3.2272 \(\int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx\)

Optimal. Leaf size=286 \[ \frac{6 c (d+e x)^{13/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{13 e^7}-\frac{2 (d+e x)^{11/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{11 e^7}+\frac{2 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac{6 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^7}+\frac{2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^3}{5 e^7}-\frac{2 c^2 (d+e x)^{15/2} (2 c d-b e)}{5 e^7}+\frac{2 c^3 (d+e x)^{17/2}}{17 e^7} \]

[Out]

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

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Rubi [A]  time = 0.424452, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{6 c (d+e x)^{13/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{13 e^7}-\frac{2 (d+e x)^{11/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{11 e^7}+\frac{2 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac{6 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^7}+\frac{2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^3}{5 e^7}-\frac{2 c^2 (d+e x)^{15/2} (2 c d-b e)}{5 e^7}+\frac{2 c^3 (d+e x)^{17/2}}{17 e^7} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 76.0032, size = 282, normalized size = 0.99 \[ \frac{2 c^{3} \left (d + e x\right )^{\frac{17}{2}}}{17 e^{7}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{15}{2}} \left (b e - 2 c d\right )}{5 e^{7}} + \frac{6 c \left (d + e x\right )^{\frac{13}{2}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{13 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{11}{2}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{11 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{3 e^{7}} + \frac{6 \left (d + e x\right )^{\frac{7}{2}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{7 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - b d e + c d^{2}\right )^{3}}{5 e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c**3*(d + e*x)**(17/2)/(17*e**7) + 2*c**2*(d + e*x)**(15/2)*(b*e - 2*c*d)/(5*e
**7) + 6*c*(d + e*x)**(13/2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(1
3*e**7) + 2*(d + e*x)**(11/2)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e
+ 10*c**2*d**2)/(11*e**7) + 2*(d + e*x)**(9/2)*(a*e**2 - b*d*e + c*d**2)*(a*c*e
**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(3*e**7) + 6*(d + e*x)**(7/2)*(b*e -
2*c*d)*(a*e**2 - b*d*e + c*d**2)**2/(7*e**7) + 2*(d + e*x)**(5/2)*(a*e**2 - b*d*
e + c*d**2)**3/(5*e**7)

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Mathematica [A]  time = 0.473981, size = 396, normalized size = 1.38 \[ \frac{2 (d+e x)^{5/2} \left (17 c e^2 \left (143 a^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+78 a b e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+3 b^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )+221 e^3 \left (231 a^3 e^3+99 a^2 b e^2 (5 e x-2 d)+11 a b^2 e \left (8 d^2-20 d e x+35 e^2 x^2\right )+b^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )-17 c^2 e \left (b \left (256 d^5-640 d^4 e x+1120 d^3 e^2 x^2-1680 d^2 e^3 x^3+2310 d e^4 x^4-3003 e^5 x^5\right )-3 a e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )+c^3 \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )\right )}{255255 e^7} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(5/2)*(c^3*(1024*d^6 - 2560*d^5*e*x + 4480*d^4*e^2*x^2 - 6720*d^3*e
^3*x^3 + 9240*d^2*e^4*x^4 - 12012*d*e^5*x^5 + 15015*e^6*x^6) + 221*e^3*(231*a^3*
e^3 + 99*a^2*b*e^2*(-2*d + 5*e*x) + 11*a*b^2*e*(8*d^2 - 20*d*e*x + 35*e^2*x^2) +
b^3*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3)) + 17*c*e^2*(143*a^2*e^
2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 78*a*b*e*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^
2 + 105*e^3*x^3) + 3*b^2*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^
3 + 1155*e^4*x^4)) - 17*c^2*e*(-3*a*e*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 -
840*d*e^3*x^3 + 1155*e^4*x^4) + b*(256*d^5 - 640*d^4*e*x + 1120*d^3*e^2*x^2 - 1
680*d^2*e^3*x^3 + 2310*d*e^4*x^4 - 3003*e^5*x^5))))/(255255*e^7)

