Optimal. Leaf size=53 \[ \frac {2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}-\frac {2 a \sqrt {x}}{b^2}+\frac {2 x^{3/2}}{3 b} \]
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Rubi [A] time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {50, 63, 205} \begin {gather*} \frac {2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}-\frac {2 a \sqrt {x}}{b^2}+\frac {2 x^{3/2}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 205
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{a+b x} \, dx &=\frac {2 x^{3/2}}{3 b}-\frac {a \int \frac {\sqrt {x}}{a+b x} \, dx}{b}\\ &=-\frac {2 a \sqrt {x}}{b^2}+\frac {2 x^{3/2}}{3 b}+\frac {a^2 \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{b^2}\\ &=-\frac {2 a \sqrt {x}}{b^2}+\frac {2 x^{3/2}}{3 b}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=-\frac {2 a \sqrt {x}}{b^2}+\frac {2 x^{3/2}}{3 b}+\frac {2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 0.92 \begin {gather*} \frac {2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}+\frac {2 \sqrt {x} (b x-3 a)}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 53, normalized size = 1.00 \begin {gather*} \frac {2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}+\frac {2 \left (b x^{3/2}-3 a \sqrt {x}\right )}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 103, normalized size = 1.94 \begin {gather*} \left [\frac {3 \, a \sqrt {-\frac {a}{b}} \log \left (\frac {b x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) + 2 \, {\left (b x - 3 \, a\right )} \sqrt {x}}{3 \, b^{2}}, \frac {2 \, {\left (3 \, a \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (b x - 3 \, a\right )} \sqrt {x}\right )}}{3 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.24, size = 45, normalized size = 0.85 \begin {gather*} \frac {2 \, a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {2 \, {\left (b^{2} x^{\frac {3}{2}} - 3 \, a b \sqrt {x}\right )}}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 43, normalized size = 0.81 \begin {gather*} \frac {2 a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}+\frac {2 x^{\frac {3}{2}}}{3 b}-\frac {2 a \sqrt {x}}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 42, normalized size = 0.79 \begin {gather*} \frac {2 \, a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {2 \, {\left (b x^{\frac {3}{2}} - 3 \, a \sqrt {x}\right )}}{3 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 37, normalized size = 0.70 \begin {gather*} \frac {2\,x^{3/2}}{3\,b}-\frac {2\,a\,\sqrt {x}}{b^2}+\frac {2\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.93, size = 105, normalized size = 1.98 \begin {gather*} \begin {cases} - \frac {i a^{\frac {3}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{b^{3} \sqrt {\frac {1}{b}}} + \frac {i a^{\frac {3}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{b^{3} \sqrt {\frac {1}{b}}} - \frac {2 a \sqrt {x}}{b^{2}} + \frac {2 x^{\frac {3}{2}}}{3 b} & \text {for}\: b \neq 0 \\\frac {2 x^{\frac {5}{2}}}{5 a} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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