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

Optimal. Leaf size=133 \[ -\frac{2 \sqrt{d+e x}}{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}-\frac{2 \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{(c d f-a e g)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.60469, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{2 \sqrt{d+e x}}{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}-\frac{2 \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{(c d f-a e g)^{3/2}} \]

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

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Rubi in Sympy [A]  time = 57.6079, size = 126, normalized size = 0.95 \[ - \frac{2 \sqrt{g} \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e g - c d f}} \right )}}{\left (a e g - c d f\right )^{\frac{3}{2}}} + \frac{2 \sqrt{d + e x}}{\left (a e g - c d f\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \]

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

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Mathematica [A]  time = 0.171734, size = 110, normalized size = 0.83 \[ \frac{2 \sqrt{d+e x} \left (\sqrt{a e g-c d f}-\sqrt{g} \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{a e g-c d f}}\right )\right )}{\sqrt{(d+e x) (a e+c d x)} (a e g-c d f)^{3/2}} \]

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]*(Sqrt[-(c*d*f) + a*e*g] - Sqrt[g]*Sqrt[a*e + c*d*x]*ArcTanh[(Sq
rt[g]*Sqrt[a*e + c*d*x])/Sqrt[-(c*d*f) + a*e*g]]))/((-(c*d*f) + a*e*g)^(3/2)*Sqr
t[(a*e + c*d*x)*(d + e*x)])

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Maple [A]  time = 0.03, size = 128, normalized size = 1. \[ -2\,{\frac{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}{\sqrt{ex+d} \left ( cdx+ae \right ) \left ( aeg-cdf \right ) \sqrt{ \left ( aeg-cdf \right ) g}} \left ( g{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) \sqrt{cdx+ae}-\sqrt{ \left ( aeg-cdf \right ) g} \right ) } \]

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*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(g*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*
g-c*d*f)*g)^(1/2))*(c*d*x+a*e)^(1/2)-((a*e*g-c*d*f)*g)^(1/2))/(e*x+d)^(1/2)/(c*d
*x+a*e)/(a*e*g-c*d*f)/((a*e*g-c*d*f)*g)^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284208, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{-\frac{g}{c d f - a e g}} \log \left (-\frac{c d e g x^{2} - c d^{2} f + 2 \, a d e g + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d f - a e g\right )} \sqrt{e x + d} \sqrt{-\frac{g}{c d f - a e g}} -{\left (c d e f -{\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x}{e g x^{2} + d f +{\left (e f + d g\right )} x}\right ) + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{a c d^{2} e f - a^{2} d e^{2} g +{\left (c^{2} d^{2} e f - a c d e^{2} g\right )} x^{2} +{\left ({\left (c^{2} d^{3} + a c d e^{2}\right )} f -{\left (a c d^{2} e + a^{2} e^{3}\right )} g\right )} x}, \frac{2 \,{\left ({\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{\frac{g}{c d f - a e g}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{g}{c d f - a e g}}}\right ) - \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}\right )}}{a c d^{2} e f - a^{2} d e^{2} g +{\left (c^{2} d^{2} e f - a c d e^{2} g\right )} x^{2} +{\left ({\left (c^{2} d^{3} + a c d e^{2}\right )} f -{\left (a c d^{2} e + a^{2} e^{3}\right )} g\right )} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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 = 0.781274, 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)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(g*x + f)),x, algorithm="giac")

[Out]

sage0*x