 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^5 (a+b x)}+\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^5 (a+b x)}-\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^5 (a+b x)} \]

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Rubi [A]  time = 0.13, 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)^{15/2}}{15 e^5 (a+b x)}-\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^5 (a+b x)}+\frac {12 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^5 (a+b x)}-\frac {8 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^5 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)) - (8*b*(b*d - a*e)^3*(d + e*
x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a + b*x)) + (12*b^2*(b*d - a*e)^2*(d + e*x)^(11/2)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x)) - (8*b^3*(b*d - a*e)*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(1
3*e^5*(a + b*x)) + (2*b^4*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*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)^{5/2} \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)^{5/2} \, 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)^{5/2} \, 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)^{5/2}}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^{7/2}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{9/2}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{11/2}}{e^4}+\frac {b^4 (d+e x)^{13/2}}{e^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 (b d-a e)^4 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}-\frac {8 b (b d-a e)^3 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac {12 b^2 (b d-a e)^2 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}-\frac {8 b^3 (b d-a e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^5 (a+b x)}+\frac {2 b^4 (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^5 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 172, normalized size = 0.65 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{7/2} \left (6435 a^4 e^4+2860 a^3 b e^3 (7 e x-2 d)+390 a^2 b^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+60 a b^3 e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+b^4 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 e^5 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(7/2)*(6435*a^4*e^4 + 2860*a^3*b*e^3*(-2*d + 7*e*x) + 390*a^2*b^2*e^2*(8*d^2 -
28*d*e*x + 63*e^2*x^2) + 60*a*b^3*e*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + b^4*(128*d^4 - 448*
d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)))/(45045*e^5*(a + b*x))

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IntegrateAlgebraic [A]  time = 51.19, size = 241, normalized size = 0.91 \begin {gather*} \frac {2 (d+e x)^{7/2} \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (6435 a^4 e^4+20020 a^3 b e^3 (d+e x)-25740 a^3 b d e^3+38610 a^2 b^2 d^2 e^2+24570 a^2 b^2 e^2 (d+e x)^2-60060 a^2 b^2 d e^2 (d+e x)-25740 a b^3 d^3 e+60060 a b^3 d^2 e (d+e x)+13860 a b^3 e (d+e x)^3-49140 a b^3 d e (d+e x)^2+6435 b^4 d^4-20020 b^4 d^3 (d+e x)+24570 b^4 d^2 (d+e x)^2+3003 b^4 (d+e x)^4-13860 b^4 d (d+e x)^3\right )}{45045 e^4 (a e+b e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*(d + e*x)^(7/2)*Sqrt[(a*e + b*e*x)^2/e^2]*(6435*b^4*d^4 - 25740*a*b^3*d^3*e + 38610*a^2*b^2*d^2*e^2 - 25740
*a^3*b*d*e^3 + 6435*a^4*e^4 - 20020*b^4*d^3*(d + e*x) + 60060*a*b^3*d^2*e*(d + e*x) - 60060*a^2*b^2*d*e^2*(d +
e*x) + 20020*a^3*b*e^3*(d + e*x) + 24570*b^4*d^2*(d + e*x)^2 - 49140*a*b^3*d*e*(d + e*x)^2 + 24570*a^2*b^2*e^
2*(d + e*x)^2 - 13860*b^4*d*(d + e*x)^3 + 13860*a*b^3*e*(d + e*x)^3 + 3003*b^4*(d + e*x)^4))/(45045*e^4*(a*e +
b*e*x))

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fricas [A]  time = 0.44, size = 377, normalized size = 1.43 \begin {gather*} \frac {2 \, {\left (3003 \, b^{4} e^{7} x^{7} + 128 \, b^{4} d^{7} - 960 \, a b^{3} d^{6} e + 3120 \, a^{2} b^{2} d^{5} e^{2} - 5720 \, a^{3} b d^{4} e^{3} + 6435 \, a^{4} d^{3} e^{4} + 231 \, {\left (31 \, b^{4} d e^{6} + 60 \, a b^{3} e^{7}\right )} x^{6} + 63 \, {\left (71 \, b^{4} d^{2} e^{5} + 540 \, a b^{3} d e^{6} + 390 \, a^{2} b^{2} e^{7}\right )} x^{5} + 35 \, {\left (b^{4} d^{3} e^{4} + 636 \, a b^{3} d^{2} e^{5} + 1794 \, a^{2} b^{2} d e^{6} + 572 \, a^{3} b e^{7}\right )} x^{4} - 5 \, {\left (8 \, b^{4} d^{4} e^{3} - 60 \, a b^{3} d^{3} e^{4} - 8814 \, a^{2} b^{2} d^{2} e^{5} - 10868 \, a^{3} b d e^{6} - 1287 \, a^{4} e^{7}\right )} x^{3} + 3 \, {\left (16 \, b^{4} d^{5} e^{2} - 120 \, a b^{3} d^{4} e^{3} + 390 \, a^{2} b^{2} d^{3} e^{4} + 14300 \, a^{3} b d^{2} e^{5} + 6435 \, a^{4} d e^{6}\right )} x^{2} - {\left (64 \, b^{4} d^{6} e - 480 \, a b^{3} d^{5} e^{2} + 1560 \, a^{2} b^{2} d^{4} e^{3} - 2860 \, a^{3} b d^{3} e^{4} - 19305 \, a^{4} d^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

