3.2110 \(\int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=374 \[ \frac{30 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^4}{11 e^7 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^5}{3 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^6}{7 e^7 (a+b x)}+\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{19/2}}{19 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2} (b d-a e)}{17 e^7 (a+b x)}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e)^2}{e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^3}{13 e^7 (a+b x)} \]

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)
) - (4*b*(b*d - a*e)^5*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a
+ b*x)) + (30*b^2*(b*d - a*e)^4*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(11*e^7*(a + b*x)) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(13*e^7*(a + b*x)) + (2*b^4*(b*d - a*e)^2*(d + e*x)^(15/2)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (12*b^5*(b*d - a*e)*(d + e*x)^(17/2)*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(17*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(19/2)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(19*e^7*(a + b*x))

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Rubi [A]  time = 0.451431, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{30 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^4}{11 e^7 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^5}{3 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^6}{7 e^7 (a+b x)}+\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{19/2}}{19 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2} (b d-a e)}{17 e^7 (a+b x)}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e)^2}{e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^3}{13 e^7 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)
) - (4*b*(b*d - a*e)^5*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a
+ b*x)) + (30*b^2*(b*d - a*e)^4*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(11*e^7*(a + b*x)) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(13*e^7*(a + b*x)) + (2*b^4*(b*d - a*e)^2*(d + e*x)^(15/2)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (12*b^5*(b*d - a*e)*(d + e*x)^(17/2)*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(17*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(19/2)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(19*e^7*(a + b*x))

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Rubi in Sympy [A]  time = 62.3786, size = 323, normalized size = 0.86 \[ \frac{2 \left (a + b x\right ) \left (d + e x\right )^{\frac{7}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{19 e} + \frac{24 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{323 e^{2}} + \frac{16 \left (5 a + 5 b x\right ) \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{1615 e^{3}} + \frac{128 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{4199 e^{4}} + \frac{256 \left (3 a + 3 b x\right ) \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{46189 e^{5}} + \frac{1024 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{138567 e^{6}} + \frac{2048 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{6} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{969969 e^{7} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

2*(a + b*x)*(d + e*x)**(7/2)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(19*e) + 24*(d
+ e*x)**(7/2)*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(323*e**2) + 16*(5
*a + 5*b*x)*(d + e*x)**(7/2)*(a*e - b*d)**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/
(1615*e**3) + 128*(d + e*x)**(7/2)*(a*e - b*d)**3*(a**2 + 2*a*b*x + b**2*x**2)**
(3/2)/(4199*e**4) + 256*(3*a + 3*b*x)*(d + e*x)**(7/2)*(a*e - b*d)**4*sqrt(a**2
+ 2*a*b*x + b**2*x**2)/(46189*e**5) + 1024*(d + e*x)**(7/2)*(a*e - b*d)**5*sqrt(
a**2 + 2*a*b*x + b**2*x**2)/(138567*e**6) + 2048*(d + e*x)**(7/2)*(a*e - b*d)**6
*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(969969*e**7*(a + b*x))

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Mathematica [A]  time = 0.521613, size = 309, normalized size = 0.83 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{7/2} \left (138567 a^6 e^6+92378 a^5 b e^5 (7 e x-2 d)+20995 a^4 b^2 e^4 \left (8 d^2-28 d e x+63 e^2 x^2\right )+6460 a^3 b^3 e^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+323 a^2 b^4 e^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+38 a b^5 e \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )+b^6 \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )\right )}{969969 e^7 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(7/2)*(138567*a^6*e^6 + 92378*a^5*b*e^5*(-2*d + 7
*e*x) + 20995*a^4*b^2*e^4*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 6460*a^3*b^3*e^3*(-1
6*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + 323*a^2*b^4*e^2*(128*d^4 - 4
48*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) + 38*a*b^5*e*(-25
6*d^5 + 896*d^4*e*x - 2016*d^3*e^2*x^2 + 3696*d^2*e^3*x^3 - 6006*d*e^4*x^4 + 900
9*e^5*x^5) + b^6*(1024*d^6 - 3584*d^5*e*x + 8064*d^4*e^2*x^2 - 14784*d^3*e^3*x^3
+ 24024*d^2*e^4*x^4 - 36036*d*e^5*x^5 + 51051*e^6*x^6)))/(969969*e^7*(a + b*x))

