3.1754 \(\int \frac{a+b x}{(c+d x) (e+f x)^{5/2}} \, dx\)

Optimal. Leaf size=119 \[ -\frac{2 (b c-a d)}{\sqrt{e+f x} (d e-c f)^2}-\frac{2 (b e-a f)}{3 f (e+f x)^{3/2} (d e-c f)}+\frac{2 \sqrt{d} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{5/2}} \]

[Out]

(-2*(b*e - a*f))/(3*f*(d*e - c*f)*(e + f*x)^(3/2)) - (2*(b*c - a*d))/((d*e - c*f
)^2*Sqrt[e + f*x]) + (2*Sqrt[d]*(b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt
[d*e - c*f]])/(d*e - c*f)^(5/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.23188, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2 (b c-a d)}{\sqrt{e+f x} (d e-c f)^2}-\frac{2 (b e-a f)}{3 f (e+f x)^{3/2} (d e-c f)}+\frac{2 \sqrt{d} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/((c + d*x)*(e + f*x)^(5/2)),x]

[Out]

(-2*(b*e - a*f))/(3*f*(d*e - c*f)*(e + f*x)^(3/2)) - (2*(b*c - a*d))/((d*e - c*f
)^2*Sqrt[e + f*x]) + (2*Sqrt[d]*(b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt
[d*e - c*f]])/(d*e - c*f)^(5/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.785, size = 104, normalized size = 0.87 \[ \frac{2 \sqrt{d} \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{\left (c f - d e\right )^{\frac{5}{2}}} + \frac{2 \left (a d - b c\right )}{\sqrt{e + f x} \left (c f - d e\right )^{2}} - \frac{2 \left (a f - b e\right )}{3 f \left (e + f x\right )^{\frac{3}{2}} \left (c f - d e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(d*x+c)/(f*x+e)**(5/2),x)

[Out]

2*sqrt(d)*(a*d - b*c)*atan(sqrt(d)*sqrt(e + f*x)/sqrt(c*f - d*e))/(c*f - d*e)**(
5/2) + 2*(a*d - b*c)/(sqrt(e + f*x)*(c*f - d*e)**2) - 2*(a*f - b*e)/(3*f*(e + f*
x)**(3/2)*(c*f - d*e))

_______________________________________________________________________________________

Mathematica [A]  time = 0.234288, size = 116, normalized size = 0.97 \[ \frac{2 \sqrt{d} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{5/2}}-\frac{2 (3 f (e+f x) (b c-a d)+(b e-a f) (d e-c f))}{3 f (e+f x)^{3/2} (d e-c f)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/((c + d*x)*(e + f*x)^(5/2)),x]

[Out]

(-2*((b*e - a*f)*(d*e - c*f) + 3*(b*c - a*d)*f*(e + f*x)))/(3*f*(d*e - c*f)^2*(e
+ f*x)^(3/2)) + (2*Sqrt[d]*(b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e
- c*f]])/(d*e - c*f)^(5/2)

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 187, normalized size = 1.6 \[ -{\frac{2\,a}{3\,cf-3\,de} \left ( fx+e \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,be}{3\, \left ( cf-de \right ) f} \left ( fx+e \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{ad}{ \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}-2\,{\frac{bc}{ \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}+2\,{\frac{{d}^{2}a}{ \left ( cf-de \right ) ^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{bdc}{ \left ( cf-de \right ) ^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(d*x+c)/(f*x+e)^(5/2),x)

[Out]

-2/3/(c*f-d*e)/(f*x+e)^(3/2)*a+2/3/f/(c*f-d*e)/(f*x+e)^(3/2)*b*e+2/(c*f-d*e)^2/(
f*x+e)^(1/2)*a*d-2/(c*f-d*e)^2/(f*x+e)^(1/2)*b*c+2*d^2/(c*f-d*e)^2/((c*f-d*e)*d)
^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a-2*d/(c*f-d*e)^2/((c*f-d*e)*
d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b*c

