Optimal. Leaf size=74 \[ -2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )+\frac {2 a \sqrt {a x^2+b x^3}}{x}+\frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.10, antiderivative size = 74, 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 = {2021, 2008, 206} \[ -2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )+\frac {2 a \sqrt {a x^2+b x^3}}{x}+\frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2021
Rubi steps
\begin {align*} \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^4} \, dx &=\frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 x^3}+a \int \frac {\sqrt {a x^2+b x^3}}{x^2} \, dx\\ &=\frac {2 a \sqrt {a x^2+b x^3}}{x}+\frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 x^3}+a^2 \int \frac {1}{\sqrt {a x^2+b x^3}} \, dx\\ &=\frac {2 a \sqrt {a x^2+b x^3}}{x}+\frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 x^3}-\left (2 a^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 {2 a \sqrt {a x^2+b x^3}}{x}+\frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 x^3}-2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 68, normalized size = 0.92 \[ \frac {2 x \sqrt {a+b x} \left (\sqrt {a+b x} (4 a+b x)-3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{3 \sqrt {x^2 (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 130, normalized size = 1.76 \[ \left [\frac {3 \, a^{\frac {3}{2}} x \log \left (\frac {b x^{2} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) + 2 \, \sqrt {b x^{3} + a x^{2}} {\left (b x + 4 \, a\right )}}{3 \, x}, \frac {2 \, {\left (3 \, \sqrt {-a} a x \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) + \sqrt {b x^{3} + a x^{2}} {\left (b x + 4 \, a\right )}\right )}}{3 \, x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 85, normalized size = 1.15 \[ \frac {2 \, a^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-a}} + \frac {2}{3} \, {\left (b x + a\right )}^{\frac {3}{2}} \mathrm {sgn}\relax (x) + 2 \, \sqrt {b x + a} a \mathrm {sgn}\relax (x) - \frac {2 \, {\left (3 \, a^{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) + 4 \, \sqrt {-a} a^{\frac {3}{2}}\right )} \mathrm {sgn}\relax (x)}{3 \, \sqrt {-a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 61, normalized size = 0.82 \[ \frac {2 \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (-3 a^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+3 \sqrt {b x +a}\, a +\left (b x +a \right )^{\frac {3}{2}}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} x^{3}} \]
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 \[ \int \frac {{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{4}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\,x^2\right )}^{3/2}}{x^4} \,d x \]
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 \[ \int \frac {\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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