3.353 \(\int (d+e x)^{7/2} \left (b x+c x^2\right )^3 \, dx\)

Optimal. Leaf size=248 \[ \frac{6 c (d+e x)^{17/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{17 e^7}-\frac{2 (d+e x)^{15/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{15 e^7}+\frac{6 d (d+e x)^{13/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{13 e^7}-\frac{6 c^2 (d+e x)^{19/2} (2 c d-b e)}{19 e^7}+\frac{2 d^3 (d+e x)^{9/2} (c d-b e)^3}{9 e^7}-\frac{6 d^2 (d+e x)^{11/2} (c d-b e)^2 (2 c d-b e)}{11 e^7}+\frac{2 c^3 (d+e x)^{21/2}}{21 e^7} \]

[Out]

(2*d^3*(c*d - b*e)^3*(d + e*x)^(9/2))/(9*e^7) - (6*d^2*(c*d - b*e)^2*(2*c*d - b*
e)*(d + e*x)^(11/2))/(11*e^7) + (6*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^
2)*(d + e*x)^(13/2))/(13*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*
e^2)*(d + e*x)^(15/2))/(15*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*
x)^(17/2))/(17*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(19/2))/(19*e^7) + (2*c^3*(
d + e*x)^(21/2))/(21*e^7)

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Rubi [A]  time = 0.363722, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{6 c (d+e x)^{17/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{17 e^7}-\frac{2 (d+e x)^{15/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{15 e^7}+\frac{6 d (d+e x)^{13/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{13 e^7}-\frac{6 c^2 (d+e x)^{19/2} (2 c d-b e)}{19 e^7}+\frac{2 d^3 (d+e x)^{9/2} (c d-b e)^3}{9 e^7}-\frac{6 d^2 (d+e x)^{11/2} (c d-b e)^2 (2 c d-b e)}{11 e^7}+\frac{2 c^3 (d+e x)^{21/2}}{21 e^7} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(7/2)*(b*x + c*x^2)^3,x]

[Out]

(2*d^3*(c*d - b*e)^3*(d + e*x)^(9/2))/(9*e^7) - (6*d^2*(c*d - b*e)^2*(2*c*d - b*
e)*(d + e*x)^(11/2))/(11*e^7) + (6*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^
2)*(d + e*x)^(13/2))/(13*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*
e^2)*(d + e*x)^(15/2))/(15*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*
x)^(17/2))/(17*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(19/2))/(19*e^7) + (2*c^3*(
d + e*x)^(21/2))/(21*e^7)

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Rubi in Sympy [A]  time = 59.6625, size = 243, normalized size = 0.98 \[ \frac{2 c^{3} \left (d + e x\right )^{\frac{21}{2}}}{21 e^{7}} + \frac{6 c^{2} \left (d + e x\right )^{\frac{19}{2}} \left (b e - 2 c d\right )}{19 e^{7}} + \frac{6 c \left (d + e x\right )^{\frac{17}{2}} \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{17 e^{7}} - \frac{2 d^{3} \left (d + e x\right )^{\frac{9}{2}} \left (b e - c d\right )^{3}}{9 e^{7}} + \frac{6 d^{2} \left (d + e x\right )^{\frac{11}{2}} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2}}{11 e^{7}} - \frac{6 d \left (d + e x\right )^{\frac{13}{2}} \left (b e - c d\right ) \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{13 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{15}{2}} \left (b e - 2 c d\right ) \left (b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{15 e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(7/2)*(c*x**2+b*x)**3,x)

[Out]

2*c**3*(d + e*x)**(21/2)/(21*e**7) + 6*c**2*(d + e*x)**(19/2)*(b*e - 2*c*d)/(19*
e**7) + 6*c*(d + e*x)**(17/2)*(b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(17*e**7) -
2*d**3*(d + e*x)**(9/2)*(b*e - c*d)**3/(9*e**7) + 6*d**2*(d + e*x)**(11/2)*(b*e
- 2*c*d)*(b*e - c*d)**2/(11*e**7) - 6*d*(d + e*x)**(13/2)*(b*e - c*d)*(b**2*e**2
- 5*b*c*d*e + 5*c**2*d**2)/(13*e**7) + 2*(d + e*x)**(15/2)*(b*e - 2*c*d)*(b**2*
e**2 - 10*b*c*d*e + 10*c**2*d**2)/(15*e**7)

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Mathematica [A]  time = 0.235877, size = 232, normalized size = 0.94 \[ \frac{2 (d+e x)^{9/2} \left (2261 b^3 e^3 \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+399 b^2 c e^2 \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )+105 b c^2 e \left (-256 d^5+1152 d^4 e x-3168 d^3 e^2 x^2+6864 d^2 e^3 x^3-12870 d e^4 x^4+21879 e^5 x^5\right )+5 c^3 \left (1024 d^6-4608 d^5 e x+12672 d^4 e^2 x^2-27456 d^3 e^3 x^3+51480 d^2 e^4 x^4-87516 d e^5 x^5+138567 e^6 x^6\right )\right )}{14549535 e^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(7/2)*(b*x + c*x^2)^3,x]

