3.1601 \(\int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=252 \[ \frac{8 c (d+e x)^{13/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{13 e^6}-\frac{2 (d+e x)^{11/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{11 e^6}+\frac{4 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{9 e^6}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^6}-\frac{2 c^2 (d+e x)^{15/2} (2 c d-b e)}{3 e^6}+\frac{4 c^3 (d+e x)^{17/2}}{17 e^6} \]

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(7/2))/(7*e^6) + (4*(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))/(9*
e^6) - (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^6) + (8*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(13
/2))/(13*e^6) - (2*c^2*(2*c*d - b*e)*(d + e*x)^(15/2))/(3*e^6) + (4*c^3*(d + e*x
)^(17/2))/(17*e^6)

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Rubi [A]  time = 0.402018, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{8 c (d+e x)^{13/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{13 e^6}-\frac{2 (d+e x)^{11/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{11 e^6}+\frac{4 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{9 e^6}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^6}-\frac{2 c^2 (d+e x)^{15/2} (2 c d-b e)}{3 e^6}+\frac{4 c^3 (d+e x)^{17/2}}{17 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(7/2))/(7*e^6) + (4*(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))/(9*
e^6) - (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^6) + (8*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(13
/2))/(13*e^6) - (2*c^2*(2*c*d - b*e)*(d + e*x)^(15/2))/(3*e^6) + (4*c^3*(d + e*x
)^(17/2))/(17*e^6)

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Rubi in Sympy [A]  time = 71.4298, size = 250, normalized size = 0.99 \[ \frac{4 c^{3} \left (d + e x\right )^{\frac{17}{2}}}{17 e^{6}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{15}{2}} \left (b e - 2 c d\right )}{3 e^{6}} + \frac{8 c \left (d + e x\right )^{\frac{13}{2}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{13 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{11}{2}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{11 e^{6}} + \frac{4 \left (d + e x\right )^{\frac{9}{2}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{9 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{7 e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*c**3*(d + e*x)**(17/2)/(17*e**6) + 2*c**2*(d + e*x)**(15/2)*(b*e - 2*c*d)/(3*e
**6) + 8*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**6) + 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**6) + 4*(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)/(9*e**6) + 2*(d + e*x)**(7/2)*(b*e -
2*c*d)*(a*e**2 - b*d*e + c*d**2)**2/(7*e**6)

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Mathematica [A]  time = 0.480673, size = 291, normalized size = 1.15 \[ \frac{2 (d+e x)^{7/2} \left (-34 c e^2 \left (143 a^2 e^2 (2 d-7 e x)-39 a b e \left (8 d^2-28 d e x+63 e^2 x^2\right )+6 b^2 \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )\right )+221 b e^3 \left (99 a^2 e^2+22 a b e (7 e x-2 d)+b^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )+17 c^2 e \left (12 a e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+b \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )-2 c^3 \left (256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right )\right )}{153153 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(7/2)*(-2*c^3*(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) + 221*b*e^3*(99*a^2*e^2 + 22*a*b*e*(-2*
d + 7*e*x) + b^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2)) - 34*c*e^2*(143*a^2*e^2*(2*d -
7*e*x) - 39*a*b*e*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 6*b^2*(16*d^3 - 56*d^2*e*x
+ 126*d*e^2*x^2 - 231*e^3*x^3)) + 17*c^2*e*(12*a*e*(-16*d^3 + 56*d^2*e*x - 126*d
*e^2*x^2 + 231*e^3*x^3) + b*(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))))/(153153*e^6)