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Maple [A]  time = 0.013, size = 393, normalized size = 1.1 \[{\frac{102102\,{x}^{6}{b}^{6}{e}^{6}+684684\,{x}^{5}a{b}^{5}{e}^{6}-72072\,{x}^{5}{b}^{6}d{e}^{5}+1939938\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-456456\,{x}^{4}a{b}^{5}d{e}^{5}+48048\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+2984520\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-1193808\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+280896\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-29568\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+2645370\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-1627920\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+651168\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-153216\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+16128\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+1293292\,x{a}^{5}b{e}^{6}-1175720\,x{a}^{4}{b}^{2}d{e}^{5}+723520\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-289408\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+68096\,xa{b}^{5}{d}^{4}{e}^{2}-7168\,x{b}^{6}{d}^{5}e+277134\,{a}^{6}{e}^{6}-369512\,{a}^{5}bd{e}^{5}+335920\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}-206720\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+82688\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-19456\,{d}^{5}a{b}^{5}e+2048\,{b}^{6}{d}^{6}}{969969\,{e}^{7} \left ( bx+a \right ) ^{5}} \left ( ex+d \right ) ^{{\frac{7}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^(5/2),x)

[Out]

2/969969*(e*x+d)^(7/2)*(51051*b^6*e^6*x^6+342342*a*b^5*e^6*x^5-36036*b^6*d*e^5*x
^5+969969*a^2*b^4*e^6*x^4-228228*a*b^5*d*e^5*x^4+24024*b^6*d^2*e^4*x^4+1492260*a
^3*b^3*e^6*x^3-596904*a^2*b^4*d*e^5*x^3+140448*a*b^5*d^2*e^4*x^3-14784*b^6*d^3*e
^3*x^3+1322685*a^4*b^2*e^6*x^2-813960*a^3*b^3*d*e^5*x^2+325584*a^2*b^4*d^2*e^4*x
^2-76608*a*b^5*d^3*e^3*x^2+8064*b^6*d^4*e^2*x^2+646646*a^5*b*e^6*x-587860*a^4*b^
2*d*e^5*x+361760*a^3*b^3*d^2*e^4*x-144704*a^2*b^4*d^3*e^3*x+34048*a*b^5*d^4*e^2*
x-3584*b^6*d^5*e*x+138567*a^6*e^6-184756*a^5*b*d*e^5+167960*a^4*b^2*d^2*e^4-1033
60*a^3*b^3*d^3*e^3+41344*a^2*b^4*d^4*e^2-9728*a*b^5*d^5*e+1024*b^6*d^6)*((b*x+a)
^2)^(5/2)/e^7/(b*x+a)^5

