3.2226 \(\int \frac{(d+e x)^2 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=146 \[ \frac{2 (d+e x)^2 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{2 (d+e x) (b e g-4 c d g+2 c e f)}{3 c e^2 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.469183, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{2 (d+e x)^2 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{2 (d+e x) (b e g-4 c d g+2 c e f)}{3 c e^2 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 56.7885, size = 134, normalized size = 0.92 \[ \frac{2 \left (d + e x\right )^{2} \left (b e g - c d g - c e f\right )}{3 c e^{2} \left (b e - 2 c d\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} + \frac{2 \left (d + e x\right ) \left (b e g - 4 c d g + 2 c e f\right )}{3 c e^{2} \left (b e - 2 c d\right )^{2} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(d + e*x)**2*(b*e*g - c*d*g - c*e*f)/(3*c*e**2*(b*e - 2*c*d)*(-b*e**2*x - c*e*
*2*x**2 + d*(-b*e + c*d))**(3/2)) + 2*(d + e*x)*(b*e*g - 4*c*d*g + 2*c*e*f)/(3*c
*e**2*(b*e - 2*c*d)**2*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d)))

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Mathematica [A]  time = 0.221185, size = 100, normalized size = 0.68 \[ \frac{2 (d+e x) \left (b e (-2 d g+3 e f+e g x)+2 c \left (d^2 g-2 d e (f+g x)+e^2 f x\right )\right )}{3 e^2 (b e-2 c d)^2 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)*(b*e*(3*e*f - 2*d*g + e*g*x) + 2*c*(d^2*g + e^2*f*x - 2*d*e*(f + g*
x))))/(3*e^2*(-2*c*d + b*e)^2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*
(d - e*x))])

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Maple [A]  time = 0.013, size = 128, normalized size = 0.9 \[ -{\frac{2\, \left ( ex+d \right ) ^{3} \left ( cex+be-cd \right ) \left ( -b{e}^{2}gx+4\,cdegx-2\,c{e}^{2}fx+2\,bdeg-3\,b{e}^{2}f-2\,c{d}^{2}g+4\,cdef \right ) }{3\,{e}^{2} \left ({b}^{2}{e}^{2}-4\,bcde+4\,{c}^{2}{d}^{2} \right ) } \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(e*x+d)^3*(c*e*x+b*e-c*d)*(-b*e^2*g*x+4*c*d*e*g*x-2*c*e^2*f*x+2*b*d*e*g-3*b
*e^2*f-2*c*d^2*g+4*c*d*e*f)/e^2/(b^2*e^2-4*b*c*d*e+4*c^2*d^2)/(-c*e^2*x^2-b*e^2*
x-b*d*e+c*d^2)^(5/2)

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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)^2*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.04438, size = 306, normalized size = 2.1 \[ \frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left ({\left (4 \, c d e - 3 \, b e^{2}\right )} f - 2 \,{\left (c d^{2} - b d e\right )} g -{\left (2 \, c e^{2} f -{\left (4 \, c d e - b e^{2}\right )} g\right )} x\right )}}{3 \,{\left (4 \, c^{4} d^{4} e^{2} - 12 \, b c^{3} d^{3} e^{3} + 13 \, b^{2} c^{2} d^{2} e^{4} - 6 \, b^{3} c d e^{5} + b^{4} e^{6} +{\left (4 \, c^{4} d^{2} e^{4} - 4 \, b c^{3} d e^{5} + b^{2} c^{2} e^{6}\right )} x^{2} - 2 \,{\left (4 \, c^{4} d^{3} e^{3} - 8 \, b c^{3} d^{2} e^{4} + 5 \, b^{2} c^{2} d e^{5} - b^{3} c e^{6}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((4*c*d*e - 3*b*e^2)*f - 2*(c*d^2
- b*d*e)*g - (2*c*e^2*f - (4*c*d*e - b*e^2)*g)*x)/(4*c^4*d^4*e^2 - 12*b*c^3*d^3
*e^3 + 13*b^2*c^2*d^2*e^4 - 6*b^3*c*d*e^5 + b^4*e^6 + (4*c^4*d^2*e^4 - 4*b*c^3*d
*e^5 + b^2*c^2*e^6)*x^2 - 2*(4*c^4*d^3*e^3 - 8*b*c^3*d^2*e^4 + 5*b^2*c^2*d*e^5 -
b^3*c*e^6)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{2} \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**2*(f + g*x)/(-(d + e*x)*(b*e - c*d + c*e*x))**(5/2), x)

