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

Optimal. Leaf size=60 \[ \frac{2 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}-\frac{2 x (b B-A c)}{b c \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0861874, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{2 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}-\frac{2 x (b B-A c)}{b c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.26774, size = 53, normalized size = 0.88 \[ \frac{2 B \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{c^{\frac{3}{2}}} + \frac{2 x \left (A c - B b\right )}{b c \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0715853, size = 79, normalized size = 1.32 \[ \frac{2 \sqrt{c} x (A c-b B)+2 b B \sqrt{x} \sqrt{b+c x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{b c^{3/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 67, normalized size = 1.1 \[ 2\,{\frac{Ax}{b\sqrt{c{x}^{2}+bx}}}-2\,{\frac{Bx}{c\sqrt{c{x}^{2}+bx}}}+{B\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)*x/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.306641, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{c x^{2} + b x} B b \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) - 2 \,{\left (B b - A c\right )} \sqrt{c} x}{\sqrt{c x^{2} + b x} b c^{\frac{3}{2}}}, \frac{2 \,{\left (\sqrt{c x^{2} + b x} B b \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (B b - A c\right )} \sqrt{-c} x\right )}}{\sqrt{c x^{2} + b x} b \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[(sqrt(c*x^2 + b*x)*B*b*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c) - 2*(B*
b - A*c)*sqrt(c)*x)/(sqrt(c*x^2 + b*x)*b*c^(3/2)), 2*(sqrt(c*x^2 + b*x)*B*b*arct
an(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (B*b - A*c)*sqrt(-c)*x)/(sqrt(c*x^2 + b*x
)*b*sqrt(-c)*c)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*(A + B*x)/(x*(b + c*x))**(3/2), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError