Optimal. Leaf size=95 \[ \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x^2+b x^3}}\right )}{4 b^{5/2}}-\frac {3 a \sqrt {a x^2+b x^3}}{4 b^2 \sqrt {x}}+\frac {\sqrt {x} \sqrt {a x^2+b x^3}}{2 b} \]
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Rubi [A] time = 0.13, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2024, 2029, 206} \begin {gather*} \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x^2+b x^3}}\right )}{4 b^{5/2}}-\frac {3 a \sqrt {a x^2+b x^3}}{4 b^2 \sqrt {x}}+\frac {\sqrt {x} \sqrt {a x^2+b x^3}}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2024
Rule 2029
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{\sqrt {a x^2+b x^3}} \, dx &=\frac {\sqrt {x} \sqrt {a x^2+b x^3}}{2 b}-\frac {(3 a) \int \frac {x^{3/2}}{\sqrt {a x^2+b x^3}} \, dx}{4 b}\\ &=-\frac {3 a \sqrt {a x^2+b x^3}}{4 b^2 \sqrt {x}}+\frac {\sqrt {x} \sqrt {a x^2+b x^3}}{2 b}+\frac {\left (3 a^2\right ) \int \frac {\sqrt {x}}{\sqrt {a x^2+b x^3}} \, dx}{8 b^2}\\ &=-\frac {3 a \sqrt {a x^2+b x^3}}{4 b^2 \sqrt {x}}+\frac {\sqrt {x} \sqrt {a x^2+b x^3}}{2 b}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {a x^2+b x^3}}\right )}{4 b^2}\\ &=-\frac {3 a \sqrt {a x^2+b x^3}}{4 b^2 \sqrt {x}}+\frac {\sqrt {x} \sqrt {a x^2+b x^3}}{2 b}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x^2+b x^3}}\right )}{4 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 90, normalized size = 0.95 \begin {gather*} \frac {3 a^{5/2} x \sqrt {\frac {b x}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+\sqrt {b} x^{3/2} \left (-3 a^2-a b x+2 b^2 x^2\right )}{4 b^{5/2} \sqrt {x^2 (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 95, normalized size = 1.00 \begin {gather*} -\frac {3 a^2 \log \left (\sqrt {a x^2+b x^3}-\sqrt {b} x^{3/2}\right )}{4 b^{5/2}}+\frac {3 a^2 \log \left (\sqrt {x}\right )}{2 b^{5/2}}+\frac {(2 b x-3 a) \sqrt {a x^2+b x^3}}{4 b^2 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 159, normalized size = 1.67 \begin {gather*} \left [\frac {3 \, a^{2} \sqrt {b} x \log \left (\frac {2 \, b x^{2} + a x + 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {b} \sqrt {x}}{x}\right ) + 2 \, \sqrt {b x^{3} + a x^{2}} {\left (2 \, b^{2} x - 3 \, a b\right )} \sqrt {x}}{8 \, b^{3} x}, -\frac {3 \, a^{2} \sqrt {-b} x \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-b}}{b x^{\frac {3}{2}}}\right ) - \sqrt {b x^{3} + a x^{2}} {\left (2 \, b^{2} x - 3 \, a b\right )} \sqrt {x}}{4 \, b^{3} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 52, normalized size = 0.55 \begin {gather*} \frac {1}{4} \, \sqrt {b x + a} \sqrt {x} {\left (\frac {2 \, x}{b} - \frac {3 \, a}{b^{2}}\right )} - \frac {3 \, a^{2} \log \left ({\left | -\sqrt {b} \sqrt {x} + \sqrt {b x + a} \right |}\right )}{4 \, b^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 92, normalized size = 0.97 \begin {gather*} \frac {\left (4 b^{\frac {7}{2}} x^{3}-2 a \,b^{\frac {5}{2}} x^{2}-6 a^{2} b^{\frac {3}{2}} x +3 \sqrt {\left (b x +a \right ) x}\, a^{2} b \ln \left (\frac {2 b x +a +2 \sqrt {b \,x^{2}+a x}\, \sqrt {b}}{2 \sqrt {b}}\right )\right ) \sqrt {x}}{8 \sqrt {b \,x^{3}+a \,x^{2}}\, b^{\frac {7}{2}}} \end {gather*}
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 \begin {gather*} \int \frac {x^{\frac {5}{2}}}{\sqrt {b x^{3} + a x^{2}}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {x^{5/2}}{\sqrt {b\,x^3+a\,x^2}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {x^{\frac {5}{2}}}{\sqrt {x^{2} \left (a + b x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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