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

Optimal. Leaf size=99 \[ -\frac{(3 b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{5/2}}+\frac{\sqrt{b x+c x^2} (3 b B-2 A c)}{b c^2}-\frac{2 x^2 (b B-A c)}{b c \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.219193, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{(3 b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{5/2}}+\frac{\sqrt{b x+c x^2} (3 b B-2 A c)}{b c^2}-\frac{2 x^2 (b B-A c)}{b c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.2525, size = 92, normalized size = 0.93 \[ \frac{2 \left (A c - \frac{3 B b}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{c^{\frac{5}{2}}} + \frac{2 x^{2} \left (A c - B b\right )}{b c \sqrt{b x + c x^{2}}} - \frac{2 \left (A c - \frac{3 B b}{2}\right ) \sqrt{b x + c x^{2}}}{b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.123661, size = 86, normalized size = 0.87 \[ \frac{\sqrt{c} x (-2 A c+3 b B+B c x)+\sqrt{x} \sqrt{b+c x} (2 A c-3 b B) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{c^{5/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 118, normalized size = 1.2 \[ -2\,{\frac{Ax}{c\sqrt{c{x}^{2}+bx}}}+{A\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}}+{\frac{{x}^{2}B}{c}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+3\,{\frac{xBb}{{c}^{2}\sqrt{c{x}^{2}+bx}}}-{\frac{3\,Bb}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/c/(c*x^2+b*x)^(1/2)*x+A/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+B
*x^2/c/(c*x^2+b*x)^(1/2)+3*B*b/c^2/(c*x^2+b*x)^(1/2)*x-3/2*B*b/c^(5/2)*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^2/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292033, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{c x^{2} + b x}{\left (3 \, B b - 2 \, A c\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) - 2 \,{\left (B c x^{2} +{\left (3 \, B b - 2 \, A c\right )} x\right )} \sqrt{c}}{2 \, \sqrt{c x^{2} + b x} c^{\frac{5}{2}}}, -\frac{\sqrt{c x^{2} + b x}{\left (3 \, B b - 2 \, A c\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (B c x^{2} +{\left (3 \, B b - 2 \, A c\right )} x\right )} \sqrt{-c}}{\sqrt{c x^{2} + b x} \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError