3.1627 \(\int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)

Optimal. Leaf size=185 \[ -\frac{12 b^5 (d+e x)^{17/2} (b d-a e)}{17 e^7}+\frac{2 b^4 (d+e x)^{15/2} (b d-a e)^2}{e^7}-\frac{40 b^3 (d+e x)^{13/2} (b d-a e)^3}{13 e^7}+\frac{30 b^2 (d+e x)^{11/2} (b d-a e)^4}{11 e^7}-\frac{4 b (d+e x)^{9/2} (b d-a e)^5}{3 e^7}+\frac{2 (d+e x)^{7/2} (b d-a e)^6}{7 e^7}+\frac{2 b^6 (d+e x)^{19/2}}{19 e^7} \]

[Out]

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

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Rubi [A]  time = 0.182906, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{12 b^5 (d+e x)^{17/2} (b d-a e)}{17 e^7}+\frac{2 b^4 (d+e x)^{15/2} (b d-a e)^2}{e^7}-\frac{40 b^3 (d+e x)^{13/2} (b d-a e)^3}{13 e^7}+\frac{30 b^2 (d+e x)^{11/2} (b d-a e)^4}{11 e^7}-\frac{4 b (d+e x)^{9/2} (b d-a e)^5}{3 e^7}+\frac{2 (d+e x)^{7/2} (b d-a e)^6}{7 e^7}+\frac{2 b^6 (d+e x)^{19/2}}{19 e^7} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 84.5476, size = 172, normalized size = 0.93 \[ \frac{2 b^{6} \left (d + e x\right )^{\frac{19}{2}}}{19 e^{7}} + \frac{12 b^{5} \left (d + e x\right )^{\frac{17}{2}} \left (a e - b d\right )}{17 e^{7}} + \frac{2 b^{4} \left (d + e x\right )^{\frac{15}{2}} \left (a e - b d\right )^{2}}{e^{7}} + \frac{40 b^{3} \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )^{3}}{13 e^{7}} + \frac{30 b^{2} \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{4}}{11 e^{7}} + \frac{4 b \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{5}}{3 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{6}}{7 e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.399254, size = 291, normalized size = 1.57 \[ \frac{2 (d+e x)^{7/2} \left (138567 a^6 e^6+92378 a^5 b e^5 (7 e x-2 d)+20995 a^4 b^2 e^4 \left (8 d^2-28 d e x+63 e^2 x^2\right )+6460 a^3 b^3 e^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+323 a^2 b^4 e^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+38 a b^5 e \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )+b^6 \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )\right )}{969969 e^7} \]

Antiderivative was successfully verified.

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

[Out]

(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*(-16*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 - 448*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*(-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) + 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)

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Maple [B]  time = 0.013, size = 377, normalized size = 2. \[{\frac{102102\,{x}^{6}{b}^{6}{e}^{6}+684684\,{x}^{5}a{b}^{5}{e}^{6}-72072\,{x}^{5}{b}^{6}d{e}^{5}+1939938\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-456456\,{x}^{4}a{b}^{5}d{e}^{5}+48048\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+2984520\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-1193808\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+280896\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-29568\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+2645370\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-1627920\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+651168\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-153216\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+16128\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+1293292\,x{a}^{5}b{e}^{6}-1175720\,x{a}^{4}{b}^{2}d{e}^{5}+723520\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-289408\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+68096\,xa{b}^{5}{d}^{4}{e}^{2}-7168\,x{b}^{6}{d}^{5}e+277134\,{a}^{6}{e}^{6}-369512\,{a}^{5}bd{e}^{5}+335920\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}-206720\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+82688\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-19456\,{d}^{5}a{b}^{5}e+2048\,{b}^{6}{d}^{6}}{969969\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^3,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)/e^7

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Maxima [A]  time = 0.73431, size = 473, normalized size = 2.56 \[ \frac{2 \,{\left (51051 \,{\left (e x + d\right )}^{\frac{19}{2}} b^{6} - 342342 \,{\left (b^{6} d - a b^{5} e\right )}{\left (e x + d\right )}^{\frac{17}{2}} + 969969 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )}{\left (e x + d\right )}^{\frac{15}{2}} - 1492260 \,{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 1322685 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 646646 \,{\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 138567 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{969969 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/969969*(51051*(e*x + d)^(19/2)*b^6 - 342342*(b^6*d - a*b^5*e)*(e*x + d)^(17/2)
+ 969969*(b^6*d^2 - 2*a*b^5*d*e + a^2*b^4*e^2)*(e*x + d)^(15/2) - 1492260*(b^6*
d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*(e*x + d)^(13/2) + 1322685*
(b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*(e
*x + d)^(11/2) - 646646*(b^6*d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b
^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5)*(e*x + d)^(9/2) + 138567*(b^6*d^6 - 6*
a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a
^5*b*d*e^5 + a^6*e^6)*(e*x + d)^(7/2))/e^7