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Maple [A]  time = 0.011, size = 495, normalized size = 1.7 \[{\frac{30030\,{c}^{3}{x}^{6}{e}^{6}+102102\,b{c}^{2}{e}^{6}{x}^{5}-24024\,{c}^{3}d{e}^{5}{x}^{5}+117810\,{x}^{4}a{c}^{2}{e}^{6}+117810\,{b}^{2}c{e}^{6}{x}^{4}-78540\,b{c}^{2}d{e}^{5}{x}^{4}+18480\,{x}^{4}{c}^{3}{d}^{2}{e}^{4}+278460\,abc{e}^{6}{x}^{3}-85680\,{x}^{3}a{c}^{2}d{e}^{5}+46410\,{b}^{3}{e}^{6}{x}^{3}-85680\,{b}^{2}cd{e}^{5}{x}^{3}+57120\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}-13440\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+170170\,{x}^{2}{a}^{2}c{e}^{6}+170170\,a{b}^{2}{e}^{6}{x}^{2}-185640\,abcd{e}^{5}{x}^{2}+57120\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-30940\,{b}^{3}d{e}^{5}{x}^{2}+57120\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}-38080\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}+8960\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}+218790\,{a}^{2}b{e}^{6}x-97240\,x{a}^{2}cd{e}^{5}-97240\,a{b}^{2}d{e}^{5}x+106080\,abc{d}^{2}{e}^{4}x-32640\,xa{c}^{2}{d}^{3}{e}^{3}+17680\,{b}^{3}{d}^{2}{e}^{4}x-32640\,{b}^{2}c{d}^{3}{e}^{3}x+21760\,b{c}^{2}{d}^{4}{e}^{2}x-5120\,{c}^{3}{d}^{5}ex+102102\,{a}^{3}{e}^{6}-87516\,{a}^{2}bd{e}^{5}+38896\,{a}^{2}c{d}^{2}{e}^{4}+38896\,a{b}^{2}{d}^{2}{e}^{4}-42432\,abc{d}^{3}{e}^{3}+13056\,{c}^{2}{d}^{4}a{e}^{2}-7072\,{b}^{3}{d}^{3}{e}^{3}+13056\,{b}^{2}c{d}^{4}{e}^{2}-8704\,b{c}^{2}{d}^{5}e+2048\,{c}^{3}{d}^{6}}{255255\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/255255*(e*x+d)^(5/2)*(15015*c^3*e^6*x^6+51051*b*c^2*e^6*x^5-12012*c^3*d*e^5*x^
5+58905*a*c^2*e^6*x^4+58905*b^2*c*e^6*x^4-39270*b*c^2*d*e^5*x^4+9240*c^3*d^2*e^4
*x^4+139230*a*b*c*e^6*x^3-42840*a*c^2*d*e^5*x^3+23205*b^3*e^6*x^3-42840*b^2*c*d*
e^5*x^3+28560*b*c^2*d^2*e^4*x^3-6720*c^3*d^3*e^3*x^3+85085*a^2*c*e^6*x^2+85085*a
*b^2*e^6*x^2-92820*a*b*c*d*e^5*x^2+28560*a*c^2*d^2*e^4*x^2-15470*b^3*d*e^5*x^2+2
8560*b^2*c*d^2*e^4*x^2-19040*b*c^2*d^3*e^3*x^2+4480*c^3*d^4*e^2*x^2+109395*a^2*b
*e^6*x-48620*a^2*c*d*e^5*x-48620*a*b^2*d*e^5*x+53040*a*b*c*d^2*e^4*x-16320*a*c^2
*d^3*e^3*x+8840*b^3*d^2*e^4*x-16320*b^2*c*d^3*e^3*x+10880*b*c^2*d^4*e^2*x-2560*c
^3*d^5*e*x+51051*a^3*e^6-43758*a^2*b*d*e^5+19448*a^2*c*d^2*e^4+19448*a*b^2*d^2*e
^4-21216*a*b*c*d^3*e^3+6528*a*c^2*d^4*e^2-3536*b^3*d^3*e^3+6528*b^2*c*d^4*e^2-43
52*b*c^2*d^5*e+1024*c^3*d^6)/e^7

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Maxima [A]  time = 0.713216, size = 549, normalized size = 1.92 \[ \frac{2 \,{\left (15015 \,{\left (e x + d\right )}^{\frac{17}{2}} c^{3} - 51051 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 58905 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 23205 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 85085 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\right )} e^{4}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 109395 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 51051 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{255255 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/255255*(15015*(e*x + d)^(17/2)*c^3 - 51051*(2*c^3*d - b*c^2*e)*(e*x + d)^(15/2
) + 58905*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(13/2) - 232
05*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 - (b^3 + 6*a*b*c)*e^3
)*(e*x + d)^(11/2) + 85085*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*(b^2*c + a*c^2)*d^2*e
^2 - (b^3 + 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)*(e*x + d)^(9/2) - 109395*(2*c^
3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(b^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*
d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4)*(e*x + d)^(7/2) + 51051*(c^3*d^6 - 3*b*c^2*d^
5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*
e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*(e*x + d)^(5/2))/e^7