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Maple [A]  time = 0.013, size = 359, normalized size = 1.4 \[{\frac{36036\,{c}^{3}{x}^{5}{e}^{5}+102102\,b{c}^{2}{e}^{5}{x}^{4}-24024\,{c}^{3}d{e}^{4}{x}^{4}+94248\,a{c}^{2}{e}^{5}{x}^{3}+94248\,{b}^{2}c{e}^{5}{x}^{3}-62832\,b{c}^{2}d{e}^{4}{x}^{3}+14784\,{c}^{3}{d}^{2}{e}^{3}{x}^{3}+167076\,abc{e}^{5}{x}^{2}-51408\,a{c}^{2}d{e}^{4}{x}^{2}+27846\,{b}^{3}{e}^{5}{x}^{2}-51408\,{b}^{2}cd{e}^{4}{x}^{2}+34272\,b{c}^{2}{d}^{2}{e}^{3}{x}^{2}-8064\,{c}^{3}{d}^{3}{e}^{2}{x}^{2}+68068\,{a}^{2}c{e}^{5}x+68068\,a{b}^{2}{e}^{5}x-74256\,abcd{e}^{4}x+22848\,a{c}^{2}{d}^{2}{e}^{3}x-12376\,{b}^{3}d{e}^{4}x+22848\,{b}^{2}c{d}^{2}{e}^{3}x-15232\,b{c}^{2}{d}^{3}{e}^{2}x+3584\,{c}^{3}{d}^{4}ex+43758\,{a}^{2}b{e}^{5}-19448\,{a}^{2}cd{e}^{4}-19448\,a{b}^{2}d{e}^{4}+21216\,abc{d}^{2}{e}^{3}-6528\,a{c}^{2}{d}^{3}{e}^{2}+3536\,{b}^{3}{d}^{2}{e}^{3}-6528\,{b}^{2}c{d}^{3}{e}^{2}+4352\,b{c}^{2}{d}^{4}e-1024\,{c}^{3}{d}^{5}}{153153\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/153153*(e*x+d)^(7/2)*(18018*c^3*e^5*x^5+51051*b*c^2*e^5*x^4-12012*c^3*d*e^4*x^
4+47124*a*c^2*e^5*x^3+47124*b^2*c*e^5*x^3-31416*b*c^2*d*e^4*x^3+7392*c^3*d^2*e^3
*x^3+83538*a*b*c*e^5*x^2-25704*a*c^2*d*e^4*x^2+13923*b^3*e^5*x^2-25704*b^2*c*d*e
^4*x^2+17136*b*c^2*d^2*e^3*x^2-4032*c^3*d^3*e^2*x^2+34034*a^2*c*e^5*x+34034*a*b^
2*e^5*x-37128*a*b*c*d*e^4*x+11424*a*c^2*d^2*e^3*x-6188*b^3*d*e^4*x+11424*b^2*c*d
^2*e^3*x-7616*b*c^2*d^3*e^2*x+1792*c^3*d^4*e*x+21879*a^2*b*e^5-9724*a^2*c*d*e^4-
9724*a*b^2*d*e^4+10608*a*b*c*d^2*e^3-3264*a*c^2*d^3*e^2+1768*b^3*d^2*e^3-3264*b^
2*c*d^3*e^2+2176*b*c^2*d^4*e-512*c^3*d^5)/e^6

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Maxima [A]  time = 0.719164, size = 416, normalized size = 1.65 \[ \frac{2 \,{\left (18018 \,{\left (e x + d\right )}^{\frac{17}{2}} c^{3} - 51051 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 47124 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 13923 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 34034 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\right )} e^{4}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 21879 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{153153 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/153153*(18018*(e*x + d)^(17/2)*c^3 - 51051*(2*c^3*d - b*c^2*e)*(e*x + d)^(15/2
) + 47124*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(13/2) - 139
23*(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) + 34034*(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) - 21879*(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))/e^6

