3.2225 \(\int \frac{(d+e x)^3 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=177 \[ \frac{g \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c^{5/2} e^2}-\frac{2 g (d+e x)}{c^2 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^3 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^3)/(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*g*(d + e*x))/(c^2*e^2*Sqrt[d*(c*d - b*e) - b*e^
2*x - c*e^2*x^2]) + (g*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*
e^2*x - c*e^2*x^2])])/(c^(5/2)*e^2)

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Rubi [A]  time = 0.639526, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{g \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c^{5/2} e^2}-\frac{2 g (d+e x)}{c^2 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^3 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^3*(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)^3)/(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*g*(d + e*x))/(c^2*e^2*Sqrt[d*(c*d - b*e) - b*e^
2*x - c*e^2*x^2]) + (g*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*
e^2*x - c*e^2*x^2])])/(c^(5/2)*e^2)

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Rubi in Sympy [A]  time = 65.9565, size = 165, normalized size = 0.93 \[ \frac{2 \left (d + e x\right )^{3} \left (b e g - c d g - c e f\right )}{3 c e^{2} \left (b e - 2 c d\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} - \frac{2 g \left (d + e x\right )}{c^{2} e^{2} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} + \frac{g \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{c^{\frac{5}{2}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(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)**3*(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*g*(d + e*x)/(c**2*e**2*sqrt(-b*e**2*x - c*
e**2*x**2 + d*(-b*e + c*d))) + g*atan(-e*(-b - 2*c*x)/(2*sqrt(c)*sqrt(-b*e**2*x
- c*e**2*x**2 + d*(-b*e + c*d))))/(c**(5/2)*e**2)

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Mathematica [C]  time = 0.895993, size = 202, normalized size = 1.14 \[ \frac{-\frac{2 \sqrt{c} (d+e x)^3 (c (d-e x)-b e) \left (3 b^2 e^2 g+4 b c e g (e x-2 d)+c^2 \left (5 d^2 g-d e (f+7 g x)-e^2 f x\right )\right )}{2 c d-b e}+3 i g (d+e x)^{5/2} (c (d-e x)-b e)^{5/2} \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{3 c^{5/2} e^2 ((d+e x) (c (d-e x)-b e))^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[c]*(d + e*x)^3*(-(b*e) + c*(d - e*x))*(3*b^2*e^2*g + 4*b*c*e*g*(-2*d +
e*x) + c^2*(5*d^2*g - e^2*f*x - d*e*(f + 7*g*x))))/(2*c*d - b*e) + (3*I)*g*(d +
e*x)^(5/2)*(-(b*e) + c*(d - e*x))^(5/2)*Log[((-I)*e*(b + 2*c*x))/Sqrt[c] + 2*Sq
rt[d + e*x]*Sqrt[-(b*e) + c*(d - e*x)]])/(3*c^(5/2)*e^2*((d + e*x)*(-(b*e) + c*(
d - e*x)))^(5/2))

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Maple [B]  time = 0.019, size = 3485, normalized size = 19.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^2/c*e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*f+3*x^2/c/(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(3/2)*d*g+1/3*d^3*f/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b
*e^2*x-b*d*e+c*d^2)^(3/2)*b*e^2-13/3*b/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*
e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*e^2*d^3*g-b/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2
*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*e^3*d^2*f-13/6*b^2/c/(-b^2*e^4+4*
b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*e^2*d^3*g-1/2*b^
2/c/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*
e^3*d^2*f-1/3*e^7*g*b^4/c^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b
*e^2*x-b*d*e+c*d^2)^(1/2)*x-3/8*e^4*g*b^4/c^3/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^
2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d+3*e^6*g*b^4/c^2/(-b^2*e^4+4*b*c*d*e^
3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d-24*e^5*g*b^2/(-b^2*e
^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^2+3/2
*e^3*g*b^3/c^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*
d^2)^(3/2)*d^2-12*e^5*g*b^3/c/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2
-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2+e^3*g/c^2*b^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^
2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+1/4*b^3/c^2/(-b^2*e^4+4*b*c*d*e^3-4*
c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*e^4*d*f-4*b^2*e^6/(-b^2*e^4+
4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d*f-2*b^3/
c*e^6/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1
/2)*d*f+2*c/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2
)^(3/2)*x*d^4*e*g-16*c^2*e^3/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^4*g-1/24*b^4/c^3/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*
e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*e^5*f+1/3*b^4/c^2*e^7/(-b^2*e^4+4*b*
c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f-1/8*e*g*b^2/c^
3*x/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+1/48*e^5*g*b^5/c^4/(-b^2*e^4+4*b*c*d*
e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-1/6*e^7*g*b^5/c^3/(-b^
2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+25/12/
e*g*b/c^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2-1/2*e*g*b/c^2*x^2/(-c*e^2*x
^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+1/2*e^3*g/c^3*b^3/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*
e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+3/4*b/c^2*x/(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(3/2)*d*g+1/4*b/c^2*x*e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*f+52/3*b
^2*e^4/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(
1/2)*d^3*g+4*b^2*e^5/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-
b*d*e+c*d^2)^(1/2)*d^2*f+3/2*x/c/e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2*g+
1/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*
d^4*e*g+104/3*b*c*e^4/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x
-b*d*e+c*d^2)^(1/2)*x*d^3*g+1/2*b^2/c/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e
^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*e^4*d*f+3*e^3*g*b^2/c/(-b^2*e^4+4*b*c*d*e^3-
4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^2+6*e^6*g*b^3/c/(-b^2*
e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d-3/4*
e^4*g*b^3/c^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(3/2)*x*d+1/2/e*g/c^3*b/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-5/3/c/e^2/(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^3*g+1/3/c/e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^
2)^(3/2)*d^2*f+8*b*c*e^5/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^
2*x-b*d*e+c*d^2)^(1/2)*x*d^2*f+1/e*g/c^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1
/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))+3/2*x/c/(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(3/2)*d*f+5/12*b/c^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d*f-8*c*e^3
/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b
*d^4*g-1/12*b^3/c^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(3/2)*x*e^5*f+2/3*b^3/c*e^7/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*f+2/3*d^3*f/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^
2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*c*e^2-16/3*d^3*f*c^2*e^4/(-b^2*e
^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x-8/3*d^3
*f*c*e^4/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)
^(1/2)*b+1/24*e^5*g*b^4/c^3/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e
^2*x-b*d*e+c*d^2)^(3/2)*x-1/24*b^2/c^3*e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*
f+1/3*e*g*x^3/c/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+1/48*e*g*b^3/c^4/(-c*e^2*
x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-11/24*g*b^2/c^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(3/2)*d-1/e*g/c^2*x/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)