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Fricas [A]  time = 0.210606, size = 836, normalized size = 2.92 \[ \frac{2 \,{\left (15015 \, c^{3} e^{8} x^{8} + 1024 \, c^{3} d^{8} - 4352 \, b c^{2} d^{7} e - 43758 \, a^{2} b d^{3} e^{5} + 51051 \, a^{3} d^{2} e^{6} + 6528 \,{\left (b^{2} c + a c^{2}\right )} d^{6} e^{2} - 3536 \,{\left (b^{3} + 6 \, a b c\right )} d^{5} e^{3} + 19448 \,{\left (a b^{2} + a^{2} c\right )} d^{4} e^{4} + 3003 \,{\left (6 \, c^{3} d e^{7} + 17 \, b c^{2} e^{8}\right )} x^{7} + 231 \,{\left (c^{3} d^{2} e^{6} + 272 \, b c^{2} d e^{7} + 255 \,{\left (b^{2} c + a c^{2}\right )} e^{8}\right )} x^{6} - 21 \,{\left (12 \, c^{3} d^{3} e^{5} - 51 \, b c^{2} d^{2} e^{6} - 3570 \,{\left (b^{2} c + a c^{2}\right )} d e^{7} - 1105 \,{\left (b^{3} + 6 \, a b c\right )} e^{8}\right )} x^{5} + 35 \,{\left (8 \, c^{3} d^{4} e^{4} - 34 \, b c^{2} d^{3} e^{5} + 51 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{6} + 884 \,{\left (b^{3} + 6 \, a b c\right )} d e^{7} + 2431 \,{\left (a b^{2} + a^{2} c\right )} e^{8}\right )} x^{4} - 5 \,{\left (64 \, c^{3} d^{5} e^{3} - 272 \, b c^{2} d^{4} e^{4} - 21879 \, a^{2} b e^{8} + 408 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{5} - 221 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{6} - 24310 \,{\left (a b^{2} + a^{2} c\right )} d e^{7}\right )} x^{3} + 3 \,{\left (128 \, c^{3} d^{6} e^{2} - 544 \, b c^{2} d^{5} e^{3} + 58344 \, a^{2} b d e^{7} + 17017 \, a^{3} e^{8} + 816 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{4} - 442 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{5} + 2431 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{6}\right )} x^{2} -{\left (512 \, c^{3} d^{7} e - 2176 \, b c^{2} d^{6} e^{2} - 21879 \, a^{2} b d^{2} e^{6} - 102102 \, a^{3} d e^{7} + 3264 \,{\left (b^{2} c + a c^{2}\right )} d^{5} e^{3} - 1768 \,{\left (b^{3} + 6 \, a b c\right )} d^{4} e^{4} + 9724 \,{\left (a b^{2} + a^{2} c\right )} d^{3} e^{5}\right )} x\right )} \sqrt{e x + d}}{255255 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/255255*(15015*c^3*e^8*x^8 + 1024*c^3*d^8 - 4352*b*c^2*d^7*e - 43758*a^2*b*d^3*
e^5 + 51051*a^3*d^2*e^6 + 6528*(b^2*c + a*c^2)*d^6*e^2 - 3536*(b^3 + 6*a*b*c)*d^
5*e^3 + 19448*(a*b^2 + a^2*c)*d^4*e^4 + 3003*(6*c^3*d*e^7 + 17*b*c^2*e^8)*x^7 +
231*(c^3*d^2*e^6 + 272*b*c^2*d*e^7 + 255*(b^2*c + a*c^2)*e^8)*x^6 - 21*(12*c^3*d
^3*e^5 - 51*b*c^2*d^2*e^6 - 3570*(b^2*c + a*c^2)*d*e^7 - 1105*(b^3 + 6*a*b*c)*e^
8)*x^5 + 35*(8*c^3*d^4*e^4 - 34*b*c^2*d^3*e^5 + 51*(b^2*c + a*c^2)*d^2*e^6 + 884
*(b^3 + 6*a*b*c)*d*e^7 + 2431*(a*b^2 + a^2*c)*e^8)*x^4 - 5*(64*c^3*d^5*e^3 - 272
*b*c^2*d^4*e^4 - 21879*a^2*b*e^8 + 408*(b^2*c + a*c^2)*d^3*e^5 - 221*(b^3 + 6*a*
b*c)*d^2*e^6 - 24310*(a*b^2 + a^2*c)*d*e^7)*x^3 + 3*(128*c^3*d^6*e^2 - 544*b*c^2
*d^5*e^3 + 58344*a^2*b*d*e^7 + 17017*a^3*e^8 + 816*(b^2*c + a*c^2)*d^4*e^4 - 442
*(b^3 + 6*a*b*c)*d^3*e^5 + 2431*(a*b^2 + a^2*c)*d^2*e^6)*x^2 - (512*c^3*d^7*e -
2176*b*c^2*d^6*e^2 - 21879*a^2*b*d^2*e^6 - 102102*a^3*d*e^7 + 3264*(b^2*c + a*c^
2)*d^5*e^3 - 1768*(b^3 + 6*a*b*c)*d^4*e^4 + 9724*(a*b^2 + a^2*c)*d^3*e^5)*x)*sqr
t(e*x + d)/e^7

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Sympy [A]  time = 14.9902, size = 1411, normalized size = 4.93 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a*
*3*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 6*a**2*b*d*(-d*(d + e*x)**(3
/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 6*a**2*b*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d +
e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 6*a**2*c*d*(d**2*(d + e*x)**(3/2)/3 -
2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 6*a**2*c*(-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 + 6*a*b**2*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*
x)**(7/2)/7)/e**3 + 6*a*b**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**3 + 12*a*b*c*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**4 + 12*a*b*c*(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*a*c**2*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**5 + 6*a*c**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 + 2*b**3*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**4 + 2*b**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**4 + 6*b**2*c*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**5 + 6*b**2*c*(-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 + 6*b*c**2*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**6 + 6*b*c**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 + 2*c**3*d*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 1
5*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 + 2*c**3*(-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)/1
3 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**7

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GIAC/XCAS [A]  time = 0.231049, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done