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

Optimal. Leaf size=150 \[ -\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (d g+e f)\right )}{c^2 d^2 \sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{2 (d+e x)^{3/2} (c d f-a e g)}{c d \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

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

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Rubi [A]  time = 0.429481, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (d g+e f)\right )}{c^2 d^2 \sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{2 (d+e x)^{3/2} (c d f-a e g)}{c d \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 46.1415, size = 139, normalized size = 0.93 \[ - \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e g - c d f\right )}{c d \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{2 \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (2 a e^{2} g - c d^{2} g - c d e f\right )}{c^{2} d^{2} \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0642961, size = 51, normalized size = 0.34 \[ \frac{2 \sqrt{d+e x} (2 a e g+c d (g x-f))}{c^2 d^2 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 66, normalized size = 0.4 \[ 2\,{\frac{ \left ( cdx+ae \right ) \left ( xcdg+2\,aeg-cdf \right ) \left ( ex+d \right ) ^{3/2}}{{c}^{2}{d}^{2} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.763476, size = 65, normalized size = 0.43 \[ -\frac{2 \, f}{\sqrt{c d x + a e} c d} + \frac{2 \,{\left (c d x + 2 \, a e\right )} g}{\sqrt{c d x + a e} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.269545, size = 119, normalized size = 0.79 \[ \frac{2 \,{\left (c d e g x^{2} - c d^{2} f + 2 \, a d e g -{\left (c d e f -{\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x\right )}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(c*d*e*g*x^2 - c*d^2*f + 2*a*d*e*g - (c*d*e*f - (c*d^2 + 2*a*e^2)*g)*x)/(sqrt(
c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*c^2*d^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((e*x+d)**(3/2)*(g*x+f)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.654246, 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*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2),x, algorithm="giac")

[Out]

sage0*x