3.1762 \(\int \frac{(a+b x)^2}{(c+d x) (e+f x)^{5/2}} \, dx\)

Optimal. Leaf size=140 \[ -\frac{2 (b e-a f) (a d f-2 b c f+b d e)}{f^2 \sqrt{e+f x} (d e-c f)^2}+\frac{2 (b e-a f)^2}{3 f^2 (e+f x)^{3/2} (d e-c f)}-\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{\sqrt{d} (d e-c f)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.361286, antiderivative size = 140, 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 \[ -\frac{2 (b e-a f) (a d f-2 b c f+b d e)}{f^2 \sqrt{e+f x} (d e-c f)^2}+\frac{2 (b e-a f)^2}{3 f^2 (e+f x)^{3/2} (d e-c f)}-\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{\sqrt{d} (d e-c f)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 70.269, size = 126, normalized size = 0.9 \[ \frac{2 \left (a f - b e\right ) \left (a d f - 2 b c f + b d e\right )}{f^{2} \sqrt{e + f x} \left (c f - d e\right )^{2}} - \frac{2 \left (a f - b e\right )^{2}}{3 f^{2} \left (e + f x\right )^{\frac{3}{2}} \left (c f - d e\right )} + \frac{2 \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{\sqrt{d} \left (c f - d e\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.397636, size = 140, normalized size = 1. \[ -\frac{2 (b e-a f) (a d f-2 b c f+b d e)}{f^2 \sqrt{e+f x} (d e-c f)^2}-\frac{2 (b e-a f)^2}{3 f^2 (e+f x)^{3/2} (c f-d e)}-\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{\sqrt{d} (d e-c f)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.023, size = 332, normalized size = 2.4 \[ -{\frac{2\,{a}^{2}}{3\,cf-3\,de} \left ( fx+e \right ) ^{-{\frac{3}{2}}}}+{\frac{4\,abe}{3\, \left ( cf-de \right ) f} \left ( fx+e \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,{b}^{2}{e}^{2}}{3\,{f}^{2} \left ( cf-de \right ) } \left ( fx+e \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{{a}^{2}d}{ \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}-4\,{\frac{abc}{ \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}+4\,{\frac{ce{b}^{2}}{f \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}-2\,{\frac{{b}^{2}d{e}^{2}}{{f}^{2} \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}+2\,{\frac{{a}^{2}{d}^{2}}{ \left ( cf-de \right ) ^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-4\,{\frac{abcd}{ \left ( cf-de \right ) ^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{{b}^{2}{c}^{2}}{ \left ( cf-de \right ) ^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3/(c*f-d*e)/(f*x+e)^(3/2)*a^2+4/3/f/(c*f-d*e)/(f*x+e)^(3/2)*a*b*e-2/3/f^2/(c*
f-d*e)/(f*x+e)^(3/2)*b^2*e^2+2/(c*f-d*e)^2/(f*x+e)^(1/2)*a^2*d-4/(c*f-d*e)^2/(f*
x+e)^(1/2)*a*b*c+4/f/(c*f-d*e)^2/(f*x+e)^(1/2)*b^2*c*e-2/f^2/(c*f-d*e)^2/(f*x+e)
^(1/2)*b^2*d*e^2+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^2-4/(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*b*c*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))*b^2*c^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)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231493, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{3} x +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e f^{2}\right )} \sqrt{f x + e} \log \left (\frac{\sqrt{d^{2} e - c d f}{\left (d f x + 2 \, d e - c f\right )} - 2 \,{\left (d^{2} e - c d f\right )} \sqrt{f x + e}}{d x + c}\right ) - 2 \,{\left (2 \, b^{2} d e^{3} + a^{2} c f^{3} -{\left (5 \, b^{2} c - 2 \, a b d\right )} e^{2} f + 4 \,{\left (a b c - a^{2} d\right )} e f^{2} + 3 \,{\left (b^{2} d e^{2} f - 2 \, b^{2} c e f^{2} +{\left (2 \, a b c - a^{2} d\right )} f^{3}\right )} x\right )} \sqrt{d^{2} e - c d f}}{3 \,{\left (d^{2} e^{3} f^{2} - 2 \, c d e^{2} f^{3} + c^{2} e f^{4} +{\left (d^{2} e^{2} f^{3} - 2 \, c d e f^{4} + c^{2} f^{5}\right )} x\right )} \sqrt{d^{2} e - c d f} \sqrt{f x + e}}, -\frac{2 \,{\left (3 \,{\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{3} x +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e f^{2}\right )} \sqrt{f x + e} \arctan \left (-\frac{d e - c f}{\sqrt{-d^{2} e + c d f} \sqrt{f x + e}}\right ) +{\left (2 \, b^{2} d e^{3} + a^{2} c f^{3} -{\left (5 \, b^{2} c - 2 \, a b d\right )} e^{2} f + 4 \,{\left (a b c - a^{2} d\right )} e f^{2} + 3 \,{\left (b^{2} d e^{2} f - 2 \, b^{2} c e f^{2} +{\left (2 \, a b c - a^{2} d\right )} f^{3}\right )} x\right )} \sqrt{-d^{2} e + c d f}\right )}}{3 \,{\left (d^{2} e^{3} f^{2} - 2 \, c d e^{2} f^{3} + c^{2} e f^{4} +{\left (d^{2} e^{2} f^{3} - 2 \, c d e f^{4} + c^{2} f^{5}\right )} x\right )} \sqrt{-d^{2} e + c d f} \sqrt{f x + e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*(3*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^3*x + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)
*e*f^2)*sqrt(f*x + e)*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^3 + a^2*c*f^3 - (5*b^2*c - 2*a
*b*d)*e^2*f + 4*(a*b*c - a^2*d)*e*f^2 + 3*(b^2*d*e^2*f - 2*b^2*c*e*f^2 + (2*a*b*
c - a^2*d)*f^3)*x)*sqrt(d^2*e - c*d*f))/((d^2*e^3*f^2 - 2*c*d*e^2*f^3 + c^2*e*f^
4 + (d^2*e^2*f^3 - 2*c*d*e*f^4 + c^2*f^5)*x)*sqrt(d^2*e - c*d*f)*sqrt(f*x + e)),
-2/3*(3*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^3*x + (b^2*c^2 - 2*a*b*c*d + a^2*d^2
)*e*f^2)*sqrt(f*x + e)*arctan(-(d*e - c*f)/(sqrt(-d^2*e + c*d*f)*sqrt(f*x + e)))
+ (2*b^2*d*e^3 + a^2*c*f^3 - (5*b^2*c - 2*a*b*d)*e^2*f + 4*(a*b*c - a^2*d)*e*f^
2 + 3*(b^2*d*e^2*f - 2*b^2*c*e*f^2 + (2*a*b*c - a^2*d)*f^3)*x)*sqrt(-d^2*e + c*d
*f))/((d^2*e^3*f^2 - 2*c*d*e^2*f^3 + c^2*e*f^4 + (d^2*e^2*f^3 - 2*c*d*e*f^4 + c^
2*f^5)*x)*sqrt(-d^2*e + c*d*f)*sqrt(f*x + e))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220839, size = 319, normalized size = 2.28 \[ \frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{{\left (c^{2} f^{2} - 2 \, c d f e + d^{2} e^{2}\right )} \sqrt{c d f - d^{2} e}} - \frac{2 \,{\left (6 \,{\left (f x + e\right )} a b c f^{2} - 3 \,{\left (f x + e\right )} a^{2} d f^{2} + a^{2} c f^{3} - 6 \,{\left (f x + e\right )} b^{2} c f e - 2 \, a b c f^{2} e - a^{2} d f^{2} e + 3 \,{\left (f x + e\right )} b^{2} d e^{2} + b^{2} c f e^{2} + 2 \, a b d f e^{2} - b^{2} d e^{3}\right )}}{3 \,{\left (c^{2} f^{4} - 2 \, c d f^{3} e + d^{2} f^{2} e^{2}\right )}{\left (f x + e\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2/((d*x + c)*(f*x + e)^(5/2)),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^2*f^2 - 2*c*d*f*e + d^2*e^2)*sqrt(c*d*f - d^2*e)) - 2/3*(6*(f*x + e)*a*b*c*f^2
- 3*(f*x + e)*a^2*d*f^2 + a^2*c*f^3 - 6*(f*x + e)*b^2*c*f*e - 2*a*b*c*f^2*e - a
^2*d*f^2*e + 3*(f*x + e)*b^2*d*e^2 + b^2*c*f*e^2 + 2*a*b*d*f*e^2 - b^2*d*e^3)/((
c^2*f^4 - 2*c*d*f^3*e + d^2*f^2*e^2)*(f*x + e)^(3/2))