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Maxima [A]  time = 0.729862, size = 1458, normalized size = 3.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/153153*(9009*b^5*e^8*x^8 - 256*b^5*d^8 + 2176*a*b^4*d^7*e - 8160*a^2*b^3*d^6*e
^2 + 17680*a^3*b^2*d^5*e^3 - 24310*a^4*b*d^4*e^4 + 21879*a^5*d^3*e^5 + 3003*(7*b
^5*d*e^7 + 17*a*b^4*e^8)*x^7 + 231*(55*b^5*d^2*e^6 + 527*a*b^4*d*e^7 + 510*a^2*b
^3*e^8)*x^6 + 63*(b^5*d^3*e^5 + 1207*a*b^4*d^2*e^6 + 4590*a^2*b^3*d*e^7 + 2210*a
^3*b^2*e^8)*x^5 - 35*(2*b^5*d^4*e^4 - 17*a*b^4*d^3*e^5 - 5406*a^2*b^3*d^2*e^6 -
10166*a^3*b^2*d*e^7 - 2431*a^4*b*e^8)*x^4 + (80*b^5*d^5*e^3 - 680*a*b^4*d^4*e^4
+ 2550*a^2*b^3*d^3*e^5 + 249730*a^3*b^2*d^2*e^6 + 230945*a^4*b*d*e^7 + 21879*a^5
*e^8)*x^3 - 3*(32*b^5*d^6*e^2 - 272*a*b^4*d^5*e^3 + 1020*a^2*b^3*d^4*e^4 - 2210*
a^3*b^2*d^3*e^5 - 60775*a^4*b*d^2*e^6 - 21879*a^5*d*e^7)*x^2 + (128*b^5*d^7*e -
1088*a*b^4*d^6*e^2 + 4080*a^2*b^3*d^5*e^3 - 8840*a^3*b^2*d^4*e^4 + 12155*a^4*b*d
^3*e^5 + 65637*a^5*d^2*e^6)*x)*sqrt(e*x + d)*a/e^6 + 2/2909907*(153153*b^5*e^9*x
^9 + 3072*b^5*d^9 - 24320*a*b^4*d^8*e + 82688*a^2*b^3*d^7*e^2 - 155040*a^3*b^2*d
^6*e^3 + 167960*a^4*b*d^5*e^4 - 92378*a^5*d^4*e^5 + 9009*(39*b^5*d*e^8 + 95*a*b^
4*e^9)*x^8 + 3003*(69*b^5*d^2*e^7 + 665*a*b^4*d*e^8 + 646*a^2*b^3*e^9)*x^7 + 231
*(3*b^5*d^3*e^6 + 5225*a*b^4*d^2*e^7 + 20026*a^2*b^3*d*e^8 + 9690*a^3*b^2*e^9)*x
^6 - 63*(12*b^5*d^4*e^5 - 95*a*b^4*d^3*e^6 - 45866*a^2*b^3*d^2*e^7 - 87210*a^3*b
^2*d*e^8 - 20995*a^4*b*e^9)*x^5 + 7*(120*b^5*d^5*e^4 - 950*a*b^4*d^4*e^5 + 3230*
a^2*b^3*d^3*e^6 + 513570*a^3*b^2*d^2*e^7 + 482885*a^4*b*d*e^8 + 46189*a^5*e^9)*x
^4 - (960*b^5*d^6*e^3 - 7600*a*b^4*d^5*e^4 + 25840*a^2*b^3*d^4*e^5 - 48450*a^3*b
^2*d^3*e^6 - 2372435*a^4*b*d^2*e^7 - 877591*a^5*d*e^8)*x^3 + 3*(384*b^5*d^7*e^2
- 3040*a*b^4*d^6*e^3 + 10336*a^2*b^3*d^5*e^4 - 19380*a^3*b^2*d^4*e^5 + 20995*a^4
*b*d^3*e^6 + 230945*a^5*d^2*e^7)*x^2 - (1536*b^5*d^8*e - 12160*a*b^4*d^7*e^2 + 4
1344*a^2*b^3*d^6*e^3 - 77520*a^3*b^2*d^5*e^4 + 83980*a^4*b*d^4*e^5 - 46189*a^5*d
^3*e^6)*x)*sqrt(e*x + d)*b/e^7