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GIAC/XCAS [A]  time = 0.294634, size = 792, normalized size = 5.42 \[ \frac{2 \, \sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}{\left ({\left ({\left (\frac{{\left (16 \, c^{3} d^{3} g e^{3} - 8 \, c^{3} d^{2} f e^{4} - 20 \, b c^{2} d^{2} g e^{4} + 8 \, b c^{2} d f e^{5} + 8 \, b^{2} c d g e^{5} - 2 \, b^{2} c f e^{6} - b^{3} g e^{6}\right )} x}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}} + \frac{3 \,{\left (8 \, c^{3} d^{4} g e^{2} - 8 \, b c^{2} d^{3} g e^{3} - 4 \, b c^{2} d^{2} f e^{4} + 2 \, b^{2} c d^{2} g e^{4} + 4 \, b^{2} c d f e^{5} - b^{3} f e^{6}\right )}}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}}\right )} x + \frac{3 \,{\left (8 \, c^{3} d^{4} f e^{2} + 4 \, b c^{2} d^{4} g e^{2} - 16 \, b c^{2} d^{3} f e^{3} - 4 \, b^{2} c d^{3} g e^{3} + 10 \, b^{2} c d^{2} f e^{4} + b^{3} d^{2} g e^{4} - 2 \, b^{3} d f e^{5}\right )}}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}}\right )} x - \frac{8 \, c^{3} d^{6} g - 16 \, c^{3} d^{5} f e - 16 \, b c^{2} d^{5} g e + 28 \, b c^{2} d^{4} f e^{2} + 10 \, b^{2} c d^{4} g e^{2} - 16 \, b^{2} c d^{3} f e^{3} - 2 \, b^{3} d^{3} g e^{3} + 3 \, b^{3} d^{2} f e^{4}}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}}\right )}}{3 \,{\left (c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*((((16*c^3*d^3*g*e^3 - 8*c^3*d^2*
f*e^4 - 20*b*c^2*d^2*g*e^4 + 8*b*c^2*d*f*e^5 + 8*b^2*c*d*g*e^5 - 2*b^2*c*f*e^6 -
b^3*g*e^6)*x/(16*c^4*d^4*e^2 - 32*b*c^3*d^3*e^3 + 24*b^2*c^2*d^2*e^4 - 8*b^3*c*
d*e^5 + b^4*e^6) + 3*(8*c^3*d^4*g*e^2 - 8*b*c^2*d^3*g*e^3 - 4*b*c^2*d^2*f*e^4 +
2*b^2*c*d^2*g*e^4 + 4*b^2*c*d*f*e^5 - b^3*f*e^6)/(16*c^4*d^4*e^2 - 32*b*c^3*d^3*
e^3 + 24*b^2*c^2*d^2*e^4 - 8*b^3*c*d*e^5 + b^4*e^6))*x + 3*(8*c^3*d^4*f*e^2 + 4*
b*c^2*d^4*g*e^2 - 16*b*c^2*d^3*f*e^3 - 4*b^2*c*d^3*g*e^3 + 10*b^2*c*d^2*f*e^4 +
b^3*d^2*g*e^4 - 2*b^3*d*f*e^5)/(16*c^4*d^4*e^2 - 32*b*c^3*d^3*e^3 + 24*b^2*c^2*d
^2*e^4 - 8*b^3*c*d*e^5 + b^4*e^6))*x - (8*c^3*d^6*g - 16*c^3*d^5*f*e - 16*b*c^2*
d^5*g*e + 28*b*c^2*d^4*f*e^2 + 10*b^2*c*d^4*g*e^2 - 16*b^2*c*d^3*f*e^3 - 2*b^3*d
^3*g*e^3 + 3*b^3*d^2*f*e^4)/(16*c^4*d^4*e^2 - 32*b*c^3*d^3*e^3 + 24*b^2*c^2*d^2*
e^4 - 8*b^3*c*d*e^5 + b^4*e^6))/(c*x^2*e^2 - c*d^2 + b*x*e^2 + b*d*e)^2