Optimal. Leaf size=166 \[ -\frac {315 b^4 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{64 a^{11/2}}+\frac {315 b^3 \sqrt {a x^2+b x^3}}{64 a^5 x^2}-\frac {105 b^2 \sqrt {a x^2+b x^3}}{32 a^4 x^3}+\frac {21 b \sqrt {a x^2+b x^3}}{8 a^3 x^4}-\frac {9 \sqrt {a x^2+b x^3}}{4 a^2 x^5}+\frac {2}{a x^3 \sqrt {a x^2+b x^3}} \]
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Rubi [A] time = 0.23, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2023, 2025, 2008, 206} \[ \frac {315 b^3 \sqrt {a x^2+b x^3}}{64 a^5 x^2}-\frac {105 b^2 \sqrt {a x^2+b x^3}}{32 a^4 x^3}-\frac {315 b^4 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{64 a^{11/2}}+\frac {21 b \sqrt {a x^2+b x^3}}{8 a^3 x^4}-\frac {9 \sqrt {a x^2+b x^3}}{4 a^2 x^5}+\frac {2}{a x^3 \sqrt {a x^2+b x^3}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2023
Rule 2025
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a x^2+b x^3\right )^{3/2}} \, dx &=\frac {2}{a x^3 \sqrt {a x^2+b x^3}}+\frac {9 \int \frac {1}{x^4 \sqrt {a x^2+b x^3}} \, dx}{a}\\ &=\frac {2}{a x^3 \sqrt {a x^2+b x^3}}-\frac {9 \sqrt {a x^2+b x^3}}{4 a^2 x^5}-\frac {(63 b) \int \frac {1}{x^3 \sqrt {a x^2+b x^3}} \, dx}{8 a^2}\\ &=\frac {2}{a x^3 \sqrt {a x^2+b x^3}}-\frac {9 \sqrt {a x^2+b x^3}}{4 a^2 x^5}+\frac {21 b \sqrt {a x^2+b x^3}}{8 a^3 x^4}+\frac {\left (105 b^2\right ) \int \frac {1}{x^2 \sqrt {a x^2+b x^3}} \, dx}{16 a^3}\\ &=\frac {2}{a x^3 \sqrt {a x^2+b x^3}}-\frac {9 \sqrt {a x^2+b x^3}}{4 a^2 x^5}+\frac {21 b \sqrt {a x^2+b x^3}}{8 a^3 x^4}-\frac {105 b^2 \sqrt {a x^2+b x^3}}{32 a^4 x^3}-\frac {\left (315 b^3\right ) \int \frac {1}{x \sqrt {a x^2+b x^3}} \, dx}{64 a^4}\\ &=\frac {2}{a x^3 \sqrt {a x^2+b x^3}}-\frac {9 \sqrt {a x^2+b x^3}}{4 a^2 x^5}+\frac {21 b \sqrt {a x^2+b x^3}}{8 a^3 x^4}-\frac {105 b^2 \sqrt {a x^2+b x^3}}{32 a^4 x^3}+\frac {315 b^3 \sqrt {a x^2+b x^3}}{64 a^5 x^2}+\frac {\left (315 b^4\right ) \int \frac {1}{\sqrt {a x^2+b x^3}} \, dx}{128 a^5}\\ &=\frac {2}{a x^3 \sqrt {a x^2+b x^3}}-\frac {9 \sqrt {a x^2+b x^3}}{4 a^2 x^5}+\frac {21 b \sqrt {a x^2+b x^3}}{8 a^3 x^4}-\frac {105 b^2 \sqrt {a x^2+b x^3}}{32 a^4 x^3}+\frac {315 b^3 \sqrt {a x^2+b x^3}}{64 a^5 x^2}-\frac {\left (315 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^3}}\right )}{64 a^5}\\ &=\frac {2}{a x^3 \sqrt {a x^2+b x^3}}-\frac {9 \sqrt {a x^2+b x^3}}{4 a^2 x^5}+\frac {21 b \sqrt {a x^2+b x^3}}{8 a^3 x^4}-\frac {105 b^2 \sqrt {a x^2+b x^3}}{32 a^4 x^3}+\frac {315 b^3 \sqrt {a x^2+b x^3}}{64 a^5 x^2}-\frac {315 b^4 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{64 a^{11/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 38, normalized size = 0.23 \[ \frac {2 b^4 x \, _2F_1\left (-\frac {1}{2},5;\frac {1}{2};\frac {b x}{a}+1\right )}{a^5 \sqrt {x^2 (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 263, normalized size = 1.58 \[ \left [\frac {315 \, {\left (b^{5} x^{6} + a b^{4} x^{5}\right )} \sqrt {a} \log \left (\frac {b x^{2} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) + 2 \, {\left (315 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} - 42 \, a^{3} b^{2} x^{2} + 24 \, a^{4} b x - 16 \, a^{5}\right )} \sqrt {b x^{3} + a x^{2}}}{128 \, {\left (a^{6} b x^{6} + a^{7} x^{5}\right )}}, \frac {315 \, {\left (b^{5} x^{6} + a b^{4} x^{5}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) + {\left (315 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} - 42 \, a^{3} b^{2} x^{2} + 24 \, a^{4} b x - 16 \, a^{5}\right )} \sqrt {b x^{3} + a x^{2}}}{64 \, {\left (a^{6} b x^{6} + a^{7} x^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 100, normalized size = 0.60 \[ -\frac {\left (b x +a \right ) \left (315 \sqrt {b x +a}\, b^{4} x^{4} \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )-315 \sqrt {a}\, b^{4} x^{4}-105 a^{\frac {3}{2}} b^{3} x^{3}+42 a^{\frac {5}{2}} b^{2} x^{2}-24 a^{\frac {7}{2}} b x +16 a^{\frac {9}{2}}\right )}{64 \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} a^{\frac {11}{2}} x} \]
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 \[ \int \frac {1}{{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.68, size = 44, normalized size = 0.27 \[ -\frac {2\,{\left (\frac {a}{b\,x}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {11}{2};\ \frac {13}{2};\ -\frac {a}{b\,x}\right )}{11\,x\,{\left (b\,x^3+a\,x^2\right )}^{3/2}} \]
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 \[ \int \frac {1}{x^{2} \left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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