3.2188 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=271 \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^3 (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (b e g-6 c d g+4 c e f)}{2 e^2 (d+e x) (2 c d-b e)}-\frac{3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-6 c d g+4 c e f)}{4 e^2}-\frac{3 (2 c d-b e) (b e g-6 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 \sqrt{c} e^2} \]

[Out]

(-3*(4*c*e*f - 6*c*d*g + b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*e^
2) - ((4*c*e*f - 6*c*d*g + b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(
2*e^2*(2*c*d - b*e)*(d + e*x)) - (2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2
*x^2)^(5/2))/(e^2*(2*c*d - b*e)*(d + e*x)^3) - (3*(2*c*d - b*e)*(4*c*e*f - 6*c*d
*g + b*e*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])])/(8*Sqrt[c]*e^2)

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Rubi [A]  time = 0.981486, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^3 (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (b e g-6 c d g+4 c e f)}{2 e^2 (d+e x) (2 c d-b e)}-\frac{3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-6 c d g+4 c e f)}{4 e^2}-\frac{3 (2 c d-b e) (b e g-6 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 \sqrt{c} e^2} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(4*c*e*f - 6*c*d*g + b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*e^
2) - ((4*c*e*f - 6*c*d*g + b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(
2*e^2*(2*c*d - b*e)*(d + e*x)) - (2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2
*x^2)^(5/2))/(e^2*(2*c*d - b*e)*(d + e*x)^3) - (3*(2*c*d - b*e)*(4*c*e*f - 6*c*d
*g + b*e*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])])/(8*Sqrt[c]*e^2)

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Rubi in Sympy [A]  time = 99.5799, size = 257, normalized size = 0.95 \[ - \frac{3 \left (\frac{b e g}{4} - \frac{3 c d g}{2} + c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{e^{2}} + \frac{\left (\frac{b e g}{2} - 3 c d g + 2 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{e^{2} \left (d + e x\right ) \left (b e - 2 c d\right )} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{e^{2} \left (d + e x\right )^{3} \left (b e - 2 c d\right )} + \frac{3 \left (b e - 2 c d\right ) \left (b e g - 6 c d g + 4 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{8 \sqrt{c} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 1.18614, size = 214, normalized size = 0.79 \[ \frac{((d+e x) (c (d-e x)-b e))^{3/2} \left (-\frac{2 ((d+e x) (5 b e g+4 c (e f-3 d g))+8 (2 c d-b e) (e f-d g)+2 c e g x (d+e x))}{(d+e x)^2 (c (d-e x)-b e)}-\frac{3 i (2 c d-b e) (b e g-6 c d g+4 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{\sqrt{c} (d+e x)^{3/2} (c (d-e x)-b e)^{3/2}}\right )}{8 e^2} \]

Antiderivative was successfully verified.

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

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*((-2*(8*(2*c*d - b*e)*(e*f - d*g) + (5
*b*e*g + 4*c*(e*f - 3*d*g))*(d + e*x) + 2*c*e*g*x*(d + e*x)))/((d + e*x)^2*(-(b*
e) + c*(d - e*x))) - ((3*I)*(2*c*d - b*e)*(4*c*e*f - 6*c*d*g + b*e*g)*Log[((-I)*
e*(b + 2*c*x))/Sqrt[c] + 2*Sqrt[d + e*x]*Sqrt[-(b*e) + c*(d - e*x)]])/(Sqrt[c]*(
d + e*x)^(3/2)*(-(b*e) + c*(d - e*x))^(3/2))))/(8*e^2)

