Optimal. Leaf size=86 \[ \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x^3+b x^4}}\right )}{4 b^{5/2}}-\frac {3 a \sqrt {a x^3+b x^4}}{4 b^2 x}+\frac {\sqrt {a x^3+b x^4}}{2 b} \]
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Rubi [A] time = 0.13, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2024, 2029, 206} \[ \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x^3+b x^4}}\right )}{4 b^{5/2}}-\frac {3 a \sqrt {a x^3+b x^4}}{4 b^2 x}+\frac {\sqrt {a x^3+b x^4}}{2 b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2024
Rule 2029
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {a x^3+b x^4}} \, dx &=\frac {\sqrt {a x^3+b x^4}}{2 b}-\frac {(3 a) \int \frac {x^2}{\sqrt {a x^3+b x^4}} \, dx}{4 b}\\ &=\frac {\sqrt {a x^3+b x^4}}{2 b}-\frac {3 a \sqrt {a x^3+b x^4}}{4 b^2 x}+\frac {\left (3 a^2\right ) \int \frac {x}{\sqrt {a x^3+b x^4}} \, dx}{8 b^2}\\ &=\frac {\sqrt {a x^3+b x^4}}{2 b}-\frac {3 a \sqrt {a x^3+b x^4}}{4 b^2 x}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a x^3+b x^4}}\right )}{4 b^2}\\ &=\frac {\sqrt {a x^3+b x^4}}{2 b}-\frac {3 a \sqrt {a x^3+b x^4}}{4 b^2 x}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x^3+b x^4}}\right )}{4 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 92, normalized size = 1.07 \[ \frac {3 a^{5/2} x^{3/2} \sqrt {\frac {b x}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+\sqrt {b} x^2 \left (-3 a^2-a b x+2 b^2 x^2\right )}{4 b^{5/2} \sqrt {x^3 (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 150, normalized size = 1.74 \[ \left [\frac {3 \, a^{2} \sqrt {b} x \log \left (\frac {2 \, b x^{2} + a x + 2 \, \sqrt {b x^{4} + a x^{3}} \sqrt {b}}{x}\right ) + 2 \, \sqrt {b x^{4} + a x^{3}} {\left (2 \, b^{2} x - 3 \, a b\right )}}{8 \, b^{3} x}, -\frac {3 \, a^{2} \sqrt {-b} x \arctan \left (\frac {\sqrt {b x^{4} + a x^{3}} \sqrt {-b}}{b x^{2}}\right ) - \sqrt {b x^{4} + a x^{3}} {\left (2 \, b^{2} x - 3 \, a b\right )}}{4 \, b^{3} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\sqrt {b x^{4} + a x^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 98, normalized size = 1.14 \[ \frac {\sqrt {\left (b x +a \right ) x}\, \left (3 a^{2} b \ln \left (\frac {2 b x +a +2 \sqrt {b \,x^{2}+a x}\, \sqrt {b}}{2 \sqrt {b}}\right )+4 \sqrt {b \,x^{2}+a x}\, b^{\frac {5}{2}} x -6 \sqrt {b \,x^{2}+a x}\, a \,b^{\frac {3}{2}}\right ) x}{8 \sqrt {b \,x^{4}+a \,x^{3}}\, b^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\sqrt {b x^{4} + a x^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3}{\sqrt {b\,x^4+a\,x^3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\sqrt {x^{3} \left (a + b x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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