[Out]

(2*(d + e*x)^(9/2)*(2261*b^3*e^3*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3
*x^3) + 399*b^2*c*e^2*(128*d^4 - 576*d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e^3*x^3
+ 6435*e^4*x^4) + 105*b*c^2*e*(-256*d^5 + 1152*d^4*e*x - 3168*d^3*e^2*x^2 + 686
4*d^2*e^3*x^3 - 12870*d*e^4*x^4 + 21879*e^5*x^5) + 5*c^3*(1024*d^6 - 4608*d^5*e*
x + 12672*d^4*e^2*x^2 - 27456*d^3*e^3*x^3 + 51480*d^2*e^4*x^4 - 87516*d*e^5*x^5
+ 138567*e^6*x^6)))/(14549535*e^7)

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Maple [A]  time = 0.011, size = 286, normalized size = 1.2 \[ -{\frac{-1385670\,{c}^{3}{x}^{6}{e}^{6}-4594590\,b{c}^{2}{e}^{6}{x}^{5}+875160\,{c}^{3}d{e}^{5}{x}^{5}-5135130\,{b}^{2}c{e}^{6}{x}^{4}+2702700\,b{c}^{2}d{e}^{5}{x}^{4}-514800\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-1939938\,{b}^{3}{e}^{6}{x}^{3}+2738736\,{b}^{2}cd{e}^{5}{x}^{3}-1441440\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}+274560\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+895356\,{b}^{3}d{e}^{5}{x}^{2}-1264032\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}+665280\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}-126720\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-325584\,{b}^{3}{d}^{2}{e}^{4}x+459648\,{b}^{2}c{d}^{3}{e}^{3}x-241920\,b{c}^{2}{d}^{4}{e}^{2}x+46080\,{c}^{3}{d}^{5}ex+72352\,{b}^{3}{d}^{3}{e}^{3}-102144\,{b}^{2}c{d}^{4}{e}^{2}+53760\,b{c}^{2}{d}^{5}e-10240\,{c}^{3}{d}^{6}}{14549535\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/14549535*(e*x+d)^(9/2)*(-692835*c^3*e^6*x^6-2297295*b*c^2*e^6*x^5+437580*c^3*
d*e^5*x^5-2567565*b^2*c*e^6*x^4+1351350*b*c^2*d*e^5*x^4-257400*c^3*d^2*e^4*x^4-9
69969*b^3*e^6*x^3+1369368*b^2*c*d*e^5*x^3-720720*b*c^2*d^2*e^4*x^3+137280*c^3*d^
3*e^3*x^3+447678*b^3*d*e^5*x^2-632016*b^2*c*d^2*e^4*x^2+332640*b*c^2*d^3*e^3*x^2
-63360*c^3*d^4*e^2*x^2-162792*b^3*d^2*e^4*x+229824*b^2*c*d^3*e^3*x-120960*b*c^2*
d^4*e^2*x+23040*c^3*d^5*e*x+36176*b^3*d^3*e^3-51072*b^2*c*d^4*e^2+26880*b*c^2*d^
5*e-5120*c^3*d^6)/e^7

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Maxima [A]  time = 0.702713, size = 366, normalized size = 1.48 \[ \frac{2 \,{\left (692835 \,{\left (e x + d\right )}^{\frac{21}{2}} c^{3} - 2297295 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{19}{2}} + 2567565 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )}{\left (e x + d\right )}^{\frac{17}{2}} - 969969 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 3357585 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 3968055 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 1616615 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{14549535 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

2/14549535*(692835*(e*x + d)^(21/2)*c^3 - 2297295*(2*c^3*d - b*c^2*e)*(e*x + d)^
(19/2) + 2567565*(5*c^3*d^2 - 5*b*c^2*d*e + b^2*c*e^2)*(e*x + d)^(17/2) - 969969
*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*(e*x + d)^(15/2) + 335
7585*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*(e*x + d)^(13/2)
- 3968055*(2*c^3*d^5 - 5*b*c^2*d^4*e + 4*b^2*c*d^3*e^2 - b^3*d^2*e^3)*(e*x + d)
^(11/2) + 1616615*(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*(e*x
+ d)^(9/2))/e^7

