3.2189 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^4} \, dx\)
Optimal. Leaf size=276 \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (d+e x)^4 (2 c d-b e)}+\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (3 b e g-8 c d g+2 c e f)}{3 e^2 (d+e x)^2 (2 c d-b e)}+\frac{c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (3 b e g-8 c d g+2 c e f)}{e^2 (2 c d-b e)}+\frac{\sqrt{c} (3 b e g-8 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 e^2} \]
[Out]
(c*(2*c*e*f - 8*c*d*g + 3*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2
*(2*c*d - b*e)) + (2*(2*c*e*f - 8*c*d*g + 3*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)*(d + e*x)^2) - (2*(e*f - d*g)*(d*(c*d - b*e
) - b*e^2*x - c*e^2*x^2)^(5/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^4) + (Sqrt[c]*(2*
c*e*f - 8*c*d*g + 3*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])])/(2*e^2)
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Rubi [A] time = 0.950973, antiderivative size = 276, 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
\[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (d+e x)^4 (2 c d-b e)}+\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (3 b e g-8 c d g+2 c e f)}{3 e^2 (d+e x)^2 (2 c d-b e)}+\frac{c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (3 b e g-8 c d g+2 c e f)}{e^2 (2 c d-b e)}+\frac{\sqrt{c} (3 b e g-8 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 e^2} \]
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)^4,x]
[Out]
(c*(2*c*e*f - 8*c*d*g + 3*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2
*(2*c*d - b*e)) + (2*(2*c*e*f - 8*c*d*g + 3*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)*(d + e*x)^2) - (2*(e*f - d*g)*(d*(c*d - b*e
) - b*e^2*x - c*e^2*x^2)^(5/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^4) + (Sqrt[c]*(2*
c*e*f - 8*c*d*g + 3*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])])/(2*e^2)
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Rubi in Sympy [A] time = 99.6822, size = 265, normalized size = 0.96 \[ \frac{2 \sqrt{c} \left (\frac{3 b e g}{4} - 2 c d g + \frac{c e f}{2}\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{e^{2}} - \frac{c \left (3 b e g - 8 c d g + 2 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{e^{2} \left (b e - 2 c d\right )} - \frac{4 \left (\frac{3 b e g}{2} - 4 c d g + c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{3 e^{2} \left (d + e x\right )^{2} \left (b e - 2 c d\right )} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{3 e^{2} \left (d + e x\right )^{4} \left (b e - 2 c d\right )} \]
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)**4,x)
[Out]
2*sqrt(c)*(3*b*e*g/4 - 2*c*d*g + c*e*f/2)*atan(-e*(-b - 2*c*x)/(2*sqrt(c)*sqrt(-
b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))))/e**2 - c*(3*b*e*g - 8*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 - 2*c*d)) - 4*(3*b*e
*g/2 - 4*c*d*g + c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(3*e**
2*(d + e*x)**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)/(3*e**2*(d + e*x)**4*(b*e - 2*c*d))
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Mathematica [C] time = 0.641418, size = 207, normalized size = 0.75 \[ \frac{i ((d+e x) (c (d-e x)-b e))^{3/2} \left (2 i \sqrt{c (d-e x)-b e} \left (c \left (19 d^2 g+d e (26 g x-4 f)+e^2 x (3 g x-8 f)\right )-2 b e (2 d g+e (f+3 g x))\right )+3 \sqrt{c} (d+e x)^{3/2} (3 b e g-8 c d g+2 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )\right )}{6 e^2 (d+e x)^3 (c (d-e x)-b e)^{3/2}} \]
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)^4,x]
[Out]
((I/6)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*((2*I)*Sqrt[-(b*e) + c*(d - e*x)
]*(-2*b*e*(2*d*g + e*(f + 3*g*x)) + c*(19*d^2*g + e^2*x*(-8*f + 3*g*x) + d*e*(-4
*f + 26*g*x))) + 3*Sqrt[c]*(2*c*e*f - 8*c*d*g + 3*b*e*g)*(d + e*x)^(3/2)*Log[((-
I)*e*(b + 2*c*x))/Sqrt[c] + 2*Sqrt[d + e*x]*Sqrt[-(b*e) + c*(d - e*x)]]))/(e^2*(
d + e*x)^3*(-(b*e) + c*(d - e*x))^(3/2))
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Maple [B] time = 0.026, size = 2773, normalized size = 10.1 \[ \text{result too large to display} \]
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)^4,x)
[Out]
16/3*c^2/(-b*e^2+2*c*d*e)^3/(d/e+x)^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x)
)^(5/2)*f+16/3*e^2*c^3/(-b*e^2+2*c*d*e)^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/
e+x))^(3/2)*f-8*g*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)-8*g/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)+3*g*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)-16/3*e*c^3/(-b*e^2+2*c*d*e)^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2
*c*d*e)*(d/e+x))^(3/2)*d*g-2*e^4*c^2/(-b*e^2+2*c*d*e)^3*b^2*(-c*(d/e+x)^2*e^2+(-
b*e^2+2*c*d*e)*(d/e+x))^(1/2)*f+2/3/e^5/(-b*e^2+2*c*d*e)/(d/e+x)^4*(-c*(d/e+x)^2
*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(5/2)*d*g+4/3/e^2*c/(-b*e^2+2*c*d*e)^2/(d/e+x)^3*
(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(5/2)*f-12*g*e*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))-16/3/e*c^2/(-b*e^2+2*c*d*e)^3/(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-4*e^4*c^3/(-b*e^2+2
*c*d*e)^3*b*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x*f+2*e^3*c^2/(-b*
e^2+2*c*d*e)^3*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d*g-e^6*c^2
/(-b*e^2+2*c*d*e)^3*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-8*e^2*c^4/(-b
*e^2+2*c*d*e)^3*d^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x*g+8*e^3*
c^4/(-b*e^2+2*c*d*e)^3*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x*f-4
*e^2*c^3/(-b*e^2+2*c*d*e)^3*d^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2
)*b*g+4*e^3*c^3/(-b*e^2+2*c*d*e)^3*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^5/(-b*e^2+2*c*d*e)^3*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+8*e^3*c^5/(-b*e^2+2*c*d*e)^3*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^4*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))-1
2*g*e*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-6*g*e*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+6*g*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-4/3/e^3*c/(-b*e^2+2*c*d*e)^2/(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-2*g/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)-12*e^4*c^4/(-b*e^2+2*c*d*e)^3*b/(c*e^2)^(1/2)*arcta
n((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+6*e^5*c^3/(-b*e^2+2*c*d*e)^3*b^2/(c*e^2)^(1/2)*arcta
n((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*g*e^3*c^2/(-b*e^2+2*c*d*e)^2*b^2/(c*e^2)^(1/2)*arcta
n((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*g*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^2-2/3/e^4/(-b*e^2+2*c*d*e)/(d/e+x)^4*(-c*(d/e+x)^2*e^2+(-b*
e^2+2*c*d*e)*(d/e+x))^(5/2)*f+4*e^3*c^3/(-b*e^2+2*c*d*e)^3*b*(-c*(d/e+x)^2*e^2+(
-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x*d*g+e^5*c^2/(-b*e^2+2*c*d*e)^3*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-6*e^4*c^3/(-b*e^2+2*c*d*e)^3*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+12*e^3*c^4/(-b*e^2+2*c*d*e)^3*b/(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^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)^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 1.11754, size = 1, normalized size = 0. \[ \left [\frac{3 \,{\left (2 \, c d^{2} e f +{\left (2 \, c e^{3} f -{\left (8 \, c d e^{2} - 3 \, b e^{3}\right )} g\right )} x^{2} -{\left (8 \, c d^{3} - 3 \, b d^{2} e\right )} g + 2 \,{\left (2 \, c d e^{2} f -{\left (8 \, c d^{2} e - 3 \, b d e^{2}\right )} g\right )} x\right )} \sqrt{-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} + 4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c e x + b e\right )} \sqrt{-c}\right ) - 4 \,{\left (3 \, c e^{2} g x^{2} - 2 \,{\left (2 \, c d e + b e^{2}\right )} f +{\left (19 \, c d^{2} - 4 \, b d e\right )} g - 2 \,{\left (4 \, c e^{2} f -{\left (13 \, c d e - 3 \, b e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{12 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, \frac{3 \,{\left (2 \, c d^{2} e f +{\left (2 \, c e^{3} f -{\left (8 \, c d e^{2} - 3 \, b e^{3}\right )} g\right )} x^{2} -{\left (8 \, c d^{3} - 3 \, b d^{2} e\right )} g + 2 \,{\left (2 \, c d e^{2} f -{\left (8 \, c d^{2} e - 3 \, b d e^{2}\right )} g\right )} x\right )} \sqrt{c} \arctan \left (\frac{2 \, c e x + b e}{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{c}}\right ) - 2 \,{\left (3 \, c e^{2} g x^{2} - 2 \,{\left (2 \, c d e + b e^{2}\right )} f +{\left (19 \, c d^{2} - 4 \, b d e\right )} g - 2 \,{\left (4 \, c e^{2} f -{\left (13 \, c d e - 3 \, b e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{6 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}\right ] \]
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)^4,x, algorithm="fricas")
[Out]
[1/12*(3*(2*c*d^2*e*f + (2*c*e^3*f - (8*c*d*e^2 - 3*b*e^3)*g)*x^2 - (8*c*d^3 - 3
*b*d^2*e)*g + 2*(2*c*d*e^2*f - (8*c*d^2*e - 3*b*d*e^2)*g)*x)*sqrt(-c)*log(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 + 4*sqrt(-c*e^2*x^2 - b*
e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) - 4*(3*c*e^2*g*x^2 - 2*(2*c*d*e
+ b*e^2)*f + (19*c*d^2 - 4*b*d*e)*g - 2*(4*c*e^2*f - (13*c*d*e - 3*b*e^2)*g)*x)
*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(e^4*x^2 + 2*d*e^3*x + d^2*e^2), 1/
6*(3*(2*c*d^2*e*f + (2*c*e^3*f - (8*c*d*e^2 - 3*b*e^3)*g)*x^2 - (8*c*d^3 - 3*b*d
^2*e)*g + 2*(2*c*d*e^2*f - (8*c*d^2*e - 3*b*d*e^2)*g)*x)*sqrt(c)*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*(3*c*e^2*g
*x^2 - 2*(2*c*d*e + b*e^2)*f + (19*c*d^2 - 4*b*d*e)*g - 2*(4*c*e^2*f - (13*c*d*e
- 3*b*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(e^4*x^2 + 2*d*e^3
*x + d^2*e^2)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{4}}\, dx \]
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)**4,x)
[Out]
Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(f + g*x)/(d + e*x)**4, x)
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GIAC/XCAS [A] time = 0.923853, 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)^4,x, algorithm="giac")
[Out]
sage0*x