Optimal. Leaf size=64 \[ 2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )-\frac {2 (a+b x)^{3/2}}{3 x^{3/2}}-\frac {2 b \sqrt {a+b x}}{\sqrt {x}} \]
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Rubi [A] time = 0.02, antiderivative size = 64, 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 = {47, 63, 217, 206} \[ 2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )-\frac {2 (a+b x)^{3/2}}{3 x^{3/2}}-\frac {2 b \sqrt {a+b x}}{\sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2}}{x^{5/2}} \, dx &=-\frac {2 (a+b x)^{3/2}}{3 x^{3/2}}+b \int \frac {\sqrt {a+b x}}{x^{3/2}} \, dx\\ &=-\frac {2 b \sqrt {a+b x}}{\sqrt {x}}-\frac {2 (a+b x)^{3/2}}{3 x^{3/2}}+b^2 \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx\\ &=-\frac {2 b \sqrt {a+b x}}{\sqrt {x}}-\frac {2 (a+b x)^{3/2}}{3 x^{3/2}}+\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 b \sqrt {a+b x}}{\sqrt {x}}-\frac {2 (a+b x)^{3/2}}{3 x^{3/2}}+\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )\\ &=-\frac {2 b \sqrt {a+b x}}{\sqrt {x}}-\frac {2 (a+b x)^{3/2}}{3 x^{3/2}}+2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 48, normalized size = 0.75 \[ -\frac {2 a \sqrt {a+b x} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-\frac {b x}{a}\right )}{3 x^{3/2} \sqrt {\frac {b x}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 109, normalized size = 1.70 \[ \left [\frac {3 \, b^{\frac {3}{2}} x^{2} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (4 \, b x + a\right )} \sqrt {b x + a} \sqrt {x}}{3 \, x^{2}}, -\frac {2 \, {\left (3 \, \sqrt {-b} b x^{2} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (4 \, b x + a\right )} \sqrt {b x + a} \sqrt {x}\right )}}{3 \, x^{2}}\right ] \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 67, normalized size = 1.05 \[ \frac {\sqrt {\left (b x +a \right ) x}\, b^{\frac {3}{2}} \ln \left (\frac {b x +\frac {a}{2}}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{\sqrt {b x +a}\, \sqrt {x}}-\frac {2 \sqrt {b x +a}\, \left (4 b x +a \right )}{3 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 67, normalized size = 1.05 \[ -b^{\frac {3}{2}} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right ) - \frac {2 \, \sqrt {b x + a} b}{\sqrt {x}} - \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,x\right )}^{3/2}}{x^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.04, size = 71, normalized size = 1.11 \[ - \frac {2 a \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 x} - \frac {8 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3} - b^{\frac {3}{2}} \log {\left (\frac {a}{b x} \right )} + 2 b^{\frac {3}{2}} \log {\left (\sqrt {\frac {a}{b x} + 1} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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