3.43 \(\int \frac{x^3}{\sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=102 \[ -\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{7/2}}+\frac{5 b^2 \sqrt{b x+c x^2}}{8 c^3}-\frac{5 b x \sqrt{b x+c x^2}}{12 c^2}+\frac{x^2 \sqrt{b x+c x^2}}{3 c} \]

[Out]

(5*b^2*Sqrt[b*x + c*x^2])/(8*c^3) - (5*b*x*Sqrt[b*x + c*x^2])/(12*c^2) + (x^2*Sq
rt[b*x + c*x^2])/(3*c) - (5*b^3*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(7/
2))

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Rubi [A]  time = 0.129006, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{7/2}}+\frac{5 b^2 \sqrt{b x+c x^2}}{8 c^3}-\frac{5 b x \sqrt{b x+c x^2}}{12 c^2}+\frac{x^2 \sqrt{b x+c x^2}}{3 c} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[b*x + c*x^2],x]

[Out]

(5*b^2*Sqrt[b*x + c*x^2])/(8*c^3) - (5*b*x*Sqrt[b*x + c*x^2])/(12*c^2) + (x^2*Sq
rt[b*x + c*x^2])/(3*c) - (5*b^3*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(7/
2))

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Rubi in Sympy [A]  time = 14.583, size = 94, normalized size = 0.92 \[ - \frac{5 b^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{8 c^{\frac{7}{2}}} + \frac{5 b^{2} \sqrt{b x + c x^{2}}}{8 c^{3}} - \frac{5 b x \sqrt{b x + c x^{2}}}{12 c^{2}} + \frac{x^{2} \sqrt{b x + c x^{2}}}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(c*x**2+b*x)**(1/2),x)

[Out]

-5*b**3*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(8*c**(7/2)) + 5*b**2*sqrt(b*x + c*x
**2)/(8*c**3) - 5*b*x*sqrt(b*x + c*x**2)/(12*c**2) + x**2*sqrt(b*x + c*x**2)/(3*
c)

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Mathematica [A]  time = 0.0637374, size = 101, normalized size = 0.99 \[ \frac{\sqrt{c} x \left (15 b^3+5 b^2 c x-2 b c^2 x^2+8 c^3 x^3\right )-15 b^3 \sqrt{x} \sqrt{b+c x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{24 c^{7/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[b*x + c*x^2],x]

[Out]

(Sqrt[c]*x*(15*b^3 + 5*b^2*c*x - 2*b*c^2*x^2 + 8*c^3*x^3) - 15*b^3*Sqrt[x]*Sqrt[
b + c*x]*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(24*c^(7/2)*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.007, size = 90, normalized size = 0.9 \[{\frac{{x}^{2}}{3\,c}\sqrt{c{x}^{2}+bx}}-{\frac{5\,bx}{12\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,{b}^{2}}{8\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(c*x^2+b*x)^(1/2),x)

[Out]

1/3*x^2*(c*x^2+b*x)^(1/2)/c-5/12*b*x*(c*x^2+b*x)^(1/2)/c^2+5/8*b^2*(c*x^2+b*x)^(
1/2)/c^3-5/16*b^3/c^(7/2)*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(x^3/sqrt(c*x^2 + b*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226526, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{3} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (8 \, c^{2} x^{2} - 10 \, b c x + 15 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{c}}{48 \, c^{\frac{7}{2}}}, -\frac{15 \, b^{3} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (8 \, c^{2} x^{2} - 10 \, b c x + 15 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}{24 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(15*b^3*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c) + 2*(8*c^2*x^2 -
10*b*c*x + 15*b^2)*sqrt(c*x^2 + b*x)*sqrt(c))/c^(7/2), -1/24*(15*b^3*arctan(sqrt
(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (8*c^2*x^2 - 10*b*c*x + 15*b^2)*sqrt(c*x^2 + b*x
)*sqrt(-c))/(sqrt(-c)*c^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(c*x**2+b*x)**(1/2),x)

[Out]

Integral(x**3/sqrt(x*(b + c*x)), x)

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GIAC/XCAS [A]  time = 0.228606, size = 104, normalized size = 1.02 \[ \frac{1}{24} \, \sqrt{c x^{2} + b x}{\left (2 \, x{\left (\frac{4 \, x}{c} - \frac{5 \, b}{c^{2}}\right )} + \frac{15 \, b^{2}}{c^{3}}\right )} + \frac{5 \, b^{3}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*sqrt(c*x^2 + b*x)*(2*x*(4*x/c - 5*b/c^2) + 15*b^2/c^3) + 5/16*b^3*ln(abs(-2
*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(7/2)