3.2271 \(\int \frac{(d+e x)^{3/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=148 \[ \frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+3 c d g+c e f)}{c^2 e^2 \sqrt{d+e x} (2 c d-b e)}+\frac{2 (d+e x)^{3/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.444631, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+3 c d g+c e f)}{c^2 e^2 \sqrt{d+e x} (2 c d-b e)}+\frac{2 (d+e x)^{3/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 55.2451, size = 136, normalized size = 0.92 \[ \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (b e g - c d g - c e f\right )}{c e^{2} \left (b e - 2 c d\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} + \frac{2 \left (2 b e g - 3 c d g - c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{c^{2} e^{2} \sqrt{d + e x} \left (b e - 2 c d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0867848, size = 60, normalized size = 0.41 \[ \frac{2 \sqrt{d+e x} (-2 b e g+2 c d g+c e (f-g x))}{c^2 e^2 \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 78, normalized size = 0.5 \[ 2\,{\frac{ \left ( cex+be-cd \right ) \left ( cegx+2\,beg-2\,cdg-fce \right ) \left ( ex+d \right ) ^{3/2}}{{c}^{2}{e}^{2} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.728315, size = 84, normalized size = 0.57 \[ \frac{2 \, f}{\sqrt{-c e x + c d - b e} c e} - \frac{2 \,{\left (c e x - 2 \, c d + 2 \, b e\right )} g}{\sqrt{-c e x + c d - b e} c^{2} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.272969, size = 131, normalized size = 0.89 \[ -\frac{2 \,{\left (c e^{2} g x^{2} - c d e f - 2 \,{\left (c d^{2} - b d e\right )} g -{\left (c e^{2} f +{\left (c d e - 2 \, b e^{2}\right )} g\right )} x\right )}}{\sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} c^{2} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.624514, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x