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Fricas [A]  time = 0.216919, size = 645, normalized size = 2.6 \[ \frac{2 \,{\left (692835 \, c^{3} e^{10} x^{10} + 5120 \, c^{3} d^{10} - 26880 \, b c^{2} d^{9} e + 51072 \, b^{2} c d^{8} e^{2} - 36176 \, b^{3} d^{7} e^{3} + 36465 \,{\left (64 \, c^{3} d e^{9} + 63 \, b c^{2} e^{10}\right )} x^{9} + 19305 \,{\left (138 \, c^{3} d^{2} e^{8} + 406 \, b c^{2} d e^{9} + 133 \, b^{2} c e^{10}\right )} x^{8} + 429 \,{\left (2420 \, c^{3} d^{3} e^{7} + 21210 \, b c^{2} d^{2} e^{8} + 20748 \, b^{2} c d e^{9} + 2261 \, b^{3} e^{10}\right )} x^{7} + 231 \,{\left (5 \, c^{3} d^{4} e^{6} + 15720 \, b c^{2} d^{3} e^{7} + 45714 \, b^{2} c d^{2} e^{8} + 14858 \, b^{3} d e^{9}\right )} x^{6} - 63 \,{\left (20 \, c^{3} d^{5} e^{5} - 105 \, b c^{2} d^{4} e^{6} - 69084 \, b^{2} c d^{3} e^{7} - 66538 \, b^{3} d^{2} e^{8}\right )} x^{5} + 35 \,{\left (40 \, c^{3} d^{6} e^{4} - 210 \, b c^{2} d^{5} e^{5} + 399 \, b^{2} c d^{4} e^{6} + 51680 \, b^{3} d^{3} e^{7}\right )} x^{4} - 5 \,{\left (320 \, c^{3} d^{7} e^{3} - 1680 \, b c^{2} d^{6} e^{4} + 3192 \, b^{2} c d^{5} e^{5} - 2261 \, b^{3} d^{4} e^{6}\right )} x^{3} + 6 \,{\left (320 \, c^{3} d^{8} e^{2} - 1680 \, b c^{2} d^{7} e^{3} + 3192 \, b^{2} c d^{6} e^{4} - 2261 \, b^{3} d^{5} e^{5}\right )} x^{2} - 8 \,{\left (320 \, c^{3} d^{9} e - 1680 \, b c^{2} d^{8} e^{2} + 3192 \, b^{2} c d^{7} e^{3} - 2261 \, b^{3} d^{6} e^{4}\right )} x\right )} \sqrt{e x + d}}{14549535 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

2/14549535*(692835*c^3*e^10*x^10 + 5120*c^3*d^10 - 26880*b*c^2*d^9*e + 51072*b^2
*c*d^8*e^2 - 36176*b^3*d^7*e^3 + 36465*(64*c^3*d*e^9 + 63*b*c^2*e^10)*x^9 + 1930
5*(138*c^3*d^2*e^8 + 406*b*c^2*d*e^9 + 133*b^2*c*e^10)*x^8 + 429*(2420*c^3*d^3*e
^7 + 21210*b*c^2*d^2*e^8 + 20748*b^2*c*d*e^9 + 2261*b^3*e^10)*x^7 + 231*(5*c^3*d
^4*e^6 + 15720*b*c^2*d^3*e^7 + 45714*b^2*c*d^2*e^8 + 14858*b^3*d*e^9)*x^6 - 63*(
20*c^3*d^5*e^5 - 105*b*c^2*d^4*e^6 - 69084*b^2*c*d^3*e^7 - 66538*b^3*d^2*e^8)*x^
5 + 35*(40*c^3*d^6*e^4 - 210*b*c^2*d^5*e^5 + 399*b^2*c*d^4*e^6 + 51680*b^3*d^3*e
^7)*x^4 - 5*(320*c^3*d^7*e^3 - 1680*b*c^2*d^6*e^4 + 3192*b^2*c*d^5*e^5 - 2261*b^
3*d^4*e^6)*x^3 + 6*(320*c^3*d^8*e^2 - 1680*b*c^2*d^7*e^3 + 3192*b^2*c*d^6*e^4 -
2261*b^3*d^5*e^5)*x^2 - 8*(320*c^3*d^9*e - 1680*b*c^2*d^8*e^2 + 3192*b^2*c*d^7*e
^3 - 2261*b^3*d^6*e^4)*x)*sqrt(e*x + d)/e^7

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Sympy [A]  time = 20.2947, size = 1741, normalized size = 7.02 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(7/2)*(c*x**2+b*x)**3,x)

[Out]

2*b**3*d**3*(-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 + 6*b**3*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**4 + 6*b**3*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**4 + 2*b**3*(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**4 + 6*b**2*c*d**3*(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
+ 18*b**2*c*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**5 + 18*b**2*c*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**5 + 6*b**
2*c*(-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**5 + 6*b*c**
2*d**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**6 + 18*b*c**2*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**6 + 18*b*c**2*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**6 + 6*b*c**2*(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) - 5
6*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(1
3/2)/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**6 + 2*c**3*d**3*(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 + 6*c**3*d**2
*(-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 + 6*c**3*d*(
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 + 2*c**3*(-d**9*(d + e*x)**(3/2)/3 + 9*d**8*(d + e*x)**(5/2)/5 -
36*d**7*(d + e*x)**(7/2)/7 + 28*d**6*(d + e*x)**(9/2)/3 - 126*d**5*(d + e*x)**(
11/2)/11 + 126*d**4*(d + e*x)**(13/2)/13 - 28*d**3*(d + e*x)**(15/2)/5 + 36*d**2
*(d + e*x)**(17/2)/17 - 9*d*(d + e*x)**(19/2)/19 + (d + e*x)**(21/2)/21)/e**7

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GIAC/XCAS [A]  time = 0.239653, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(e*x + d)^(7/2),x, algorithm="giac")

[Out]

Done