3.2382 \(\int \frac{(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=118 \[ \frac{16 \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

(-2*(d + e*x)^2*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x
^2)^(3/2)) + (16*(c*d^2 - b*d*e + a*e^2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^
2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.134856, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{16 \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(d + e*x)^2*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x
^2)^(3/2)) + (16*(c*d^2 - b*d*e + a*e^2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^
2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 20.4545, size = 114, normalized size = 0.97 \[ \frac{2 \left (d + e x\right )^{2} \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{3 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{8 \left (4 a e - 2 b d + x \left (2 b e - 4 c d\right )\right ) \left (a e^{2} - b d e + c d^{2}\right )}{3 \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(d + e*x)**2*(2*a*e - b*d + x*(b*e - 2*c*d))/(3*(-4*a*c + b**2)*(a + b*x + c*x
**2)**(3/2)) - 8*(4*a*e - 2*b*d + x*(2*b*e - 4*c*d))*(a*e**2 - b*d*e + c*d**2)/(
3*(-4*a*c + b**2)**2*sqrt(a + b*x + c*x**2))

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Mathematica [A]  time = 0.291056, size = 190, normalized size = 1.61 \[ \frac{2 \left (12 b (d-e x) \left (2 a^2 e^2+a c (d-e x)^2+2 c^2 d^2 x^2\right )-8 \left (2 a^3 e^3+3 a^2 c e \left (d^2+e^2 x^2\right )-3 a c^2 d x \left (d^2+e^2 x^2\right )-2 c^3 d^3 x^3\right )-6 b^2 \left (d^2-6 d e x+e^2 x^2\right ) (a e-c d x)+b^3 \left (-d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-6*b^2*(a*e - c*d*x)*(d^2 - 6*d*e*x + e^2*x^2) + b^3*(-d^3 - 9*d^2*e*x + 9*d
*e^2*x^2 + e^3*x^3) + 12*b*(d - e*x)*(2*a^2*e^2 + 2*c^2*d^2*x^2 + a*c*(d - e*x)^
2) - 8*(2*a^3*e^3 - 2*c^3*d^3*x^3 + 3*a^2*c*e*(d^2 + e^2*x^2) - 3*a*c^2*d*x*(d^2
+ e^2*x^2))))/(3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2))

