3.173 \(\int \frac{A+B x}{x^{3/2} \left (b x+c x^2\right )} \, dx\)

Optimal. Leaf size=69 \[ -\frac{2 \sqrt{c} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}-\frac{2 (b B-A c)}{b^2 \sqrt{x}}-\frac{2 A}{3 b x^{3/2}} \]

[Out]

(-2*A)/(3*b*x^(3/2)) - (2*(b*B - A*c))/(b^2*Sqrt[x]) - (2*Sqrt[c]*(b*B - A*c)*Ar
cTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/b^(5/2)

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Rubi [A]  time = 0.104422, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{2 \sqrt{c} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}-\frac{2 (b B-A c)}{b^2 \sqrt{x}}-\frac{2 A}{3 b x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(3/2)*(b*x + c*x^2)),x]

[Out]

(-2*A)/(3*b*x^(3/2)) - (2*(b*B - A*c))/(b^2*Sqrt[x]) - (2*Sqrt[c]*(b*B - A*c)*Ar
cTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/b^(5/2)

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Rubi in Sympy [A]  time = 12.6528, size = 65, normalized size = 0.94 \[ - \frac{2 A}{3 b x^{\frac{3}{2}}} + \frac{2 \left (A c - B b\right )}{b^{2} \sqrt{x}} + \frac{2 \sqrt{c} \left (A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(3/2)/(c*x**2+b*x),x)

[Out]

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

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Mathematica [A]  time = 0.12429, size = 64, normalized size = 0.93 \[ \frac{2 \sqrt{c} (A c-b B) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}-\frac{2 (A (b-3 c x)+3 b B x)}{3 b^2 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(3/2)*(b*x + c*x^2)),x]

[Out]

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

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Maple [A]  time = 0.016, size = 78, normalized size = 1.1 \[ -{\frac{2\,A}{3\,b}{x}^{-{\frac{3}{2}}}}+2\,{\frac{Ac}{\sqrt{x}{b}^{2}}}-2\,{\frac{B}{b\sqrt{x}}}+2\,{\frac{A{c}^{2}}{{b}^{2}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) }-2\,{\frac{Bc}{b\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(3/2)/(c*x^2+b*x),x)

[Out]

-2/3*A/b/x^(3/2)+2/x^(1/2)/b^2*A*c-2/x^(1/2)/b*B+2*c^2/b^2/(b*c)^(1/2)*arctan(c*
x^(1/2)/(b*c)^(1/2))*A-2*c/b/(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^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.299279, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (B b - A c\right )} x^{\frac{3}{2}} \sqrt{-\frac{c}{b}} \log \left (\frac{c x + 2 \, b \sqrt{x} \sqrt{-\frac{c}{b}} - b}{c x + b}\right ) + 2 \, A b + 6 \,{\left (B b - A c\right )} x}{3 \, b^{2} x^{\frac{3}{2}}}, \frac{2 \,{\left (3 \,{\left (B b - A c\right )} x^{\frac{3}{2}} \sqrt{\frac{c}{b}} \arctan \left (\frac{b \sqrt{\frac{c}{b}}}{c \sqrt{x}}\right ) - A b - 3 \,{\left (B b - A c\right )} x\right )}}{3 \, b^{2} x^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x)*x^(3/2)),x, algorithm="fricas")

[Out]

[-1/3*(3*(B*b - A*c)*x^(3/2)*sqrt(-c/b)*log((c*x + 2*b*sqrt(x)*sqrt(-c/b) - b)/(
c*x + b)) + 2*A*b + 6*(B*b - A*c)*x)/(b^2*x^(3/2)), 2/3*(3*(B*b - A*c)*x^(3/2)*s
qrt(c/b)*arctan(b*sqrt(c/b)/(c*sqrt(x))) - A*b - 3*(B*b - A*c)*x)/(b^2*x^(3/2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{x^{\frac{5}{2}} \left (b + c x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**(3/2)/(c*x**2+b*x),x)

[Out]

Integral((A + B*x)/(x**(5/2)*(b + c*x)), x)

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GIAC/XCAS [A]  time = 0.271203, size = 74, normalized size = 1.07 \[ -\frac{2 \,{\left (B b c - A c^{2}\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} b^{2}} - \frac{2 \,{\left (3 \, B b x - 3 \, A c x + A b\right )}}{3 \, b^{2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x)*x^(3/2)),x, algorithm="giac")

[Out]

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