3.1629 \(\int \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)

Optimal. Leaf size=187 \[ -\frac{12 b^5 (d+e x)^{13/2} (b d-a e)}{13 e^7}+\frac{30 b^4 (d+e x)^{11/2} (b d-a e)^2}{11 e^7}-\frac{40 b^3 (d+e x)^{9/2} (b d-a e)^3}{9 e^7}+\frac{30 b^2 (d+e x)^{7/2} (b d-a e)^4}{7 e^7}-\frac{12 b (d+e x)^{5/2} (b d-a e)^5}{5 e^7}+\frac{2 (d+e x)^{3/2} (b d-a e)^6}{3 e^7}+\frac{2 b^6 (d+e x)^{15/2}}{15 e^7} \]

[Out]

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

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Rubi [A]  time = 0.175293, antiderivative size = 187, 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 \[ -\frac{12 b^5 (d+e x)^{13/2} (b d-a e)}{13 e^7}+\frac{30 b^4 (d+e x)^{11/2} (b d-a e)^2}{11 e^7}-\frac{40 b^3 (d+e x)^{9/2} (b d-a e)^3}{9 e^7}+\frac{30 b^2 (d+e x)^{7/2} (b d-a e)^4}{7 e^7}-\frac{12 b (d+e x)^{5/2} (b d-a e)^5}{5 e^7}+\frac{2 (d+e x)^{3/2} (b d-a e)^6}{3 e^7}+\frac{2 b^6 (d+e x)^{15/2}}{15 e^7} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 79.3353, size = 173, normalized size = 0.93 \[ \frac{2 b^{6} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{7}} + \frac{12 b^{5} \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )}{13 e^{7}} + \frac{30 b^{4} \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{2}}{11 e^{7}} + \frac{40 b^{3} \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{3}}{9 e^{7}} + \frac{30 b^{2} \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{4}}{7 e^{7}} + \frac{12 b \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{5}}{5 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{6}}{3 e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**3*(e*x+d)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.223663, size = 291, normalized size = 1.56 \[ \frac{2 (d+e x)^{3/2} \left (15015 a^6 e^6+18018 a^5 b e^5 (3 e x-2 d)+6435 a^4 b^2 e^4 \left (8 d^2-12 d e x+15 e^2 x^2\right )+2860 a^3 b^3 e^3 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+195 a^2 b^4 e^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+30 a b^5 e \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )+b^6 \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )}{45045 e^7} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(3/2)*(15015*a^6*e^6 + 18018*a^5*b*e^5*(-2*d + 3*e*x) + 6435*a^4*b^
2*e^4*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 2860*a^3*b^3*e^3*(-16*d^3 + 24*d^2*e*x -
30*d*e^2*x^2 + 35*e^3*x^3) + 195*a^2*b^4*e^2*(128*d^4 - 192*d^3*e*x + 240*d^2*e
^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + 30*a*b^5*e*(-256*d^5 + 384*d^4*e*x - 480
*d^3*e^2*x^2 + 560*d^2*e^3*x^3 - 630*d*e^4*x^4 + 693*e^5*x^5) + b^6*(1024*d^6 -
1536*d^5*e*x + 1920*d^4*e^2*x^2 - 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^4 - 2772*d*e
^5*x^5 + 3003*e^6*x^6)))/(45045*e^7)

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Maple [B]  time = 0.013, size = 377, normalized size = 2. \[{\frac{6006\,{x}^{6}{b}^{6}{e}^{6}+41580\,{x}^{5}a{b}^{5}{e}^{6}-5544\,{x}^{5}{b}^{6}d{e}^{5}+122850\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-37800\,{x}^{4}a{b}^{5}d{e}^{5}+5040\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+200200\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-109200\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+33600\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-4480\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+193050\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-171600\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+93600\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-28800\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+3840\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+108108\,x{a}^{5}b{e}^{6}-154440\,x{a}^{4}{b}^{2}d{e}^{5}+137280\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-74880\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+23040\,xa{b}^{5}{d}^{4}{e}^{2}-3072\,x{b}^{6}{d}^{5}e+30030\,{a}^{6}{e}^{6}-72072\,{a}^{5}bd{e}^{5}+102960\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}-91520\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+49920\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-15360\,{d}^{5}a{b}^{5}e+2048\,{b}^{6}{d}^{6}}{45045\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^3*(e*x+d)^(1/2),x)

[Out]

2/45045*(e*x+d)^(3/2)*(3003*b^6*e^6*x^6+20790*a*b^5*e^6*x^5-2772*b^6*d*e^5*x^5+6
1425*a^2*b^4*e^6*x^4-18900*a*b^5*d*e^5*x^4+2520*b^6*d^2*e^4*x^4+100100*a^3*b^3*e
^6*x^3-54600*a^2*b^4*d*e^5*x^3+16800*a*b^5*d^2*e^4*x^3-2240*b^6*d^3*e^3*x^3+9652
5*a^4*b^2*e^6*x^2-85800*a^3*b^3*d*e^5*x^2+46800*a^2*b^4*d^2*e^4*x^2-14400*a*b^5*
d^3*e^3*x^2+1920*b^6*d^4*e^2*x^2+54054*a^5*b*e^6*x-77220*a^4*b^2*d*e^5*x+68640*a
^3*b^3*d^2*e^4*x-37440*a^2*b^4*d^3*e^3*x+11520*a*b^5*d^4*e^2*x-1536*b^6*d^5*e*x+
15015*a^6*e^6-36036*a^5*b*d*e^5+51480*a^4*b^2*d^2*e^4-45760*a^3*b^3*d^3*e^3+2496
0*a^2*b^4*d^4*e^2-7680*a*b^5*d^5*e+1024*b^6*d^6)/e^7

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Maxima [A]  time = 0.740617, size = 473, normalized size = 2.53 \[ \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} b^{6} - 20790 \,{\left (b^{6} d - a b^{5} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 61425 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 100100 \,{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 96525 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 54054 \,{\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 15015 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{45045 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(3003*(e*x + d)^(15/2)*b^6 - 20790*(b^6*d - a*b^5*e)*(e*x + d)^(13/2) +
61425*(b^6*d^2 - 2*a*b^5*d*e + a^2*b^4*e^2)*(e*x + d)^(11/2) - 100100*(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)^(9/2) + 96525*(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)^
(7/2) - 54054*(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)^(5/2) + 15015*(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)^(3/2))/e^7

