3.1753 \(\int \frac{a+b x}{(c+d x) (e+f x)^{3/2}} \, dx\)

Optimal. Leaf size=88 \[ \frac{2 (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{\sqrt{d} (d e-c f)^{3/2}}-\frac{2 (b e-a f)}{f \sqrt{e+f x} (d e-c f)} \]

[Out]

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

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Rubi [A]  time = 0.171358, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{2 (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{\sqrt{d} (d e-c f)^{3/2}}-\frac{2 (b e-a f)}{f \sqrt{e+f x} (d e-c f)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.9762, size = 76, normalized size = 0.86 \[ - \frac{2 \left (a f - b e\right )}{f \sqrt{e + f x} \left (c f - d e\right )} - \frac{2 \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{\sqrt{d} \left (c f - d e\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.35999, size = 88, normalized size = 1. \[ \frac{2 \left (\frac{(a d-b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{\sqrt{d} \sqrt{d e-c f}}+\frac{b e-a f}{f \sqrt{e+f x}}\right )}{c f-d e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 142, normalized size = 1.6 \[ -2\,{\frac{ad}{ \left ( cf-de \right ) \sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{bc}{ \left ( cf-de \right ) \sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{a}{ \left ( cf-de \right ) \sqrt{fx+e}}}+2\,{\frac{be}{ \left ( cf-de \right ) f\sqrt{fx+e}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(c*f-d*e)/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a*d
+2/(c*f-d*e)/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b*c
-2/(c*f-d*e)/(f*x+e)^(1/2)*a+2/f/(c*f-d*e)/(f*x+e)^(1/2)*b*e

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219316, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b c - a d\right )} \sqrt{f x + e} f \log \left (\frac{\sqrt{d^{2} e - c d f}{\left (d f x + 2 \, d e - c f\right )} + 2 \,{\left (d^{2} e - c d f\right )} \sqrt{f x + e}}{d x + c}\right ) - 2 \, \sqrt{d^{2} e - c d f}{\left (b e - a f\right )}}{\sqrt{d^{2} e - c d f}{\left (d e f - c f^{2}\right )} \sqrt{f x + e}}, \frac{2 \,{\left ({\left (b c - a d\right )} \sqrt{f x + e} f \arctan \left (-\frac{d e - c f}{\sqrt{-d^{2} e + c d f} \sqrt{f x + e}}\right ) - \sqrt{-d^{2} e + c d f}{\left (b e - a f\right )}\right )}}{\sqrt{-d^{2} e + c d f}{\left (d e f - c f^{2}\right )} \sqrt{f x + e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.215174, size = 127, normalized size = 1.44 \[ \frac{2 \,{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e}{\left (c f - d e\right )}} - \frac{2 \,{\left (a f - b e\right )}}{{\left (c f^{2} - d f e\right )} \sqrt{f x + e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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