3.1761 \(\int \frac{(a+b x)^2}{(c+d x) (e+f x)^{3/2}} \, dx\)

Optimal. Leaf size=112 \[ -\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{3/2} (d e-c f)^{3/2}}+\frac{2 (b e-a f)^2}{f^2 \sqrt{e+f x} (d e-c f)}+\frac{2 b^2 \sqrt{e+f x}}{d f^2} \]

[Out]

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

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Rubi [A]  time = 0.280463, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{3/2} (d e-c f)^{3/2}}+\frac{2 (b e-a f)^2}{f^2 \sqrt{e+f x} (d e-c f)}+\frac{2 b^2 \sqrt{e+f x}}{d f^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 58.9068, size = 153, normalized size = 1.37 \[ - \frac{2 \sqrt{e + f x} \left (a f - b e\right ) \left (a d f - 2 b c f + b d e\right )}{f^{2} \left (c f - d e\right )^{2}} - \frac{2 \left (a f - b e\right )^{2}}{f^{2} \sqrt{e + f x} \left (c f - d e\right )} + \frac{2 \sqrt{e + f x} \left (a d - b c\right )^{2}}{d \left (c f - d e\right )^{2}} - \frac{2 \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{3}{2}} \left (c f - d e\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.354946, size = 108, normalized size = 0.96 \[ \frac{2 \sqrt{e+f x} \left (\frac{(b e-a f)^2}{(e+f x) (d e-c f)}+\frac{b^2}{d}\right )}{f^2}-\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{3/2} (d e-c f)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.02, size = 249, normalized size = 2.2 \[ 2\,{\frac{{b}^{2}\sqrt{fx+e}}{d{f}^{2}}}-2\,{\frac{d{a}^{2}}{ \left ( cf-de \right ) \sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+4\,{\frac{abc}{ \left ( cf-de \right ) \sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{{b}^{2}{c}^{2}}{ \left ( cf-de \right ) d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{{a}^{2}}{ \left ( cf-de \right ) \sqrt{fx+e}}}+4\,{\frac{abe}{ \left ( cf-de \right ) f\sqrt{fx+e}}}-2\,{\frac{{b}^{2}{e}^{2}}{{f}^{2} \left ( cf-de \right ) \sqrt{fx+e}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b^2*(f*x+e)^(1/2)/d/f^2-2/(c*f-d*e)*d/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)
*d/((c*f-d*e)*d)^(1/2))*a^2+4/(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*b*c-2/(c*f-d*e)/d/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(
1/2)*d/((c*f-d*e)*d)^(1/2))*b^2*c^2-2/(c*f-d*e)/(f*x+e)^(1/2)*a^2+4/f/(c*f-d*e)/
(f*x+e)^(1/2)*a*b*e-2/f^2/(c*f-d*e)/(f*x+e)^(1/2)*b^2*e^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((b*x + a)^2/((d*x + c)*(f*x + e)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220759, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{f x + e} f^{2} \log \left (\frac{\sqrt{d^{2} e - c d f}{\left (d f x + 2 \, d e - c f\right )} + 2 \,{\left (d^{2} e - c d f\right )} \sqrt{f x + e}}{d x + c}\right ) - 2 \,{\left (2 \, b^{2} d e^{2} + a^{2} d f^{2} -{\left (b^{2} c + 2 \, a b d\right )} e f +{\left (b^{2} d e f - b^{2} c f^{2}\right )} x\right )} \sqrt{d^{2} e - c d f}}{{\left (d^{2} e f^{2} - c d f^{3}\right )} \sqrt{d^{2} e - c d f} \sqrt{f x + e}}, -\frac{2 \,{\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{f x + e} f^{2} \arctan \left (-\frac{d e - c f}{\sqrt{-d^{2} e + c d f} \sqrt{f x + e}}\right ) -{\left (2 \, b^{2} d e^{2} + a^{2} d f^{2} -{\left (b^{2} c + 2 \, a b d\right )} e f +{\left (b^{2} d e f - b^{2} c f^{2}\right )} x\right )} \sqrt{-d^{2} e + c d f}\right )}}{{\left (d^{2} e f^{2} - c d f^{3}\right )} \sqrt{-d^{2} e + c d f} \sqrt{f x + e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.21739, size = 174, normalized size = 1.55 \[ -\frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{{\left (c d f - d^{2} e\right )}^{\frac{3}{2}}} - \frac{2 \,{\left (a^{2} f^{2} - 2 \, a b f e + b^{2} e^{2}\right )}}{{\left (c f^{3} - d f^{2} e\right )} \sqrt{f x + e}} + \frac{2 \, \sqrt{f x + e} b^{2}}{d f^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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