3.310 \(\int \frac{x^4}{\sqrt{a x^3+b x^4}} \, dx\)

Optimal. Leaf size=112 \[ -\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^3+b x^4}}\right )}{8 b^{7/2}}+\frac{5 a^2 \sqrt{a x^3+b x^4}}{8 b^3 x}-\frac{5 a \sqrt{a x^3+b x^4}}{12 b^2}+\frac{x \sqrt{a x^3+b x^4}}{3 b} \]

[Out]

(-5*a*Sqrt[a*x^3 + b*x^4])/(12*b^2) + (5*a^2*Sqrt[a*x^3 + b*x^4])/(8*b^3*x) + (x
*Sqrt[a*x^3 + b*x^4])/(3*b) - (5*a^3*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a*x^3 + b*x^4]])
/(8*b^(7/2))

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Rubi [A]  time = 0.310282, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^3+b x^4}}\right )}{8 b^{7/2}}+\frac{5 a^2 \sqrt{a x^3+b x^4}}{8 b^3 x}-\frac{5 a \sqrt{a x^3+b x^4}}{12 b^2}+\frac{x \sqrt{a x^3+b x^4}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[a*x^3 + b*x^4],x]

[Out]

(-5*a*Sqrt[a*x^3 + b*x^4])/(12*b^2) + (5*a^2*Sqrt[a*x^3 + b*x^4])/(8*b^3*x) + (x
*Sqrt[a*x^3 + b*x^4])/(3*b) - (5*a^3*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a*x^3 + b*x^4]])
/(8*b^(7/2))

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Rubi in Sympy [A]  time = 25.8014, size = 100, normalized size = 0.89 \[ - \frac{5 a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a x^{3} + b x^{4}}} \right )}}{8 b^{\frac{7}{2}}} + \frac{5 a^{2} \sqrt{a x^{3} + b x^{4}}}{8 b^{3} x} - \frac{5 a \sqrt{a x^{3} + b x^{4}}}{12 b^{2}} + \frac{x \sqrt{a x^{3} + b x^{4}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**4+a*x**3)**(1/2),x)

[Out]

-5*a**3*atanh(sqrt(b)*x**2/sqrt(a*x**3 + b*x**4))/(8*b**(7/2)) + 5*a**2*sqrt(a*x
**3 + b*x**4)/(8*b**3*x) - 5*a*sqrt(a*x**3 + b*x**4)/(12*b**2) + x*sqrt(a*x**3 +
b*x**4)/(3*b)

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Mathematica [A]  time = 0.0734323, size = 105, normalized size = 0.94 \[ \frac{\sqrt{b} x^2 \left (15 a^3+5 a^2 b x-2 a b^2 x^2+8 b^3 x^3\right )-15 a^3 x^{3/2} \sqrt{a+b x} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{24 b^{7/2} \sqrt{x^3 (a+b x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/Sqrt[a*x^3 + b*x^4],x]

[Out]

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

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Maple [A]  time = 0.01, size = 120, normalized size = 1.1 \[{\frac{x}{48}\sqrt{x \left ( bx+a \right ) } \left ( 16\,{x}^{2}\sqrt{b{x}^{2}+ax}{b}^{7/2}-20\,\sqrt{b{x}^{2}+ax}{b}^{5/2}xa+30\,\sqrt{b{x}^{2}+ax}{b}^{3/2}{a}^{2}-15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{b{x}^{2}+ax}\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{3}b \right ){\frac{1}{\sqrt{b{x}^{4}+a{x}^{3}}}}{b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^4+a*x^3)^(1/2),x)

[Out]

1/48*x*(x*(b*x+a))^(1/2)*(16*x^2*(b*x^2+a*x)^(1/2)*b^(7/2)-20*(b*x^2+a*x)^(1/2)*
b^(5/2)*x*a+30*(b*x^2+a*x)^(1/2)*b^(3/2)*a^2-15*ln(1/2*(2*(b*x^2+a*x)^(1/2)*b^(1
/2)+2*b*x+a)/b^(1/2))*a^3*b)/(b*x^4+a*x^3)^(1/2)/b^(9/2)

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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(b*x^4 + a*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238075, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{3} \sqrt{b} x \log \left (\frac{{\left (2 \, b x^{2} + a x\right )} \sqrt{b} - 2 \, \sqrt{b x^{4} + a x^{3}} b}{x}\right ) + 2 \,{\left (8 \, b^{3} x^{2} - 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt{b x^{4} + a x^{3}}}{48 \, b^{4} x}, \frac{15 \, a^{3} \sqrt{-b} x \arctan \left (\frac{\sqrt{b x^{4} + a x^{3}} \sqrt{-b}}{b x^{2}}\right ) +{\left (8 \, b^{3} x^{2} - 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt{b x^{4} + a x^{3}}}{24 \, b^{4} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(b*x^4 + a*x^3),x, algorithm="fricas")

[Out]

[1/48*(15*a^3*sqrt(b)*x*log(((2*b*x^2 + a*x)*sqrt(b) - 2*sqrt(b*x^4 + a*x^3)*b)/
x) + 2*(8*b^3*x^2 - 10*a*b^2*x + 15*a^2*b)*sqrt(b*x^4 + a*x^3))/(b^4*x), 1/24*(1
5*a^3*sqrt(-b)*x*arctan(sqrt(b*x^4 + a*x^3)*sqrt(-b)/(b*x^2)) + (8*b^3*x^2 - 10*
a*b^2*x + 15*a^2*b)*sqrt(b*x^4 + a*x^3))/(b^4*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{x^{3} \left (a + b x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**4+a*x**3)**(1/2),x)

[Out]

Integral(x**4/sqrt(x**3*(a + b*x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{b x^{4} + a x^{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(b*x^4 + a*x^3),x, algorithm="giac")

[Out]

integrate(x^4/sqrt(b*x^4 + a*x^3), x)