2/45045*(3003*b^4*e^7*x^7 + 128*b^4*d^7 - 960*a*b^3*d^6*e + 3120*a^2*b^2*d^5*e^2 - 5720*a^3*b*d^4*e^3 + 6435*a
^4*d^3*e^4 + 231*(31*b^4*d*e^6 + 60*a*b^3*e^7)*x^6 + 63*(71*b^4*d^2*e^5 + 540*a*b^3*d*e^6 + 390*a^2*b^2*e^7)*x
^5 + 35*(b^4*d^3*e^4 + 636*a*b^3*d^2*e^5 + 1794*a^2*b^2*d*e^6 + 572*a^3*b*e^7)*x^4 - 5*(8*b^4*d^4*e^3 - 60*a*b
^3*d^3*e^4 - 8814*a^2*b^2*d^2*e^5 - 10868*a^3*b*d*e^6 - 1287*a^4*e^7)*x^3 + 3*(16*b^4*d^5*e^2 - 120*a*b^3*d^4*
e^3 + 390*a^2*b^2*d^3*e^4 + 14300*a^3*b*d^2*e^5 + 6435*a^4*d*e^6)*x^2 - (64*b^4*d^6*e - 480*a*b^3*d^5*e^2 + 15
60*a^2*b^2*d^4*e^3 - 2860*a^3*b*d^3*e^4 - 19305*a^4*d^2*e^5)*x)*sqrt(e*x + d)/e^5

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giac [B]  time = 0.32, size = 1397, normalized size = 5.29

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(5/2)*(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^3*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^3*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^3*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^3*e^(-4)*sgn(b*x + a) + 36036*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*
b*d^2*e^(-1)*sgn(b*x + a) + 23166*(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^2*e^(-2)*sgn(b*x + a) + 1716*(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^2*e^(-3)*sgn(b*x + a) + 195*(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^2*e^(-4)*sgn(b*x + a) + 45045*sqrt(x*e + d)*a^4*d^3*sgn(b*x + a) + 45045*
((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^4*d^2*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^3*b*d*e^(-1)*sgn(b*x + a) + 2574*(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*d*e^(-2)
*sgn(b*x + a) + 780*(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*d*e^(-3)*sgn(b*x + a) + 45*(231*(x*e + d)^(1
3/2) - 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*d*e^(-4)*sgn(b*x + a) + 9009*(3*(x*e + d)^(5/2) - 1
0*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^4*d*sgn(b*x + a) + 572*(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^3*b*e^(-1)*sgn(b*x + a) + 39
0*(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^2*b^2*e^(-2)*sgn(b*x + a) + 60*(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)*a*b^3*e^(-3)*sgn(b*x + a) + 7*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/
2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5
/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*b^4*e^(-4)*sgn(b*x + a) + 1287*(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^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 {7}{2}} \left (3003 b^{4} e^{4} x^{4}+13860 a \,b^{3} e^{4} x^{3}-1848 b^{4} d \,e^{3} x^{3}+24570 a^{2} b^{2} e^{4} x^{2}-7560 a \,b^{3} d \,e^{3} x^{2}+1008 b^{4} d^{2} e^{2} x^{2}+20020 a^{3} b \,e^{4} x -10920 a^{2} b^{2} d \,e^{3} x +3360 a \,b^{3} d^{2} e^{2} x -448 b^{4} d^{3} e x +6435 a^{4} e^{4}-5720 a^{3} b d \,e^{3}+3120 a^{2} b^{2} d^{2} e^{2}-960 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{45045 \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)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