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Fricas [A]  time = 0.208747, size = 857, normalized size = 4.63 \[ \frac{2 \,{\left (51051 \, b^{6} e^{9} x^{9} + 1024 \, b^{6} d^{9} - 9728 \, a b^{5} d^{8} e + 41344 \, a^{2} b^{4} d^{7} e^{2} - 103360 \, a^{3} b^{3} d^{6} e^{3} + 167960 \, a^{4} b^{2} d^{5} e^{4} - 184756 \, a^{5} b d^{4} e^{5} + 138567 \, a^{6} d^{3} e^{6} + 9009 \,{\left (13 \, b^{6} d e^{8} + 38 \, a b^{5} e^{9}\right )} x^{8} + 3003 \,{\left (23 \, b^{6} d^{2} e^{7} + 266 \, a b^{5} d e^{8} + 323 \, a^{2} b^{4} e^{9}\right )} x^{7} + 231 \,{\left (b^{6} d^{3} e^{6} + 2090 \, a b^{5} d^{2} e^{7} + 10013 \, a^{2} b^{4} d e^{8} + 6460 \, a^{3} b^{3} e^{9}\right )} x^{6} - 63 \,{\left (4 \, b^{6} d^{4} e^{5} - 38 \, a b^{5} d^{3} e^{6} - 22933 \, a^{2} b^{4} d^{2} e^{7} - 58140 \, a^{3} b^{3} d e^{8} - 20995 \, a^{4} b^{2} e^{9}\right )} x^{5} + 7 \,{\left (40 \, b^{6} d^{5} e^{4} - 380 \, a b^{5} d^{4} e^{5} + 1615 \, a^{2} b^{4} d^{3} e^{6} + 342380 \, a^{3} b^{3} d^{2} e^{7} + 482885 \, a^{4} b^{2} d e^{8} + 92378 \, a^{5} b e^{9}\right )} x^{4} -{\left (320 \, b^{6} d^{6} e^{3} - 3040 \, a b^{5} d^{5} e^{4} + 12920 \, a^{2} b^{4} d^{4} e^{5} - 32300 \, a^{3} b^{3} d^{3} e^{6} - 2372435 \, a^{4} b^{2} d^{2} e^{7} - 1755182 \, a^{5} b d e^{8} - 138567 \, a^{6} e^{9}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{7} e^{2} - 1216 \, a b^{5} d^{6} e^{3} + 5168 \, a^{2} b^{4} d^{5} e^{4} - 12920 \, a^{3} b^{3} d^{4} e^{5} + 20995 \, a^{4} b^{2} d^{3} e^{6} + 461890 \, a^{5} b d^{2} e^{7} + 138567 \, a^{6} d e^{8}\right )} x^{2} -{\left (512 \, b^{6} d^{8} e - 4864 \, a b^{5} d^{7} e^{2} + 20672 \, a^{2} b^{4} d^{6} e^{3} - 51680 \, a^{3} b^{3} d^{5} e^{4} + 83980 \, a^{4} b^{2} d^{4} e^{5} - 92378 \, a^{5} b d^{3} e^{6} - 415701 \, a^{6} d^{2} e^{7}\right )} x\right )} \sqrt{e x + d}}{969969 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(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 [A]  time = 18.1829, size = 1671, normalized size = 9.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**6*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4
*a**6*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 2*a**6*(d**2*(d + e*x)*
*(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e + 12*a**5*b*d**2*(-d*(
d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 24*a**5*b*d*(d**2*(d + e*x)**(3/2
)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 12*a**5*b*(-d**3*(d +
e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**
(9/2)/9)/e**2 + 30*a**4*b**2*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2
)/5 + (d + e*x)**(7/2)/7)/e**3 + 60*a**4*b**2*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d*
*2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 30*a
**4*b**2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)
**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**3 + 40*a**3*b**3*d
**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)
/7 + (d + e*x)**(9/2)/9)/e**4 + 80*a**3*b**3*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3
*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d +
e*x)**(11/2)/11)/e**4 + 40*a**3*b**3*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)*
*(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)
**(11/2)/11 + (d + e*x)**(13/2)/13)/e**4 + 30*a**2*b**4*d**2*(d**4*(d + e*x)**(3
/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(
9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 60*a**2*b**4*d*(-d**5*(d + e*x)**(3/2)/3 +
d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9
- 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 30*a**2*b**4*(d**6*(d
+ e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d
**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13
+ (d + e*x)**(15/2)/15)/e**5 + 12*a*b**5*d**2*(-d**5*(d + e*x)**(3/2)/3 + d**4*
(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d
*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**6 + 24*a*b**5*d*(d**6*(d + e*x)
**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d
+ e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d +
e*x)**(15/2)/15)/e**6 + 12*a*b**5*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)*
*(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e
*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d +
e*x)**(17/2)/17)/e**6 + 2*b**6*d**2*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)*
*(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d
+ e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**7 + 4*b*
*6*d*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(
7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d +
e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**7 + 2*b**6
*(d**8*(d + e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2)
- 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)*
*(13/2)/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)
**(19/2)/19)/e**7

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GIAC/XCAS [A]  time = 0.242442, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Done