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Fricas [A]  time = 0.28345, size = 797, normalized size = 3.16 \[ \frac{2 \,{\left (18018 \, c^{3} e^{8} x^{8} - 512 \, c^{3} d^{8} + 2176 \, b c^{2} d^{7} e + 21879 \, a^{2} b d^{3} e^{5} - 3264 \,{\left (b^{2} c + a c^{2}\right )} d^{6} e^{2} + 1768 \,{\left (b^{3} + 6 \, a b c\right )} d^{5} e^{3} - 9724 \,{\left (a b^{2} + a^{2} c\right )} d^{4} e^{4} + 3003 \,{\left (14 \, c^{3} d e^{7} + 17 \, b c^{2} e^{8}\right )} x^{7} + 231 \,{\left (110 \, c^{3} d^{2} e^{6} + 527 \, b c^{2} d e^{7} + 204 \,{\left (b^{2} c + a c^{2}\right )} e^{8}\right )} x^{6} + 63 \,{\left (2 \, c^{3} d^{3} e^{5} + 1207 \, b c^{2} d^{2} e^{6} + 1836 \,{\left (b^{2} c + a c^{2}\right )} d e^{7} + 221 \,{\left (b^{3} + 6 \, a b c\right )} e^{8}\right )} x^{5} - 7 \,{\left (20 \, c^{3} d^{4} e^{4} - 85 \, b c^{2} d^{3} e^{5} - 10812 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{6} - 5083 \,{\left (b^{3} + 6 \, a b c\right )} d e^{7} - 4862 \,{\left (a b^{2} + a^{2} c\right )} e^{8}\right )} x^{4} +{\left (160 \, c^{3} d^{5} e^{3} - 680 \, b c^{2} d^{4} e^{4} + 21879 \, a^{2} b e^{8} + 1020 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{5} + 24973 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{6} + 92378 \,{\left (a b^{2} + a^{2} c\right )} d e^{7}\right )} x^{3} - 3 \,{\left (64 \, c^{3} d^{6} e^{2} - 272 \, b c^{2} d^{5} e^{3} - 21879 \, a^{2} b d e^{7} + 408 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{4} - 221 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{5} - 24310 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{6}\right )} x^{2} +{\left (256 \, c^{3} d^{7} e - 1088 \, b c^{2} d^{6} e^{2} + 65637 \, a^{2} b d^{2} e^{6} + 1632 \,{\left (b^{2} c + a c^{2}\right )} d^{5} e^{3} - 884 \,{\left (b^{3} + 6 \, a b c\right )} d^{4} e^{4} + 4862 \,{\left (a b^{2} + a^{2} c\right )} d^{3} e^{5}\right )} x\right )} \sqrt{e x + d}}{153153 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/153153*(18018*c^3*e^8*x^8 - 512*c^3*d^8 + 2176*b*c^2*d^7*e + 21879*a^2*b*d^3*e
^5 - 3264*(b^2*c + a*c^2)*d^6*e^2 + 1768*(b^3 + 6*a*b*c)*d^5*e^3 - 9724*(a*b^2 +
a^2*c)*d^4*e^4 + 3003*(14*c^3*d*e^7 + 17*b*c^2*e^8)*x^7 + 231*(110*c^3*d^2*e^6
+ 527*b*c^2*d*e^7 + 204*(b^2*c + a*c^2)*e^8)*x^6 + 63*(2*c^3*d^3*e^5 + 1207*b*c^
2*d^2*e^6 + 1836*(b^2*c + a*c^2)*d*e^7 + 221*(b^3 + 6*a*b*c)*e^8)*x^5 - 7*(20*c^
3*d^4*e^4 - 85*b*c^2*d^3*e^5 - 10812*(b^2*c + a*c^2)*d^2*e^6 - 5083*(b^3 + 6*a*b
*c)*d*e^7 - 4862*(a*b^2 + a^2*c)*e^8)*x^4 + (160*c^3*d^5*e^3 - 680*b*c^2*d^4*e^4
+ 21879*a^2*b*e^8 + 1020*(b^2*c + a*c^2)*d^3*e^5 + 24973*(b^3 + 6*a*b*c)*d^2*e^
6 + 92378*(a*b^2 + a^2*c)*d*e^7)*x^3 - 3*(64*c^3*d^6*e^2 - 272*b*c^2*d^5*e^3 - 2
1879*a^2*b*d*e^7 + 408*(b^2*c + a*c^2)*d^4*e^4 - 221*(b^3 + 6*a*b*c)*d^3*e^5 - 2
4310*(a*b^2 + a^2*c)*d^2*e^6)*x^2 + (256*c^3*d^7*e - 1088*b*c^2*d^6*e^2 + 65637*
a^2*b*d^2*e^6 + 1632*(b^2*c + a*c^2)*d^5*e^3 - 884*(b^3 + 6*a*b*c)*d^4*e^4 + 486
2*(a*b^2 + a^2*c)*d^3*e^5)*x)*sqrt(e*x + d)/e^6

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Sympy [A]  time = 21.5187, size = 1860, normalized size = 7.38 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*b*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) +
4*a**2*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 2*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 + 4*a**2*c*d**2*
(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 8*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**2 + 4*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**2 + 4*a*b**2*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e*
*2 + 8*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**2 + 4*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**2 + 12*a*b*c*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 + 24*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**3 + 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)/1
1)/e**3 + 8*a*c**2*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 + 16*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**4 + 8*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**4 + 2*b**3*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 + 4*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**3 + 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**3 + 8*b**2*c*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 + 16*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**4 + 8*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**4 + 10*b*c**2*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 + 20*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**5 + 10*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**5 + 4*c**3*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 + 8*c**3*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 + 4*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)**(1
1/2)/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

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GIAC/XCAS [A]  time = 0.30828, 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)^2*(2*c*x + b)*(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Done