_______________________________________________________________________________________

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)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.22321, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, b d e^{2} + 2 \, a c f^{2} + 6 \,{\left (b c - a d\right )} f^{2} x + 4 \,{\left (b c - 2 \, a d\right )} e f + 3 \,{\left ({\left (b c - a d\right )} f^{2} x +{\left (b c - a d\right )} e f\right )} \sqrt{f x + e} \sqrt{\frac{d}{d e - c f}} \log \left (\frac{d f x + 2 \, d e - c f - 2 \,{\left (d e - c f\right )} \sqrt{f x + e} \sqrt{\frac{d}{d e - c f}}}{d x + c}\right )}{3 \,{\left (d^{2} e^{3} f - 2 \, c d e^{2} f^{2} + c^{2} e f^{3} +{\left (d^{2} e^{2} f^{2} - 2 \, c d e f^{3} + c^{2} f^{4}\right )} x\right )} \sqrt{f x + e}}, -\frac{2 \,{\left (b d e^{2} + a c f^{2} + 3 \,{\left (b c - a d\right )} f^{2} x + 2 \,{\left (b c - 2 \, a d\right )} e f - 3 \,{\left ({\left (b c - a d\right )} f^{2} x +{\left (b c - a d\right )} e f\right )} \sqrt{f x + e} \sqrt{-\frac{d}{d e - c f}} \arctan \left (-\frac{{\left (d e - c f\right )} \sqrt{-\frac{d}{d e - c f}}}{\sqrt{f x + e} d}\right )\right )}}{3 \,{\left (d^{2} e^{3} f - 2 \, c d e^{2} f^{2} + c^{2} e f^{3} +{\left (d^{2} e^{2} f^{2} - 2 \, c d e f^{3} + c^{2} f^{4}\right )} x\right )} \sqrt{f x + e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((d*x + c)*(f*x + e)^(5/2)),x, algorithm="fricas")

[Out]

[-1/3*(2*b*d*e^2 + 2*a*c*f^2 + 6*(b*c - a*d)*f^2*x + 4*(b*c - 2*a*d)*e*f + 3*((b
*c - a*d)*f^2*x + (b*c - a*d)*e*f)*sqrt(f*x + e)*sqrt(d/(d*e - c*f))*log((d*f*x
+ 2*d*e - c*f - 2*(d*e - c*f)*sqrt(f*x + e)*sqrt(d/(d*e - c*f)))/(d*x + c)))/((d
^2*e^3*f - 2*c*d*e^2*f^2 + c^2*e*f^3 + (d^2*e^2*f^2 - 2*c*d*e*f^3 + c^2*f^4)*x)*
sqrt(f*x + e)), -2/3*(b*d*e^2 + a*c*f^2 + 3*(b*c - a*d)*f^2*x + 2*(b*c - 2*a*d)*
e*f - 3*((b*c - a*d)*f^2*x + (b*c - a*d)*e*f)*sqrt(f*x + e)*sqrt(-d/(d*e - c*f))
*arctan(-(d*e - c*f)*sqrt(-d/(d*e - c*f))/(sqrt(f*x + e)*d)))/((d^2*e^3*f - 2*c*
d*e^2*f^2 + c^2*e*f^3 + (d^2*e^2*f^2 - 2*c*d*e*f^3 + c^2*f^4)*x)*sqrt(f*x + e))]

_______________________________________________________________________________________

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((b*x+a)/(d*x+c)/(f*x+e)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.219759, size = 216, normalized size = 1.82 \[ -\frac{2 \,{\left (b c d - a d^{2}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{{\left (c^{2} f^{2} - 2 \, c d f e + d^{2} e^{2}\right )} \sqrt{c d f - d^{2} e}} - \frac{2 \,{\left (3 \,{\left (f x + e\right )} b c f - 3 \,{\left (f x + e\right )} a d f + a c f^{2} - b c f e - a d f e + b d e^{2}\right )}}{3 \,{\left (c^{2} f^{3} - 2 \, c d f^{2} e + d^{2} f e^{2}\right )}{\left (f x + e\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((d*x + c)*(f*x + e)^(5/2)),x, algorithm="giac")

[Out]

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