Optimal. Leaf size=77 \[ \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{5/2}}-\frac {3 a \sqrt {x} \sqrt {a+b x}}{4 b^2}+\frac {x^{3/2} \sqrt {a+b x}}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {50, 63, 217, 206} \begin {gather*} \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{5/2}}-\frac {3 a \sqrt {x} \sqrt {a+b x}}{4 b^2}+\frac {x^{3/2} \sqrt {a+b x}}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx &=\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {(3 a) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{4 b}\\ &=-\frac {3 a \sqrt {x} \sqrt {a+b x}}{4 b^2}+\frac {x^{3/2} \sqrt {a+b x}}{2 b}+\frac {\left (3 a^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{8 b^2}\\ &=-\frac {3 a \sqrt {x} \sqrt {a+b x}}{4 b^2}+\frac {x^{3/2} \sqrt {a+b x}}{2 b}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{4 b^2}\\ &=-\frac {3 a \sqrt {x} \sqrt {a+b x}}{4 b^2}+\frac {x^{3/2} \sqrt {a+b x}}{2 b}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^2}\\ &=-\frac {3 a \sqrt {x} \sqrt {a+b x}}{4 b^2}+\frac {x^{3/2} \sqrt {a+b x}}{2 b}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 85, normalized size = 1.10 \begin {gather*} \frac {3 a^{5/2} \sqrt {\frac {b x}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+\sqrt {b} \sqrt {x} \left (-3 a^2-a b x+2 b^2 x^2\right )}{4 b^{5/2} \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 69, normalized size = 0.90 \begin {gather*} \frac {\sqrt {a+b x} \left (2 b x^{3/2}-3 a \sqrt {x}\right )}{4 b^2}-\frac {3 a^2 \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{4 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 119, normalized size = 1.55 \begin {gather*} \left [\frac {3 \, a^{2} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (2 \, b^{2} x - 3 \, a b\right )} \sqrt {b x + a} \sqrt {x}}{8 \, b^{3}}, -\frac {3 \, a^{2} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (2 \, b^{2} x - 3 \, a b\right )} \sqrt {b x + a} \sqrt {x}}{4 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 1.09 \begin {gather*} \frac {\sqrt {b x +a}\, x^{\frac {3}{2}}}{2 b}+\frac {3 \sqrt {\left (b x +a \right ) x}\, a^{2} \ln \left (\frac {b x +\frac {a}{2}}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 \sqrt {b x +a}\, b^{\frac {5}{2}} \sqrt {x}}-\frac {3 \sqrt {b x +a}\, a \sqrt {x}}{4 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.87, size = 112, normalized size = 1.45 \begin {gather*} -\frac {3 \, a^{2} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{8 \, b^{\frac {5}{2}}} + \frac {\frac {5 \, \sqrt {b x + a} a^{2} b}{\sqrt {x}} - \frac {3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2}}{x^{\frac {3}{2}}}}{4 \, {\left (b^{4} - \frac {2 \, {\left (b x + a\right )} b^{3}}{x} + \frac {{\left (b x + a\right )}^{2} b^{2}}{x^{2}}\right )}} \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^{3/2}}{\sqrt {a+b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.30, size = 100, normalized size = 1.30 \begin {gather*} - \frac {3 a^{\frac {3}{2}} \sqrt {x}}{4 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {\sqrt {a} x^{\frac {3}{2}}}{4 b \sqrt {1 + \frac {b x}{a}}} + \frac {3 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 b^{\frac {5}{2}}} + \frac {x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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