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Maple [B]  time = 0.024, size = 2535, normalized size = 9.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*e*c^2/(-b*e^2+2*c*d*e)^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*f+
2*g/e^3/(-b*e^2+2*c*d*e)/(d/e+x)^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(
5/2)+2*g/e*c/(-b*e^2+2*c*d*e)*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)+
8*c^2/(-b*e^2+2*c*d*e)^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*d*g-2
/e^3/(-b*e^2+2*c*d*e)/(d/e+x)^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(5/2
)*f-3/4*g*e/(-b*e^2+2*c*d*e)*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/
2)+3*e^3*c/(-b*e^2+2*c*d*e)^2*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1
/2)*f+3*g*c^3/(-b*e^2+2*c*d*e)*d^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2
*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))+3*g*
c^2/(-b*e^2+2*c*d*e)*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x-3/8*g
*e^3/(-b*e^2+2*c*d*e)*b^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+
2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))+3/2*g*c/(-b*e
^2+2*c*d*e)*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*b+2/e^4/(-b*e^2+
2*c*d*e)/(d/e+x)^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(5/2)*d*g-8/e*c/(
-b*e^2+2*c*d*e)^2/(d/e+x)^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(5/2)*f+
12*e*c^4/(-b*e^2+2*c*d*e)^2*d^4/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-
b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*g-12*e^
2*c^4/(-b*e^2+2*c*d*e)^2*d^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e
^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*f-3/2*g*e*
c/(-b*e^2+2*c*d*e)*b*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x-3*e^2*c
/(-b*e^2+2*c*d*e)^2*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d*g+3/
2*e^5*c/(-b*e^2+2*c*d*e)^2*b^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b
*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*f+12*e*c
^3/(-b*e^2+2*c*d*e)^2*d^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x*g-
12*e^2*c^3/(-b*e^2+2*c*d*e)^2*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2
)*x*f+6*e*c^2/(-b*e^2+2*c*d*e)^2*d^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))
^(1/2)*b*g-6*e^2*c^2/(-b*e^2+2*c*d*e)^2*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/
e+x))^(1/2)*b*f+8/e^2*c/(-b*e^2+2*c*d*e)^2/(d/e+x)^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2
*c*d*e)*(d/e+x))^(5/2)*d*g+6*e^3*c^2/(-b*e^2+2*c*d*e)^2*b*(-c*(d/e+x)^2*e^2+(-b*
e^2+2*c*d*e)*(d/e+x))^(1/2)*x*f+9*e^3*c^2/(-b*e^2+2*c*d*e)^2*b^2/(c*e^2)^(1/2)*a
rctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2
+2*c*d*e)*(d/e+x))^(1/2))*d^2*g+18*e^3*c^3/(-b*e^2+2*c*d*e)^2*b/(c*e^2)^(1/2)*ar
ctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+
2*c*d*e)*(d/e+x))^(1/2))*d^2*f-9/2*g*e*c^2/(-b*e^2+2*c*d*e)*b/(c*e^2)^(1/2)*arct
an((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*
c*d*e)*(d/e+x))^(1/2))*d^2-9*e^4*c^2/(-b*e^2+2*c*d*e)^2*b^2/(c*e^2)^(1/2)*arctan
((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*
d*e)*(d/e+x))^(1/2))*d*f-6*e^2*c^2/(-b*e^2+2*c*d*e)^2*b*(-c*(d/e+x)^2*e^2+(-b*e^
2+2*c*d*e)*(d/e+x))^(1/2)*x*d*g-3/2*e^4*c/(-b*e^2+2*c*d*e)^2*b^3/(c*e^2)^(1/2)*a
rctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2
+2*c*d*e)*(d/e+x))^(1/2))*d*g+9/4*g*e^2*c/(-b*e^2+2*c*d*e)*b^2/(c*e^2)^(1/2)*arc
tan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2
*c*d*e)*(d/e+x))^(1/2))*d-18*e^2*c^3/(-b*e^2+2*c*d*e)^2*b/(c*e^2)^(1/2)*arctan((
c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*
e)*(d/e+x))^(1/2))*d^3*g

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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((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(g*x + f)/(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.0707, size = 1, normalized size = 0. \[ \left [-\frac{4 \,{\left (2 \, c e^{2} g x^{2} + 4 \,{\left (5 \, c d e - 2 \, b e^{2}\right )} f -{\left (28 \, c d^{2} - 13 \, b d e\right )} g +{\left (4 \, c e^{2} f - 5 \,{\left (2 \, c d e - b e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{-c} + 3 \,{\left (4 \,{\left (2 \, c^{2} d^{2} e - b c d e^{2}\right )} f -{\left (12 \, c^{2} d^{3} - 8 \, b c d^{2} e + b^{2} d e^{2}\right )} g +{\left (4 \,{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} f -{\left (12 \, c^{2} d^{2} e - 8 \, b c d e^{2} + b^{2} e^{3}\right )} g\right )} x\right )} \log \left (4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c^{2} e x + b c e\right )} +{\left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} \sqrt{-c}\right )}{16 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{-c}}, -\frac{2 \,{\left (2 \, c e^{2} g x^{2} + 4 \,{\left (5 \, c d e - 2 \, b e^{2}\right )} f -{\left (28 \, c d^{2} - 13 \, b d e\right )} g +{\left (4 \, c e^{2} f - 5 \,{\left (2 \, c d e - b e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{c} + 3 \,{\left (4 \,{\left (2 \, c^{2} d^{2} e - b c d e^{2}\right )} f -{\left (12 \, c^{2} d^{3} - 8 \, b c d^{2} e + b^{2} d e^{2}\right )} g +{\left (4 \,{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} f -{\left (12 \, c^{2} d^{2} e - 8 \, b c d e^{2} + b^{2} e^{3}\right )} g\right )} x\right )} \arctan \left (\frac{2 \, c e x + b e}{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{c}}\right )}{8 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(4*(2*c*e^2*g*x^2 + 4*(5*c*d*e - 2*b*e^2)*f - (28*c*d^2 - 13*b*d*e)*g + (
4*c*e^2*f - 5*(2*c*d*e - b*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)
*sqrt(-c) + 3*(4*(2*c^2*d^2*e - b*c*d*e^2)*f - (12*c^2*d^3 - 8*b*c*d^2*e + b^2*d
*e^2)*g + (4*(2*c^2*d*e^2 - b*c*e^3)*f - (12*c^2*d^2*e - 8*b*c*d*e^2 + b^2*e^3)*
g)*x)*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)))/((e^3*x
+ d*e^2)*sqrt(-c)), -1/8*(2*(2*c*e^2*g*x^2 + 4*(5*c*d*e - 2*b*e^2)*f - (28*c*d^2
- 13*b*d*e)*g + (4*c*e^2*f - 5*(2*c*d*e - b*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*
x + c*d^2 - b*d*e)*sqrt(c) + 3*(4*(2*c^2*d^2*e - b*c*d*e^2)*f - (12*c^2*d^3 - 8*
b*c*d^2*e + b^2*d*e^2)*g + (4*(2*c^2*d*e^2 - b*c*e^3)*f - (12*c^2*d^2*e - 8*b*c*
d*e^2 + b^2*e^3)*g)*x)*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))))/((e^3*x + d*e^2)*sqrt(c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(f + g*x)/(d + e*x)**3, x)

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GIAC/XCAS [A]  time = 0.806094, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x