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Fricas [A]  time = 0.207802, size = 603, normalized size = 3.22 \[ \frac{2 \,{\left (3003 \, b^{6} e^{7} x^{7} + 1024 \, b^{6} d^{7} - 7680 \, a b^{5} d^{6} e + 24960 \, a^{2} b^{4} d^{5} e^{2} - 45760 \, a^{3} b^{3} d^{4} e^{3} + 51480 \, a^{4} b^{2} d^{3} e^{4} - 36036 \, a^{5} b d^{2} e^{5} + 15015 \, a^{6} d e^{6} + 231 \,{\left (b^{6} d e^{6} + 90 \, a b^{5} e^{7}\right )} x^{6} - 63 \,{\left (4 \, b^{6} d^{2} e^{5} - 30 \, a b^{5} d e^{6} - 975 \, a^{2} b^{4} e^{7}\right )} x^{5} + 35 \,{\left (8 \, b^{6} d^{3} e^{4} - 60 \, a b^{5} d^{2} e^{5} + 195 \, a^{2} b^{4} d e^{6} + 2860 \, a^{3} b^{3} e^{7}\right )} x^{4} - 5 \,{\left (64 \, b^{6} d^{4} e^{3} - 480 \, a b^{5} d^{3} e^{4} + 1560 \, a^{2} b^{4} d^{2} e^{5} - 2860 \, a^{3} b^{3} d e^{6} - 19305 \, a^{4} b^{2} e^{7}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{5} e^{2} - 960 \, a b^{5} d^{4} e^{3} + 3120 \, a^{2} b^{4} d^{3} e^{4} - 5720 \, a^{3} b^{3} d^{2} e^{5} + 6435 \, a^{4} b^{2} d e^{6} + 18018 \, a^{5} b e^{7}\right )} x^{2} -{\left (512 \, b^{6} d^{6} e - 3840 \, a b^{5} d^{5} e^{2} + 12480 \, a^{2} b^{4} d^{4} e^{3} - 22880 \, a^{3} b^{3} d^{3} e^{4} + 25740 \, a^{4} b^{2} d^{2} e^{5} - 18018 \, a^{5} b d e^{6} - 15015 \, a^{6} e^{7}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(3003*b^6*e^7*x^7 + 1024*b^6*d^7 - 7680*a*b^5*d^6*e + 24960*a^2*b^4*d^5*
e^2 - 45760*a^3*b^3*d^4*e^3 + 51480*a^4*b^2*d^3*e^4 - 36036*a^5*b*d^2*e^5 + 1501
5*a^6*d*e^6 + 231*(b^6*d*e^6 + 90*a*b^5*e^7)*x^6 - 63*(4*b^6*d^2*e^5 - 30*a*b^5*
d*e^6 - 975*a^2*b^4*e^7)*x^5 + 35*(8*b^6*d^3*e^4 - 60*a*b^5*d^2*e^5 + 195*a^2*b^
4*d*e^6 + 2860*a^3*b^3*e^7)*x^4 - 5*(64*b^6*d^4*e^3 - 480*a*b^5*d^3*e^4 + 1560*a
^2*b^4*d^2*e^5 - 2860*a^3*b^3*d*e^6 - 19305*a^4*b^2*e^7)*x^3 + 3*(128*b^6*d^5*e^
2 - 960*a*b^5*d^4*e^3 + 3120*a^2*b^4*d^3*e^4 - 5720*a^3*b^3*d^2*e^5 + 6435*a^4*b
^2*d*e^6 + 18018*a^5*b*e^7)*x^2 - (512*b^6*d^6*e - 3840*a*b^5*d^5*e^2 + 12480*a^
2*b^4*d^4*e^3 - 22880*a^3*b^3*d^3*e^4 + 25740*a^4*b^2*d^2*e^5 - 18018*a^5*b*d*e^
6 - 15015*a^6*e^7)*x)*sqrt(e*x + d)/e^7

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Sympy [A]  time = 4.62047, size = 422, normalized size = 2.26 \[ \frac{2 \left (\frac{b^{6} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{6}} + \frac{\left (d + e x\right )^{\frac{13}{2}} \left (6 a b^{5} e - 6 b^{6} d\right )}{13 e^{6}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (15 a^{2} b^{4} e^{2} - 30 a b^{5} d e + 15 b^{6} d^{2}\right )}{11 e^{6}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (20 a^{3} b^{3} e^{3} - 60 a^{2} b^{4} d e^{2} + 60 a b^{5} d^{2} e - 20 b^{6} d^{3}\right )}{9 e^{6}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (15 a^{4} b^{2} e^{4} - 60 a^{3} b^{3} d e^{3} + 90 a^{2} b^{4} d^{2} e^{2} - 60 a b^{5} d^{3} e + 15 b^{6} d^{4}\right )}{7 e^{6}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (6 a^{5} b e^{5} - 30 a^{4} b^{2} d e^{4} + 60 a^{3} b^{3} d^{2} e^{3} - 60 a^{2} b^{4} d^{3} e^{2} + 30 a b^{5} d^{4} e - 6 b^{6} d^{5}\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{6} e^{6} - 6 a^{5} b d e^{5} + 15 a^{4} b^{2} d^{2} e^{4} - 20 a^{3} b^{3} d^{3} e^{3} + 15 a^{2} b^{4} d^{4} e^{2} - 6 a b^{5} d^{5} e + b^{6} d^{6}\right )}{3 e^{6}}\right )}{e} \]

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)**(1/2),x)

[Out]