2/45045*(e*x+d)^(7/2)*(3003*b^4*e^4*x^4+13860*a*b^3*e^4*x^3-1848*b^4*d*e^3*x^3+24570*a^2*b^2*e^4*x^2-7560*a*b^
3*d*e^3*x^2+1008*b^4*d^2*e^2*x^2+20020*a^3*b*e^4*x-10920*a^2*b^2*d*e^3*x+3360*a*b^3*d^2*e^2*x-448*b^4*d^3*e*x+
6435*a^4*e^4-5720*a^3*b*d*e^3+3120*a^2*b^2*d^2*e^2-960*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 = 0.85, size = 592, normalized size = 2.24 \begin {gather*} \frac {2 \, {\left (231 \, b^{3} e^{6} x^{6} - 16 \, b^{3} d^{6} + 104 \, a b^{2} d^{5} e - 286 \, a^{2} b d^{4} e^{2} + 429 \, a^{3} d^{3} e^{3} + 63 \, {\left (9 \, b^{3} d e^{5} + 13 \, a b^{2} e^{6}\right )} x^{5} + 7 \, {\left (53 \, b^{3} d^{2} e^{4} + 299 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} + {\left (5 \, b^{3} d^{3} e^{3} + 1469 \, a b^{2} d^{2} e^{4} + 2717 \, a^{2} b d e^{5} + 429 \, a^{3} e^{6}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{4} e^{2} - 13 \, a b^{2} d^{3} e^{3} - 715 \, a^{2} b d^{2} e^{4} - 429 \, a^{3} d e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{5} e - 52 \, a b^{2} d^{4} e^{2} + 143 \, a^{2} b d^{3} e^{3} + 1287 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d} a}{3003 \, e^{4}} + \frac {2 \, {\left (3003 \, b^{3} e^{7} x^{7} + 128 \, b^{3} d^{7} - 720 \, a b^{2} d^{6} e + 1560 \, a^{2} b d^{5} e^{2} - 1430 \, a^{3} d^{4} e^{3} + 231 \, {\left (31 \, b^{3} d e^{6} + 45 \, a b^{2} e^{7}\right )} x^{6} + 63 \, {\left (71 \, b^{3} d^{2} e^{5} + 405 \, a b^{2} d e^{6} + 195 \, a^{2} b e^{7}\right )} x^{5} + 35 \, {\left (b^{3} d^{3} e^{4} + 477 \, a b^{2} d^{2} e^{5} + 897 \, a^{2} b d e^{6} + 143 \, a^{3} e^{7}\right )} x^{4} - 5 \, {\left (8 \, b^{3} d^{4} e^{3} - 45 \, a b^{2} d^{3} e^{4} - 4407 \, a^{2} b d^{2} e^{5} - 2717 \, a^{3} d e^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{3} d^{5} e^{2} - 90 \, a b^{2} d^{4} e^{3} + 195 \, a^{2} b d^{3} e^{4} + 3575 \, a^{3} d^{2} e^{5}\right )} x^{2} - {\left (64 \, b^{3} d^{6} e - 360 \, a b^{2} d^{5} e^{2} + 780 \, a^{2} b d^{4} e^{3} - 715 \, a^{3} d^{3} e^{4}\right )} x\right )} \sqrt {e x + d} b}{45045 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

2/3003*(231*b^3*e^6*x^6 - 16*b^3*d^6 + 104*a*b^2*d^5*e - 286*a^2*b*d^4*e^2 + 429*a^3*d^3*e^3 + 63*(9*b^3*d*e^5
+ 13*a*b^2*e^6)*x^5 + 7*(53*b^3*d^2*e^4 + 299*a*b^2*d*e^5 + 143*a^2*b*e^6)*x^4 + (5*b^3*d^3*e^3 + 1469*a*b^2*
d^2*e^4 + 2717*a^2*b*d*e^5 + 429*a^3*e^6)*x^3 - 3*(2*b^3*d^4*e^2 - 13*a*b^2*d^3*e^3 - 715*a^2*b*d^2*e^4 - 429*
a^3*d*e^5)*x^2 + (8*b^3*d^5*e - 52*a*b^2*d^4*e^2 + 143*a^2*b*d^3*e^3 + 1287*a^3*d^2*e^4)*x)*sqrt(e*x + d)*a/e^
4 + 2/45045*(3003*b^3*e^7*x^7 + 128*b^3*d^7 - 720*a*b^2*d^6*e + 1560*a^2*b*d^5*e^2 - 1430*a^3*d^4*e^3 + 231*(3
1*b^3*d*e^6 + 45*a*b^2*e^7)*x^6 + 63*(71*b^3*d^2*e^5 + 405*a*b^2*d*e^6 + 195*a^2*b*e^7)*x^5 + 35*(b^3*d^3*e^4
+ 477*a*b^2*d^2*e^5 + 897*a^2*b*d*e^6 + 143*a^3*e^7)*x^4 - 5*(8*b^3*d^4*e^3 - 45*a*b^2*d^3*e^4 - 4407*a^2*b*d^
2*e^5 - 2717*a^3*d*e^6)*x^3 + 3*(16*b^3*d^5*e^2 - 90*a*b^2*d^4*e^3 + 195*a^2*b*d^3*e^4 + 3575*a^3*d^2*e^5)*x^2
- (64*b^3*d^6*e - 360*a*b^2*d^5*e^2 + 780*a^2*b*d^4*e^3 - 715*a^3*d^3*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 )}^{5/2}\,{\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)^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)

[Out]

int((a + b*x)*(d + e*x)^(5/2)*(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)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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