Optimal. Leaf size=61 \[ \frac {a^2 x \log (a+b x)}{b^3 \sqrt {c x^2}}-\frac {a x^2}{b^2 \sqrt {c x^2}}+\frac {x^3}{2 b \sqrt {c x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 43} \[ \frac {a^2 x \log (a+b x)}{b^3 \sqrt {c x^2}}-\frac {a x^2}{b^2 \sqrt {c x^2}}+\frac {x^3}{2 b \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {c x^2} (a+b x)} \, dx &=\frac {x \int \frac {x^2}{a+b x} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx}{\sqrt {c x^2}}\\ &=-\frac {a x^2}{b^2 \sqrt {c x^2}}+\frac {x^3}{2 b \sqrt {c x^2}}+\frac {a^2 x \log (a+b x)}{b^3 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.64 \[ \frac {x \left (2 a^2 \log (a+b x)+b x (b x-2 a)\right )}{2 b^3 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 42, normalized size = 0.69 \[ \frac {{\left (b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{2 \, b^{3} c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 67, normalized size = 1.10 \[ \frac {1}{2} \, \sqrt {c x^{2}} {\left (\frac {x}{b c} - \frac {2 \, a}{b^{2} c}\right )} - \frac {a^{2} \log \left ({\left | -{\left (\sqrt {c} x - \sqrt {c x^{2}}\right )} b \sqrt {c} - 2 \, a c \right |}\right )}{b^{3} \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 38, normalized size = 0.62 \[ \frac {\left (b^{2} x^{2}+2 a^{2} \ln \left (b x +a \right )-2 a b x \right ) x}{2 \sqrt {c \,x^{2}}\, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.49, size = 100, normalized size = 1.64 \[ \frac {x^{2}}{2 \, b \sqrt {c}} + \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a^{2} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{3} \sqrt {c}} + \frac {2 \, a x}{b^{2} \sqrt {c}} + \frac {a^{2} \log \left (b x\right )}{b^{3} \sqrt {c}} - \frac {3 \, \sqrt {c x^{2}} a}{b^{2} c} + \frac {3 \, a^{2}}{2 \, b^{3} \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^3}{\sqrt {c\,x^2}\,\left (a+b\,x\right )} \,d x \]
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 \[ \int \frac {x^{3}}{\sqrt {c x^{2}} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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