3.42 \(\int \frac{x^4}{\sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=128 \[ \frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{9/2}}-\frac{35 b^3 \sqrt{b x+c x^2}}{64 c^4}+\frac{35 b^2 x \sqrt{b x+c x^2}}{96 c^3}-\frac{7 b x^2 \sqrt{b x+c x^2}}{24 c^2}+\frac{x^3 \sqrt{b x+c x^2}}{4 c} \]

[Out]

(-35*b^3*Sqrt[b*x + c*x^2])/(64*c^4) + (35*b^2*x*Sqrt[b*x + c*x^2])/(96*c^3) - (
7*b*x^2*Sqrt[b*x + c*x^2])/(24*c^2) + (x^3*Sqrt[b*x + c*x^2])/(4*c) + (35*b^4*Ar
cTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(64*c^(9/2))

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Rubi [A]  time = 0.179116, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{9/2}}-\frac{35 b^3 \sqrt{b x+c x^2}}{64 c^4}+\frac{35 b^2 x \sqrt{b x+c x^2}}{96 c^3}-\frac{7 b x^2 \sqrt{b x+c x^2}}{24 c^2}+\frac{x^3 \sqrt{b x+c x^2}}{4 c} \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[b*x + c*x^2],x]

[Out]

(-35*b^3*Sqrt[b*x + c*x^2])/(64*c^4) + (35*b^2*x*Sqrt[b*x + c*x^2])/(96*c^3) - (
7*b*x^2*Sqrt[b*x + c*x^2])/(24*c^2) + (x^3*Sqrt[b*x + c*x^2])/(4*c) + (35*b^4*Ar
cTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(64*c^(9/2))

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Rubi in Sympy [A]  time = 20.1891, size = 119, normalized size = 0.93 \[ \frac{35 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{64 c^{\frac{9}{2}}} - \frac{35 b^{3} \sqrt{b x + c x^{2}}}{64 c^{4}} + \frac{35 b^{2} x \sqrt{b x + c x^{2}}}{96 c^{3}} - \frac{7 b x^{2} \sqrt{b x + c x^{2}}}{24 c^{2}} + \frac{x^{3} \sqrt{b x + c x^{2}}}{4 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(c*x**2+b*x)**(1/2),x)

[Out]

35*b**4*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(64*c**(9/2)) - 35*b**3*sqrt(b*x + c
*x**2)/(64*c**4) + 35*b**2*x*sqrt(b*x + c*x**2)/(96*c**3) - 7*b*x**2*sqrt(b*x +
c*x**2)/(24*c**2) + x**3*sqrt(b*x + c*x**2)/(4*c)

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Mathematica [A]  time = 0.0891658, size = 112, normalized size = 0.88 \[ \frac{105 b^4 \sqrt{x} \sqrt{b+c x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )+\sqrt{c} x \left (-105 b^4-35 b^3 c x+14 b^2 c^2 x^2-8 b c^3 x^3+48 c^4 x^4\right )}{192 c^{9/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/Sqrt[b*x + c*x^2],x]

[Out]

(Sqrt[c]*x*(-105*b^4 - 35*b^3*c*x + 14*b^2*c^2*x^2 - 8*b*c^3*x^3 + 48*c^4*x^4) +
105*b^4*Sqrt[x]*Sqrt[b + c*x]*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(192*c^(9
/2)*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.007, size = 112, normalized size = 0.9 \[{\frac{{x}^{3}}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{7\,b{x}^{2}}{24\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{35\,{b}^{2}x}{96\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{35\,{b}^{3}}{64\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{35\,{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(c*x^2+b*x)^(1/2),x)

[Out]

1/4*x^3*(c*x^2+b*x)^(1/2)/c-7/24*b*x^2*(c*x^2+b*x)^(1/2)/c^2+35/96*b^2*x*(c*x^2+
b*x)^(1/2)/c^3-35/64*b^3*(c*x^2+b*x)^(1/2)/c^4+35/128*b^4/c^(9/2)*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^4/sqrt(c*x^2 + b*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23005, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, b^{4} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (48 \, c^{3} x^{3} - 56 \, b c^{2} x^{2} + 70 \, b^{2} c x - 105 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{c}}{384 \, c^{\frac{9}{2}}}, \frac{105 \, b^{4} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (48 \, c^{3} x^{3} - 56 \, b c^{2} x^{2} + 70 \, b^{2} c x - 105 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}{192 \, \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(c*x^2 + b*x),x, algorithm="fricas")

[Out]

[1/384*(105*b^4*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c) + 2*(48*c^3*x^3
- 56*b*c^2*x^2 + 70*b^2*c*x - 105*b^3)*sqrt(c*x^2 + b*x)*sqrt(c))/c^(9/2), 1/19
2*(105*b^4*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (48*c^3*x^3 - 56*b*c^2*x^2
+ 70*b^2*c*x - 105*b^3)*sqrt(c*x^2 + b*x)*sqrt(-c))/(sqrt(-c)*c^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{x \left (b + c x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(c*x**2+b*x)**(1/2),x)

[Out]

Integral(x**4/sqrt(x*(b + c*x)), x)

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GIAC/XCAS [A]  time = 0.2269, size = 120, normalized size = 0.94 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \, x{\left (\frac{6 \, x}{c} - \frac{7 \, b}{c^{2}}\right )} + \frac{35 \, b^{2}}{c^{3}}\right )} x - \frac{105 \, b^{3}}{c^{4}}\right )} - \frac{35 \, b^{4}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(c*x^2 + b*x),x, algorithm="giac")

[Out]

1/192*sqrt(c*x^2 + b*x)*(2*(4*x*(6*x/c - 7*b/c^2) + 35*b^2/c^3)*x - 105*b^3/c^4)
- 35/128*b^4*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(9/2)