3.174 \(\int \frac{A+B x}{x^{5/2} \left (b x+c x^2\right )} \, dx\)

Optimal. Leaf size=90 \[ \frac{2 c^{3/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{7/2}}+\frac{2 c (b B-A c)}{b^3 \sqrt{x}}-\frac{2 (b B-A c)}{3 b^2 x^{3/2}}-\frac{2 A}{5 b x^{5/2}} \]

[Out]

(-2*A)/(5*b*x^(5/2)) - (2*(b*B - A*c))/(3*b^2*x^(3/2)) + (2*c*(b*B - A*c))/(b^3*
Sqrt[x]) + (2*c^(3/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/b^(7/2)

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Rubi [A]  time = 0.128155, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{2 c^{3/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{7/2}}+\frac{2 c (b B-A c)}{b^3 \sqrt{x}}-\frac{2 (b B-A c)}{3 b^2 x^{3/2}}-\frac{2 A}{5 b x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(5/2)*(b*x + c*x^2)),x]

[Out]

(-2*A)/(5*b*x^(5/2)) - (2*(b*B - A*c))/(3*b^2*x^(3/2)) + (2*c*(b*B - A*c))/(b^3*
Sqrt[x]) + (2*c^(3/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/b^(7/2)

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Rubi in Sympy [A]  time = 15.9893, size = 85, normalized size = 0.94 \[ - \frac{2 A}{5 b x^{\frac{5}{2}}} + \frac{2 \left (A c - B b\right )}{3 b^{2} x^{\frac{3}{2}}} - \frac{2 c \left (A c - B b\right )}{b^{3} \sqrt{x}} - \frac{2 c^{\frac{3}{2}} \left (A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(5/2)/(c*x**2+b*x),x)

[Out]

-2*A/(5*b*x**(5/2)) + 2*(A*c - B*b)/(3*b**2*x**(3/2)) - 2*c*(A*c - B*b)/(b**3*sq
rt(x)) - 2*c**(3/2)*(A*c - B*b)*atan(sqrt(c)*sqrt(x)/sqrt(b))/b**(7/2)

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Mathematica [A]  time = 0.143349, size = 83, normalized size = 0.92 \[ \frac{2 c^{3/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{7/2}}-\frac{2 \left (A \left (3 b^2-5 b c x+15 c^2 x^2\right )+5 b B x (b-3 c x)\right )}{15 b^3 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(5/2)*(b*x + c*x^2)),x]

[Out]

(-2*(5*b*B*x*(b - 3*c*x) + A*(3*b^2 - 5*b*c*x + 15*c^2*x^2)))/(15*b^3*x^(5/2)) +
(2*c^(3/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/b^(7/2)

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Maple [A]  time = 0.017, size = 102, normalized size = 1.1 \[ -{\frac{2\,A}{5\,b}{x}^{-{\frac{5}{2}}}}+{\frac{2\,Ac}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}}-{\frac{2\,B}{3\,b}{x}^{-{\frac{3}{2}}}}-2\,{\frac{A{c}^{2}}{{b}^{3}\sqrt{x}}}+2\,{\frac{Bc}{{b}^{2}\sqrt{x}}}-2\,{\frac{A{c}^{3}}{{b}^{3}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) }+2\,{\frac{B{c}^{2}}{{b}^{2}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(5/2)/(c*x^2+b*x),x)

[Out]

-2/5*A/b/x^(5/2)+2/3/x^(3/2)/b^2*A*c-2/3/x^(3/2)/b*B-2/b^3*c^2/x^(1/2)*A+2/b^2*c
/x^(1/2)*B-2*c^3/b^3/(b*c)^(1/2)*arctan(c*x^(1/2)/(b*c)^(1/2))*A+2*c^2/b^2/(b*c)
^(1/2)*arctan(c*x^(1/2)/(b*c)^(1/2))*B

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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)/((c*x^2 + b*x)*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.288849, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (B b c - A c^{2}\right )} x^{\frac{5}{2}} \sqrt{-\frac{c}{b}} \log \left (\frac{c x - 2 \, b \sqrt{x} \sqrt{-\frac{c}{b}} - b}{c x + b}\right ) + 6 \, A b^{2} - 30 \,{\left (B b c - A c^{2}\right )} x^{2} + 10 \,{\left (B b^{2} - A b c\right )} x}{15 \, b^{3} x^{\frac{5}{2}}}, -\frac{2 \,{\left (15 \,{\left (B b c - A c^{2}\right )} x^{\frac{5}{2}} \sqrt{\frac{c}{b}} \arctan \left (\frac{b \sqrt{\frac{c}{b}}}{c \sqrt{x}}\right ) + 3 \, A b^{2} - 15 \,{\left (B b c - A c^{2}\right )} x^{2} + 5 \,{\left (B b^{2} - A b c\right )} x\right )}}{15 \, b^{3} x^{\frac{5}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x)*x^(5/2)),x, algorithm="fricas")

[Out]

[-1/15*(15*(B*b*c - A*c^2)*x^(5/2)*sqrt(-c/b)*log((c*x - 2*b*sqrt(x)*sqrt(-c/b)
- b)/(c*x + b)) + 6*A*b^2 - 30*(B*b*c - A*c^2)*x^2 + 10*(B*b^2 - A*b*c)*x)/(b^3*
x^(5/2)), -2/15*(15*(B*b*c - A*c^2)*x^(5/2)*sqrt(c/b)*arctan(b*sqrt(c/b)/(c*sqrt
(x))) + 3*A*b^2 - 15*(B*b*c - A*c^2)*x^2 + 5*(B*b^2 - A*b*c)*x)/(b^3*x^(5/2))]

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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)/x**(5/2)/(c*x**2+b*x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.270702, size = 108, normalized size = 1.2 \[ \frac{2 \,{\left (B b c^{2} - A c^{3}\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} b^{3}} + \frac{2 \,{\left (15 \, B b c x^{2} - 15 \, A c^{2} x^{2} - 5 \, B b^{2} x + 5 \, A b c x - 3 \, A b^{2}\right )}}{15 \, b^{3} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x)*x^(5/2)),x, algorithm="giac")

[Out]

2*(B*b*c^2 - A*c^3)*arctan(c*sqrt(x)/sqrt(b*c))/(sqrt(b*c)*b^3) + 2/15*(15*B*b*c
*x^2 - 15*A*c^2*x^2 - 5*B*b^2*x + 5*A*b*c*x - 3*A*b^2)/(b^3*x^(5/2))