Optimal. Leaf size=109 \[ \frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{8 a^{3/2}}-\frac {b^2 \sqrt {a x^2+b x^3}}{8 a x^2}-\frac {b \sqrt {a x^2+b x^3}}{4 x^3}-\frac {\left (a x^2+b x^3\right )^{3/2}}{3 x^6} \]
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Rubi [A] time = 0.13, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2020, 2025, 2008, 206} \[ \frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{8 a^{3/2}}-\frac {b^2 \sqrt {a x^2+b x^3}}{8 a x^2}-\frac {b \sqrt {a x^2+b x^3}}{4 x^3}-\frac {\left (a x^2+b x^3\right )^{3/2}}{3 x^6} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2020
Rule 2025
Rubi steps
\begin {align*} \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^7} \, dx &=-\frac {\left (a x^2+b x^3\right )^{3/2}}{3 x^6}+\frac {1}{2} b \int \frac {\sqrt {a x^2+b x^3}}{x^4} \, dx\\ &=-\frac {b \sqrt {a x^2+b x^3}}{4 x^3}-\frac {\left (a x^2+b x^3\right )^{3/2}}{3 x^6}+\frac {1}{8} b^2 \int \frac {1}{x \sqrt {a x^2+b x^3}} \, dx\\ &=-\frac {b \sqrt {a x^2+b x^3}}{4 x^3}-\frac {b^2 \sqrt {a x^2+b x^3}}{8 a x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{3 x^6}-\frac {b^3 \int \frac {1}{\sqrt {a x^2+b x^3}} \, dx}{16 a}\\ &=-\frac {b \sqrt {a x^2+b x^3}}{4 x^3}-\frac {b^2 \sqrt {a x^2+b x^3}}{8 a x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{3 x^6}+\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^3}}\right )}{8 a}\\ &=-\frac {b \sqrt {a x^2+b x^3}}{4 x^3}-\frac {b^2 \sqrt {a x^2+b x^3}}{8 a x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{3 x^6}+\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{8 a^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 42, normalized size = 0.39 \[ \frac {2 b^3 \left (x^2 (a+b x)\right )^{5/2} \, _2F_1\left (\frac {5}{2},4;\frac {7}{2};\frac {b x}{a}+1\right )}{5 a^4 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 175, normalized size = 1.61 \[ \left [\frac {3 \, \sqrt {a} b^{3} x^{4} \log \left (\frac {b x^{2} + 2 \, a x + 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) - 2 \, {\left (3 \, a b^{2} x^{2} + 14 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {b x^{3} + a x^{2}}}{48 \, a^{2} x^{4}}, -\frac {3 \, \sqrt {-a} b^{3} x^{4} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) + {\left (3 \, a b^{2} x^{2} + 14 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {b x^{3} + a x^{2}}}{24 \, a^{2} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 92, normalized size = 0.84 \[ -\frac {\frac {3 \, b^{4} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-a} a} + \frac {3 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{4} \mathrm {sgn}\relax (x) + 8 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{4} \mathrm {sgn}\relax (x) - 3 \, \sqrt {b x + a} a^{2} b^{4} \mathrm {sgn}\relax (x)}{a b^{3} x^{3}}}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 87, normalized size = 0.80 \[ -\frac {\left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (-3 a \,b^{3} x^{3} \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )-3 \sqrt {b x +a}\, a^{\frac {7}{2}}+8 \left (b x +a \right )^{\frac {3}{2}} a^{\frac {5}{2}}+3 \left (b x +a \right )^{\frac {5}{2}} a^{\frac {3}{2}}\right )}{24 \left (b x +a \right )^{\frac {3}{2}} a^{\frac {5}{2}} x^{6}} \]
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 \[ \int \frac {{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{7}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\,x^2\right )}^{3/2}}{x^7} \,d x \]
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 \[ \int \frac {\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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