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Maple [B]  time = 0.013, size = 296, normalized size = 2.5 \[ -{\frac{24\,abc{e}^{3}{x}^{3}-48\,a{c}^{2}d{e}^{2}{x}^{3}-2\,{b}^{3}{e}^{3}{x}^{3}-12\,{b}^{2}cd{e}^{2}{x}^{3}+48\,b{c}^{2}{d}^{2}e{x}^{3}-32\,{x}^{3}{c}^{3}{d}^{3}+48\,{a}^{2}c{e}^{3}{x}^{2}+12\,{x}^{2}a{b}^{2}{e}^{3}-72\,abcd{e}^{2}{x}^{2}-18\,{x}^{2}{b}^{3}d{e}^{2}+72\,{b}^{2}c{d}^{2}e{x}^{2}-48\,b{c}^{2}{d}^{3}{x}^{2}+48\,x{a}^{2}b{e}^{3}-72\,xa{b}^{2}d{e}^{2}+72\,abc{d}^{2}ex-48\,a{c}^{2}{d}^{3}x+18\,x{b}^{3}{d}^{2}e-12\,{b}^{2}c{d}^{3}x+32\,{a}^{3}{e}^{3}-48\,{a}^{2}bd{e}^{2}+48\,{a}^{2}c{d}^{2}e+12\,a{b}^{2}{d}^{2}e-24\,abc{d}^{3}+2\,{b}^{3}{d}^{3}}{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3/(c*x^2+b*x+a)^(3/2)*(12*a*b*c*e^3*x^3-24*a*c^2*d*e^2*x^3-b^3*e^3*x^3-6*b^2*
c*d*e^2*x^3+24*b*c^2*d^2*e*x^3-16*c^3*d^3*x^3+24*a^2*c*e^3*x^2+6*a*b^2*e^3*x^2-3
6*a*b*c*d*e^2*x^2-9*b^3*d*e^2*x^2+36*b^2*c*d^2*e*x^2-24*b*c^2*d^3*x^2+24*a^2*b*e
^3*x-36*a*b^2*d*e^2*x+36*a*b*c*d^2*e*x-24*a*c^2*d^3*x+9*b^3*d^2*e*x-6*b^2*c*d^3*
x+16*a^3*e^3-24*a^2*b*d*e^2+24*a^2*c*d^2*e+6*a*b^2*d^2*e-12*a*b*c*d^3+b^3*d^3)/(
16*a^2*c^2-8*a*b^2*c+b^4)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.383975, size = 494, normalized size = 4.19 \[ \frac{2 \,{\left (24 \, a^{2} b d e^{2} - 16 \, a^{3} e^{3} -{\left (b^{3} - 12 \, a b c\right )} d^{3} - 6 \,{\left (a b^{2} + 4 \, a^{2} c\right )} d^{2} e +{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \,{\left (b^{2} c + 4 \, a c^{2}\right )} d e^{2} +{\left (b^{3} - 12 \, a b c\right )} e^{3}\right )} x^{3} + 3 \,{\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \,{\left (b^{3} + 4 \, a b c\right )} d e^{2} - 2 \,{\left (a b^{2} + 4 \, a^{2} c\right )} e^{3}\right )} x^{2} + 3 \,{\left (12 \, a b^{2} d e^{2} - 8 \, a^{2} b e^{3} + 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} d^{3} - 3 \,{\left (b^{3} + 4 \, a b c\right )} d^{2} e\right )} x\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(24*a^2*b*d*e^2 - 16*a^3*e^3 - (b^3 - 12*a*b*c)*d^3 - 6*(a*b^2 + 4*a^2*c)*d^
2*e + (16*c^3*d^3 - 24*b*c^2*d^2*e + 6*(b^2*c + 4*a*c^2)*d*e^2 + (b^3 - 12*a*b*c
)*e^3)*x^3 + 3*(8*b*c^2*d^3 - 12*b^2*c*d^2*e + 3*(b^3 + 4*a*b*c)*d*e^2 - 2*(a*b^
2 + 4*a^2*c)*e^3)*x^2 + 3*(12*a*b^2*d*e^2 - 8*a^2*b*e^3 + 2*(b^2*c + 4*a*c^2)*d^
3 - 3*(b^3 + 4*a*b*c)*d^2*e)*x)*sqrt(c*x^2 + b*x + a)/(a^2*b^4 - 8*a^3*b^2*c + 1
6*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 +
16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c
+ 16*a^3*b*c^2)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21927, size = 474, normalized size = 4.02 \[ \frac{{\left ({\left (\frac{{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + 24 \, a c^{2} d e^{2} + b^{3} e^{3} - 12 \, a b c e^{3}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \, b^{3} d e^{2} + 12 \, a b c d e^{2} - 2 \, a b^{2} e^{3} - 8 \, a^{2} c e^{3}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \,{\left (2 \, b^{2} c d^{3} + 8 \, a c^{2} d^{3} - 3 \, b^{3} d^{2} e - 12 \, a b c d^{2} e + 12 \, a b^{2} d e^{2} - 8 \, a^{2} b e^{3}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{3} d^{3} - 12 \, a b c d^{3} + 6 \, a b^{2} d^{2} e + 24 \, a^{2} c d^{2} e - 24 \, a^{2} b d e^{2} + 16 \, a^{3} e^{3}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((((16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + 24*a*c^2*d*e^2 + b^3*e^3 -
12*a*b*c*e^3)*x/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4) + 3*(8*b*c^2*d^3 - 12*b^2*
c*d^2*e + 3*b^3*d*e^2 + 12*a*b*c*d*e^2 - 2*a*b^2*e^3 - 8*a^2*c*e^3)/(b^4*c^2 - 8
*a*b^2*c^3 + 16*a^2*c^4))*x + 3*(2*b^2*c*d^3 + 8*a*c^2*d^3 - 3*b^3*d^2*e - 12*a*
b*c*d^2*e + 12*a*b^2*d*e^2 - 8*a^2*b*e^3)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*
x - (b^3*d^3 - 12*a*b*c*d^3 + 6*a*b^2*d^2*e + 24*a^2*c*d^2*e - 24*a^2*b*d*e^2 +
16*a^3*e^3)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/(c*x^2 + b*x + a)^(3/2)