Optimal. Leaf size=73 \[ \frac {3 b \sqrt {a x^2+b x^3}}{x}-3 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{x^4} \]
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Rubi [A] time = 0.09, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2020, 2021, 2008, 206} \[ -\frac {\left (a x^2+b x^3\right )^{3/2}}{x^4}+\frac {3 b \sqrt {a x^2+b x^3}}{x}-3 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2020
Rule 2021
Rubi steps
\begin {align*} \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^5} \, dx &=-\frac {\left (a x^2+b x^3\right )^{3/2}}{x^4}+\frac {1}{2} (3 b) \int \frac {\sqrt {a x^2+b x^3}}{x^2} \, dx\\ &=\frac {3 b \sqrt {a x^2+b x^3}}{x}-\frac {\left (a x^2+b x^3\right )^{3/2}}{x^4}+\frac {1}{2} (3 a b) \int \frac {1}{\sqrt {a x^2+b x^3}} \, dx\\ &=\frac {3 b \sqrt {a x^2+b x^3}}{x}-\frac {\left (a x^2+b x^3\right )^{3/2}}{x^4}-(3 a b) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^3}}\right )\\ &=\frac {3 b \sqrt {a x^2+b x^3}}{x}-\frac {\left (a x^2+b x^3\right )^{3/2}}{x^4}-3 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 40, normalized size = 0.55 \[ \frac {2 b \left (x^2 (a+b x)\right )^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {b x}{a}+1\right )}{5 a^2 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 136, normalized size = 1.86 \[ \left [\frac {3 \, \sqrt {a} b x^{2} \log \left (\frac {b x^{2} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) + 2 \, \sqrt {b x^{3} + a x^{2}} {\left (2 \, b x - a\right )}}{2 \, x^{2}}, \frac {3 \, \sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) + \sqrt {b x^{3} + a x^{2}} {\left (2 \, b x - a\right )}}{x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 62, normalized size = 0.85 \[ \frac {\frac {3 \, a b^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-a}} + 2 \, \sqrt {b x + a} b^{2} \mathrm {sgn}\relax (x) - \frac {\sqrt {b x + a} a b \mathrm {sgn}\relax (x)}{x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 72, normalized size = 0.99 \[ -\frac {\left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (3 a b x \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )-2 \sqrt {b x +a}\, \sqrt {a}\, b x +\sqrt {b x +a}\, a^{\frac {3}{2}}\right )}{\left (b x +a \right )^{\frac {3}{2}} \sqrt {a}\, x^{4}} \]
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 \[ \int \frac {{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{5}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\,x^2\right )}^{3/2}}{x^5} \,d x \]
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 \[ \int \frac {\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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