Optimal. Leaf size=81 \[ -\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{4 \sqrt {a}}-\frac {3 b \sqrt {a x^2+b x^3}}{4 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{2 x^5} \]
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Rubi [A] time = 0.09, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2020, 2008, 206} \[ -\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{4 \sqrt {a}}-\frac {3 b \sqrt {a x^2+b x^3}}{4 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{2 x^5} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2020
Rubi steps
\begin {align*} \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^6} \, dx &=-\frac {\left (a x^2+b x^3\right )^{3/2}}{2 x^5}+\frac {1}{4} (3 b) \int \frac {\sqrt {a x^2+b x^3}}{x^3} \, dx\\ &=-\frac {3 b \sqrt {a x^2+b x^3}}{4 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{2 x^5}+\frac {1}{8} \left (3 b^2\right ) \int \frac {1}{\sqrt {a x^2+b x^3}} \, dx\\ &=-\frac {3 b \sqrt {a x^2+b x^3}}{4 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{2 x^5}-\frac {1}{4} \left (3 b^2\right ) \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}}{4 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{2 x^5}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{4 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 72, normalized size = 0.89 \[ -\frac {2 a^2+3 b^2 x^2 \sqrt {\frac {b x}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b x}{a}+1}\right )+7 a b x+5 b^2 x^2}{4 x \sqrt {x^2 (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 154, normalized size = 1.90 \[ \left [\frac {3 \, \sqrt {a} b^{2} x^{3} \log \left (\frac {b x^{2} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) - 2 \, \sqrt {b x^{3} + a x^{2}} {\left (5 \, a b x + 2 \, a^{2}\right )}}{8 \, a x^{3}}, \frac {3 \, \sqrt {-a} b^{2} x^{3} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) - \sqrt {b x^{3} + a x^{2}} {\left (5 \, a b x + 2 \, a^{2}\right )}}{4 \, a x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 70, normalized size = 0.86 \[ \frac {\frac {3 \, b^{3} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-a}} - \frac {5 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{3} \mathrm {sgn}\relax (x) - 3 \, \sqrt {b x + a} a b^{3} \mathrm {sgn}\relax (x)}{b^{2} x^{2}}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 74, normalized size = 0.91 \[ -\frac {\left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (3 b^{2} x^{2} \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )-3 \sqrt {b x +a}\, a^{\frac {3}{2}}+5 \left (b x +a \right )^{\frac {3}{2}} \sqrt {a}\right )}{4 \left (b x +a \right )^{\frac {3}{2}} \sqrt {a}\, x^{5}} \]
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 \[ \int \frac {{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{6}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\,x^2\right )}^{3/2}}{x^6} \,d x \]
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 \[ \int \frac {\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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