Optimal. Leaf size=112 \[ -\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x^3+b x^4}}\right )}{8 b^{7/2}}+\frac {5 a^2 \sqrt {a x^3+b x^4}}{8 b^3 x}-\frac {5 a \sqrt {a x^3+b x^4}}{12 b^2}+\frac {x \sqrt {a x^3+b x^4}}{3 b} \]
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Rubi [A] time = 0.18, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2024, 2029, 206} \[ \frac {5 a^2 \sqrt {a x^3+b x^4}}{8 b^3 x}-\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x^3+b x^4}}\right )}{8 b^{7/2}}-\frac {5 a \sqrt {a x^3+b x^4}}{12 b^2}+\frac {x \sqrt {a x^3+b x^4}}{3 b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2024
Rule 2029
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {a x^3+b x^4}} \, dx &=\frac {x \sqrt {a x^3+b x^4}}{3 b}-\frac {(5 a) \int \frac {x^3}{\sqrt {a x^3+b x^4}} \, dx}{6 b}\\ &=-\frac {5 a \sqrt {a x^3+b x^4}}{12 b^2}+\frac {x \sqrt {a x^3+b x^4}}{3 b}+\frac {\left (5 a^2\right ) \int \frac {x^2}{\sqrt {a x^3+b x^4}} \, dx}{8 b^2}\\ &=-\frac {5 a \sqrt {a x^3+b x^4}}{12 b^2}+\frac {5 a^2 \sqrt {a x^3+b x^4}}{8 b^3 x}+\frac {x \sqrt {a x^3+b x^4}}{3 b}-\frac {\left (5 a^3\right ) \int \frac {x}{\sqrt {a x^3+b x^4}} \, dx}{16 b^3}\\ &=-\frac {5 a \sqrt {a x^3+b x^4}}{12 b^2}+\frac {5 a^2 \sqrt {a x^3+b x^4}}{8 b^3 x}+\frac {x \sqrt {a x^3+b x^4}}{3 b}-\frac {\left (5 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a x^3+b x^4}}\right )}{8 b^3}\\ &=-\frac {5 a \sqrt {a x^3+b x^4}}{12 b^2}+\frac {5 a^2 \sqrt {a x^3+b x^4}}{8 b^3 x}+\frac {x \sqrt {a x^3+b x^4}}{3 b}-\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x^3+b x^4}}\right )}{8 b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 94, normalized size = 0.84 \[ \frac {\sqrt {x^3 (a+b x)} \left (\sqrt {b} \sqrt {x} \left (15 a^2-10 a b x+8 b^2 x^2\right )-\frac {15 a^{5/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {\frac {b x}{a}+1}}\right )}{24 b^{7/2} x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 171, normalized size = 1.53 \[ \left [\frac {15 \, a^{3} \sqrt {b} x \log \left (\frac {2 \, b x^{2} + a x - 2 \, \sqrt {b x^{4} + a x^{3}} \sqrt {b}}{x}\right ) + 2 \, {\left (8 \, b^{3} x^{2} - 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {b x^{4} + a x^{3}}}{48 \, b^{4} x}, \frac {15 \, a^{3} \sqrt {-b} x \arctan \left (\frac {\sqrt {b x^{4} + a x^{3}} \sqrt {-b}}{b x^{2}}\right ) + {\left (8 \, b^{3} x^{2} - 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {b x^{4} + a x^{3}}}{24 \, b^{4} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\sqrt {b x^{4} + a x^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 120, normalized size = 1.07 \[ \frac {\sqrt {\left (b x +a \right ) x}\, \left (16 \sqrt {b \,x^{2}+a x}\, b^{\frac {7}{2}} x^{2}-15 a^{3} b \ln \left (\frac {2 b x +a +2 \sqrt {b \,x^{2}+a x}\, \sqrt {b}}{2 \sqrt {b}}\right )-20 \sqrt {b \,x^{2}+a x}\, a \,b^{\frac {5}{2}} x +30 \sqrt {b \,x^{2}+a x}\, a^{2} b^{\frac {3}{2}}\right ) x}{48 \sqrt {b \,x^{4}+a \,x^{3}}\, b^{\frac {9}{2}}} \]
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 \[ \int \frac {x^{4}}{\sqrt {b x^{4} + a x^{3}}}\,{d x} \]
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 \[ \int \frac {x^4}{\sqrt {b\,x^4+a\,x^3}} \,d x \]
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 \[ \int \frac {x^{4}}{\sqrt {x^{3} \left (a + b x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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