2*(b**6*(d + e*x)**(15/2)/(15*e**6) + (d + e*x)**(13/2)*(6*a*b**5*e - 6*b**6*d)/
(13*e**6) + (d + e*x)**(11/2)*(15*a**2*b**4*e**2 - 30*a*b**5*d*e + 15*b**6*d**2)
/(11*e**6) + (d + e*x)**(9/2)*(20*a**3*b**3*e**3 - 60*a**2*b**4*d*e**2 + 60*a*b*
*5*d**2*e - 20*b**6*d**3)/(9*e**6) + (d + e*x)**(7/2)*(15*a**4*b**2*e**4 - 60*a*
*3*b**3*d*e**3 + 90*a**2*b**4*d**2*e**2 - 60*a*b**5*d**3*e + 15*b**6*d**4)/(7*e*
*6) + (d + e*x)**(5/2)*(6*a**5*b*e**5 - 30*a**4*b**2*d*e**4 + 60*a**3*b**3*d**2*
e**3 - 60*a**2*b**4*d**3*e**2 + 30*a*b**5*d**4*e - 6*b**6*d**5)/(5*e**6) + (d +
e*x)**(3/2)*(a**6*e**6 - 6*a**5*b*d*e**5 + 15*a**4*b**2*d**2*e**4 - 20*a**3*b**3
*d**3*e**3 + 15*a**2*b**4*d**4*e**2 - 6*a*b**5*d**5*e + b**6*d**6)/(3*e**6))/e

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GIAC/XCAS [A]  time = 0.21965, size = 603, normalized size = 3.22 \[ \frac{2}{45045} \,{\left (18018 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{5} b e^{\left (-1\right )} + 6435 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} a^{4} b^{2} e^{\left (-14\right )} + 2860 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} a^{3} b^{3} e^{\left (-27\right )} + 195 \,{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{40} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{40} + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{40} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{40} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{40}\right )} a^{2} b^{4} e^{\left (-44\right )} + 30 \,{\left (693 \,{\left (x e + d\right )}^{\frac{13}{2}} e^{60} - 4095 \,{\left (x e + d\right )}^{\frac{11}{2}} d e^{60} + 10010 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} e^{60} - 12870 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} e^{60} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} e^{60} - 3003 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5} e^{60}\right )} a b^{5} e^{\left (-65\right )} +{\left (3003 \,{\left (x e + d\right )}^{\frac{15}{2}} e^{84} - 20790 \,{\left (x e + d\right )}^{\frac{13}{2}} d e^{84} + 61425 \,{\left (x e + d\right )}^{\frac{11}{2}} d^{2} e^{84} - 100100 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{3} e^{84} + 96525 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{4} e^{84} - 54054 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{5} e^{84} + 15015 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{6} e^{84}\right )} b^{6} e^{\left (-90\right )} + 15015 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{6}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(18018*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^5*b*e^(-1) + 6435*(15
*(x*e + d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e + d)^(3/2)*d^2*e^12)
*a^4*b^2*e^(-14) + 2860*(35*(x*e + d)^(9/2)*e^24 - 135*(x*e + d)^(7/2)*d*e^24 +
189*(x*e + d)^(5/2)*d^2*e^24 - 105*(x*e + d)^(3/2)*d^3*e^24)*a^3*b^3*e^(-27) + 1
95*(315*(x*e + d)^(11/2)*e^40 - 1540*(x*e + d)^(9/2)*d*e^40 + 2970*(x*e + d)^(7/
2)*d^2*e^40 - 2772*(x*e + d)^(5/2)*d^3*e^40 + 1155*(x*e + d)^(3/2)*d^4*e^40)*a^2
*b^4*e^(-44) + 30*(693*(x*e + d)^(13/2)*e^60 - 4095*(x*e + d)^(11/2)*d*e^60 + 10
010*(x*e + d)^(9/2)*d^2*e^60 - 12870*(x*e + d)^(7/2)*d^3*e^60 + 9009*(x*e + d)^(
5/2)*d^4*e^60 - 3003*(x*e + d)^(3/2)*d^5*e^60)*a*b^5*e^(-65) + (3003*(x*e + d)^(
15/2)*e^84 - 20790*(x*e + d)^(13/2)*d*e^84 + 61425*(x*e + d)^(11/2)*d^2*e^84 - 1
00100*(x*e + d)^(9/2)*d^3*e^84 + 96525*(x*e + d)^(7/2)*d^4*e^84 - 54054*(x*e + d
)^(5/2)*d^5*e^84 + 15015*(x*e + d)^(3/2)*d^6*e^84)*b^6*e^(-90) + 15015*(x*e + d)
^(3/2)*a^6)*e^(-1)