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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)^3*(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 = 0.854251, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (c^{2} d e f -{\left (5 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} g +{\left (c^{2} e^{2} f +{\left (7 \, c^{2} d e - 4 \, b c e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{-c} + 3 \,{\left ({\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} g x^{2} - 2 \,{\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} g x +{\left (2 \, c^{3} d^{3} - 5 \, b c^{2} d^{2} e + 4 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} g\right )} \log \left (4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c^{2} e x + b c e\right )} +{\left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} \sqrt{-c}\right )}{6 \,{\left (2 \, c^{5} d^{3} e^{2} - 5 \, b c^{4} d^{2} e^{3} + 4 \, b^{2} c^{3} d e^{4} - b^{3} c^{2} e^{5} +{\left (2 \, c^{5} d e^{4} - b c^{4} e^{5}\right )} x^{2} - 2 \,{\left (2 \, c^{5} d^{2} e^{3} - 3 \, b c^{4} d e^{4} + b^{2} c^{3} e^{5}\right )} x\right )} \sqrt{-c}}, \frac{2 \,{\left (c^{2} d e f -{\left (5 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} g +{\left (c^{2} e^{2} f +{\left (7 \, c^{2} d e - 4 \, b c e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{c} + 3 \,{\left ({\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} g x^{2} - 2 \,{\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} g x +{\left (2 \, c^{3} d^{3} - 5 \, b c^{2} d^{2} e + 4 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} g\right )} \arctan \left (\frac{2 \, c e x + b e}{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{c}}\right )}{3 \,{\left (2 \, c^{5} d^{3} e^{2} - 5 \, b c^{4} d^{2} e^{3} + 4 \, b^{2} c^{3} d e^{4} - b^{3} c^{2} e^{5} +{\left (2 \, c^{5} d e^{4} - b c^{4} e^{5}\right )} x^{2} - 2 \,{\left (2 \, c^{5} d^{2} e^{3} - 3 \, b c^{4} d e^{4} + b^{2} c^{3} e^{5}\right )} x\right )} \sqrt{c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(4*(c^2*d*e*f - (5*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*g + (c^2*e^2*f + (7*c^2
*d*e - 4*b*c*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-c) + 3*
((2*c^3*d*e^2 - b*c^2*e^3)*g*x^2 - 2*(2*c^3*d^2*e - 3*b*c^2*d*e^2 + b^2*c*e^3)*g
*x + (2*c^3*d^3 - 5*b*c^2*d^2*e + 4*b^2*c*d*e^2 - b^3*e^3)*g)*log(4*sqrt(-c*e^2*
x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c^2*e*x + b*c*e) + (8*c^2*e^2*x^2 + 8*b*c*e^2*
x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*sqrt(-c)))/((2*c^5*d^3*e^2 - 5*b*c^4*d^2*e^
3 + 4*b^2*c^3*d*e^4 - b^3*c^2*e^5 + (2*c^5*d*e^4 - b*c^4*e^5)*x^2 - 2*(2*c^5*d^2
*e^3 - 3*b*c^4*d*e^4 + b^2*c^3*e^5)*x)*sqrt(-c)), 1/3*(2*(c^2*d*e*f - (5*c^2*d^2
- 8*b*c*d*e + 3*b^2*e^2)*g + (c^2*e^2*f + (7*c^2*d*e - 4*b*c*e^2)*g)*x)*sqrt(-c
*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(c) + 3*((2*c^3*d*e^2 - b*c^2*e^3)*g*x^2
- 2*(2*c^3*d^2*e - 3*b*c^2*d*e^2 + b^2*c*e^3)*g*x + (2*c^3*d^3 - 5*b*c^2*d^2*e
+ 4*b^2*c*d*e^2 - b^3*e^3)*g)*arctan(1/2*(2*c*e*x + b*e)/(sqrt(-c*e^2*x^2 - b*e^
2*x + c*d^2 - b*d*e)*sqrt(c))))/((2*c^5*d^3*e^2 - 5*b*c^4*d^2*e^3 + 4*b^2*c^3*d*
e^4 - b^3*c^2*e^5 + (2*c^5*d*e^4 - b*c^4*e^5)*x^2 - 2*(2*c^5*d^2*e^3 - 3*b*c^4*d
*e^4 + b^2*c^3*e^5)*x)*sqrt(c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3} \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

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GIAC/XCAS [A]  time = 0.32338, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done