3.2187 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=278 \[ \frac{(2 c d-b e)^2 (-b e g-4 c d g+6 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16 c^{3/2} e^2}+\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2 (2 c d-b e)}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-b e g-4 c d g+6 c e f)}{3 e^2 (2 c d-b e)}+\frac{(b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-b e g-4 c d g+6 c e f)}{8 c e} \]

[Out]

((6*c*e*f - 4*c*d*g - b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^
2])/(8*c*e) + ((6*c*e*f - 4*c*d*g - b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(3/2))/(3*e^2*(2*c*d - b*e)) + (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)^2) + ((2*c*d - b*e)^2*(6*c*e*f - 4*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])])/(16*c^(3/2)*e^2)

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Rubi [A]  time = 0.900945, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114 \[ \frac{(2 c d-b e)^2 (-b e g-4 c d g+6 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16 c^{3/2} e^2}+\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2 (2 c d-b e)}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-b e g-4 c d g+6 c e f)}{3 e^2 (2 c d-b e)}+\frac{(b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-b e g-4 c d g+6 c e f)}{8 c e} \]

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)^2,x]

[Out]

((6*c*e*f - 4*c*d*g - b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^
2])/(8*c*e) + ((6*c*e*f - 4*c*d*g - b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(3/2))/(3*e^2*(2*c*d - b*e)) + (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)^2) + ((2*c*d - b*e)^2*(6*c*e*f - 4*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])])/(16*c^(3/2)*e^2)

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Rubi in Sympy [A]  time = 79.4997, size = 257, normalized size = 0.92 \[ \frac{\left (b e g + 4 c d g - 6 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{3 e^{2} \left (b e - 2 c d\right )} + \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{e^{2} \left (d + e x\right )^{2} \left (b e - 2 c d\right )} - \frac{\left (b + 2 c x\right ) \left (b e g + 4 c d g - 6 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{8 c e} - \frac{\left (b e - 2 c d\right )^{2} \left (b e g + 4 c d g - 6 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{16 c^{\frac{3}{2}} e^{2}} \]

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)**2,x)

[Out]

(b*e*g + 4*c*d*g - 6*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(3
*e**2*(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)**2*(b*e - 2*c*d)) - (b + 2*c*x)*(b*e*g + 4*c*d*g - 6*c*e*
f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(8*c*e) - (b*e - 2*c*d)**2*(b*
e*g + 4*c*d*g - 6*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))))/(16*c**(3/2)*e**2)

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Mathematica [C]  time = 1.11393, size = 231, normalized size = 0.83 \[ \frac{((d+e x) (c (d-e x)-b e))^{3/2} \left (-\frac{2 \sqrt{c} \left (3 b^2 e^2 g+2 b c e (-14 d g+15 e f+7 e g x)+4 c^2 \left (10 d^2 g-6 d e (2 f+g x)+e^2 x (3 f+2 g x)\right )\right )}{(d+e x) (c (d-e x)-b e)}-\frac{3 i (b e-2 c d)^2 (b e g+4 c d g-6 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{(d+e x)^{3/2} (c (d-e x)-b e)^{3/2}}\right )}{48 c^{3/2} e^2} \]

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)^2,x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*((-2*Sqrt[c]*(3*b^2*e^2*g + 2*b*c*e*(1
5*e*f - 14*d*g + 7*e*g*x) + 4*c^2*(10*d^2*g - 6*d*e*(2*f + g*x) + e^2*x*(3*f + 2
*g*x))))/((d + e*x)*(-(b*e) + c*(d - e*x))) - ((3*I)*(-2*c*d + b*e)^2*(-6*c*e*f
+ 4*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)]])/((d + e*x)^(3/2)*(-(b*e) + c*(d - e*x))^(3/2))))/(48*c^(3/2)*
e^2)

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Maple [B]  time = 0.025, size = 2174, normalized size = 7.8 \[ \text{result too large to display} \]

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)^2,x)

[Out]

1/3*g/e^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)-3/4*e^2/(-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)*f+3/8*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))*d*g+3*e*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*f+3/2*e*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*f+3*e*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))*f+1/2*g/e*d^3*c^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))+3/8*g*e*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
-3/4*g*b*c/(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^2-2/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)*d*g-3*c^3/(-b*e^2+2*c
*d*e)*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-1/16*g*e^2*b^3/c/(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))-2/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)*d*g+3/4*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)*d*g+1/2*g/e*d*c*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e
+x))^(1/2)*x-3/8*e^4/(-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
))*f-3*c^2/(-b*e^2+2*c*d*e)*d^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2
)*x*g-3/2*c/(-b*e^2+2*c*d*e)*d^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/
2)*b*g-3/2*e^2*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*f+9/2*e*c^2/(-b*e^2+2*c*d*e)*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+2*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)*f-1
/4*g*b*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x-1/8*g*b^2/c*(-c*(d/e+
x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)+2/e^2/(-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)*f+9/4*e^3*c/(-b*e^2+2*c*d*e)*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-9/4*e^2*c/(-b*e^2+2*c*d*e)*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^2*g-9/2*e^2*c^2/(-b*e^2+2*c*d*e)*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^2*f+3/2*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*d*g+1/4*g/e*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c
*d*e)*(d/e+x))^(1/2)*b

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.777662, size = 1, normalized size = 0. \[ \left [-\frac{4 \,{\left (8 \, c^{2} e^{2} g x^{2} - 6 \,{\left (8 \, c^{2} d e - 5 \, b c e^{2}\right )} f +{\left (40 \, c^{2} d^{2} - 28 \, b c d e + 3 \, b^{2} e^{2}\right )} g + 2 \,{\left (6 \, c^{2} e^{2} f -{\left (12 \, c^{2} d e - 7 \, b c e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{-c} + 3 \,{\left (6 \,{\left (4 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f -{\left (16 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + b^{3} e^{3}\right )} g\right )} \log \left (-4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c^{2} e x + b c e\right )} +{\left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} \sqrt{-c}\right )}{96 \, \sqrt{-c} c e^{2}}, -\frac{2 \,{\left (8 \, c^{2} e^{2} g x^{2} - 6 \,{\left (8 \, c^{2} d e - 5 \, b c e^{2}\right )} f +{\left (40 \, c^{2} d^{2} - 28 \, b c d e + 3 \, b^{2} e^{2}\right )} g + 2 \,{\left (6 \, c^{2} e^{2} f -{\left (12 \, c^{2} d e - 7 \, b c e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{c} - 3 \,{\left (6 \,{\left (4 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f -{\left (16 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + b^{3} e^{3}\right )} g\right )} \arctan \left (\frac{2 \, c e x + b e}{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{c}}\right )}{48 \, c^{\frac{3}{2}} e^{2}}\right ] \]

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)^2,x, algorithm="fricas")

[Out]

[-1/96*(4*(8*c^2*e^2*g*x^2 - 6*(8*c^2*d*e - 5*b*c*e^2)*f + (40*c^2*d^2 - 28*b*c*
d*e + 3*b^2*e^2)*g + 2*(6*c^2*e^2*f - (12*c^2*d*e - 7*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*(6*(4*c^3*d^2*e - 4*b*c^2*d*e^2 + b
^2*c*e^3)*f - (16*c^3*d^3 - 12*b*c^2*d^2*e + 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)))/(sqrt(-c)*c*e^2), -1/48*(2*(8*c^2*e^
2*g*x^2 - 6*(8*c^2*d*e - 5*b*c*e^2)*f + (40*c^2*d^2 - 28*b*c*d*e + 3*b^2*e^2)*g
+ 2*(6*c^2*e^2*f - (12*c^2*d*e - 7*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*(6*(4*c^3*d^2*e - 4*b*c^2*d*e^2 + b^2*c*e^3)*f - (16*c^
3*d^3 - 12*b*c^2*d^2*e + 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))))/(c^(3/2)*e^2)]

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

[Out]

Timed out

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

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)^2,x, algorithm="giac")

[Out]

Timed out