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Fricas [A]  time = 0.293521, size = 857, normalized size = 2.29 \[ \frac{2 \,{\left (51051 \, b^{6} e^{9} x^{9} + 1024 \, b^{6} d^{9} - 9728 \, a b^{5} d^{8} e + 41344 \, a^{2} b^{4} d^{7} e^{2} - 103360 \, a^{3} b^{3} d^{6} e^{3} + 167960 \, a^{4} b^{2} d^{5} e^{4} - 184756 \, a^{5} b d^{4} e^{5} + 138567 \, a^{6} d^{3} e^{6} + 9009 \,{\left (13 \, b^{6} d e^{8} + 38 \, a b^{5} e^{9}\right )} x^{8} + 3003 \,{\left (23 \, b^{6} d^{2} e^{7} + 266 \, a b^{5} d e^{8} + 323 \, a^{2} b^{4} e^{9}\right )} x^{7} + 231 \,{\left (b^{6} d^{3} e^{6} + 2090 \, a b^{5} d^{2} e^{7} + 10013 \, a^{2} b^{4} d e^{8} + 6460 \, a^{3} b^{3} e^{9}\right )} x^{6} - 63 \,{\left (4 \, b^{6} d^{4} e^{5} - 38 \, a b^{5} d^{3} e^{6} - 22933 \, a^{2} b^{4} d^{2} e^{7} - 58140 \, a^{3} b^{3} d e^{8} - 20995 \, a^{4} b^{2} e^{9}\right )} x^{5} + 7 \,{\left (40 \, b^{6} d^{5} e^{4} - 380 \, a b^{5} d^{4} e^{5} + 1615 \, a^{2} b^{4} d^{3} e^{6} + 342380 \, a^{3} b^{3} d^{2} e^{7} + 482885 \, a^{4} b^{2} d e^{8} + 92378 \, a^{5} b e^{9}\right )} x^{4} -{\left (320 \, b^{6} d^{6} e^{3} - 3040 \, a b^{5} d^{5} e^{4} + 12920 \, a^{2} b^{4} d^{4} e^{5} - 32300 \, a^{3} b^{3} d^{3} e^{6} - 2372435 \, a^{4} b^{2} d^{2} e^{7} - 1755182 \, a^{5} b d e^{8} - 138567 \, a^{6} e^{9}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{7} e^{2} - 1216 \, a b^{5} d^{6} e^{3} + 5168 \, a^{2} b^{4} d^{5} e^{4} - 12920 \, a^{3} b^{3} d^{4} e^{5} + 20995 \, a^{4} b^{2} d^{3} e^{6} + 461890 \, a^{5} b d^{2} e^{7} + 138567 \, a^{6} d e^{8}\right )} x^{2} -{\left (512 \, b^{6} d^{8} e - 4864 \, a b^{5} d^{7} e^{2} + 20672 \, a^{2} b^{4} d^{6} e^{3} - 51680 \, a^{3} b^{3} d^{5} e^{4} + 83980 \, a^{4} b^{2} d^{4} e^{5} - 92378 \, a^{5} b d^{3} e^{6} - 415701 \, a^{6} d^{2} e^{7}\right )} x\right )} \sqrt{e x + d}}{969969 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/969969*(51051*b^6*e^9*x^9 + 1024*b^6*d^9 - 9728*a*b^5*d^8*e + 41344*a^2*b^4*d^
7*e^2 - 103360*a^3*b^3*d^6*e^3 + 167960*a^4*b^2*d^5*e^4 - 184756*a^5*b*d^4*e^5 +
138567*a^6*d^3*e^6 + 9009*(13*b^6*d*e^8 + 38*a*b^5*e^9)*x^8 + 3003*(23*b^6*d^2*
e^7 + 266*a*b^5*d*e^8 + 323*a^2*b^4*e^9)*x^7 + 231*(b^6*d^3*e^6 + 2090*a*b^5*d^2
*e^7 + 10013*a^2*b^4*d*e^8 + 6460*a^3*b^3*e^9)*x^6 - 63*(4*b^6*d^4*e^5 - 38*a*b^
5*d^3*e^6 - 22933*a^2*b^4*d^2*e^7 - 58140*a^3*b^3*d*e^8 - 20995*a^4*b^2*e^9)*x^5
+ 7*(40*b^6*d^5*e^4 - 380*a*b^5*d^4*e^5 + 1615*a^2*b^4*d^3*e^6 + 342380*a^3*b^3
*d^2*e^7 + 482885*a^4*b^2*d*e^8 + 92378*a^5*b*e^9)*x^4 - (320*b^6*d^6*e^3 - 3040
*a*b^5*d^5*e^4 + 12920*a^2*b^4*d^4*e^5 - 32300*a^3*b^3*d^3*e^6 - 2372435*a^4*b^2
*d^2*e^7 - 1755182*a^5*b*d*e^8 - 138567*a^6*e^9)*x^3 + 3*(128*b^6*d^7*e^2 - 1216
*a*b^5*d^6*e^3 + 5168*a^2*b^4*d^5*e^4 - 12920*a^3*b^3*d^4*e^5 + 20995*a^4*b^2*d^
3*e^6 + 461890*a^5*b*d^2*e^7 + 138567*a^6*d*e^8)*x^2 - (512*b^6*d^8*e - 4864*a*b
^5*d^7*e^2 + 20672*a^2*b^4*d^6*e^3 - 51680*a^3*b^3*d^5*e^4 + 83980*a^4*b^2*d^4*e
^5 - 92378*a^5*b*d^3*e^6 - 415701*a^6*d^2*e^7)*x)*sqrt(e*x + d)/e^7

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.369552, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)*(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Done