Optimal. Leaf size=115 \[ \frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{8 a^{7/2}}-\frac {5 b^2 \sqrt {a x^2+b x^3}}{8 a^3 x^2}+\frac {5 b \sqrt {a x^2+b x^3}}{12 a^2 x^3}-\frac {\sqrt {a x^2+b x^3}}{3 a x^4} \]
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Rubi [A] time = 0.14, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2025, 2008, 206} \begin {gather*} -\frac {5 b^2 \sqrt {a x^2+b x^3}}{8 a^3 x^2}+\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{8 a^{7/2}}+\frac {5 b \sqrt {a x^2+b x^3}}{12 a^2 x^3}-\frac {\sqrt {a x^2+b x^3}}{3 a x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2025
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {a x^2+b x^3}} \, dx &=-\frac {\sqrt {a x^2+b x^3}}{3 a x^4}-\frac {(5 b) \int \frac {1}{x^2 \sqrt {a x^2+b x^3}} \, dx}{6 a}\\ &=-\frac {\sqrt {a x^2+b x^3}}{3 a x^4}+\frac {5 b \sqrt {a x^2+b x^3}}{12 a^2 x^3}+\frac {\left (5 b^2\right ) \int \frac {1}{x \sqrt {a x^2+b x^3}} \, dx}{8 a^2}\\ &=-\frac {\sqrt {a x^2+b x^3}}{3 a x^4}+\frac {5 b \sqrt {a x^2+b x^3}}{12 a^2 x^3}-\frac {5 b^2 \sqrt {a x^2+b x^3}}{8 a^3 x^2}-\frac {\left (5 b^3\right ) \int \frac {1}{\sqrt {a x^2+b x^3}} \, dx}{16 a^3}\\ &=-\frac {\sqrt {a x^2+b x^3}}{3 a x^4}+\frac {5 b \sqrt {a x^2+b x^3}}{12 a^2 x^3}-\frac {5 b^2 \sqrt {a x^2+b x^3}}{8 a^3 x^2}+\frac {\left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^3}}\right )}{8 a^3}\\ &=-\frac {\sqrt {a x^2+b x^3}}{3 a x^4}+\frac {5 b \sqrt {a x^2+b x^3}}{12 a^2 x^3}-\frac {5 b^2 \sqrt {a x^2+b x^3}}{8 a^3 x^2}+\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{8 a^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 40, normalized size = 0.35 \begin {gather*} \frac {2 b^3 \sqrt {x^2 (a+b x)} \, _2F_1\left (\frac {1}{2},4;\frac {3}{2};\frac {b x}{a}+1\right )}{a^4 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 80, normalized size = 0.70 \begin {gather*} \frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{8 a^{7/2}}+\frac {\left (-8 a^2+10 a b x-15 b^2 x^2\right ) \sqrt {a x^2+b x^3}}{24 a^3 x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 175, normalized size = 1.52 \begin {gather*} \left [\frac {15 \, \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 (15 \, a b^{2} x^{2} - 10 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {b x^{3} + a x^{2}}}{48 \, a^{4} x^{4}}, -\frac {15 \, \sqrt {-a} b^{3} x^{4} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) + {\left (15 \, a b^{2} x^{2} - 10 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {b x^{3} + a x^{2}}}{24 \, a^{4} x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 95, normalized size = 0.83 \begin {gather*} -\frac {\sqrt {b x +a}\, \left (-15 a \,b^{3} x^{3} \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+15 \sqrt {b x +a}\, a^{\frac {3}{2}} b^{2} x^{2}-10 \sqrt {b x +a}\, a^{\frac {5}{2}} b x +8 \sqrt {b x +a}\, a^{\frac {7}{2}}\right )}{24 \sqrt {b \,x^{3}+a \,x^{2}}\, a^{\frac {9}{2}} x^{2}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {b x^{3} + a x^{2}} x^{3}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^3\,\sqrt {b\,x^3+a\,x^2}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{3} \sqrt {x^{2} \left (a + b x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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