Optimal. Leaf size=64 \[ -\frac {a^2 x}{b^3 \sqrt {c x^2} (a+b x)}-\frac {2 a x \log (a+b x)}{b^3 \sqrt {c x^2}}+\frac {x^2}{b^2 \sqrt {c x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 64, 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} \[ -\frac {a^2 x}{b^3 \sqrt {c x^2} (a+b x)}-\frac {2 a x \log (a+b x)}{b^3 \sqrt {c x^2}}+\frac {x^2}{b^2 \sqrt {c x^2}} \]
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)^2} \, dx &=\frac {x \int \frac {x^2}{(a+b x)^2} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {1}{b^2}+\frac {a^2}{b^2 (a+b x)^2}-\frac {2 a}{b^2 (a+b x)}\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {x^2}{b^2 \sqrt {c x^2}}-\frac {a^2 x}{b^3 \sqrt {c x^2} (a+b x)}-\frac {2 a x \log (a+b x)}{b^3 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 52, normalized size = 0.81 \[ \frac {x \left (-a^2+a b x-2 a (a+b x) \log (a+b x)+b^2 x^2\right )}{b^3 \sqrt {c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 59, normalized size = 0.92 \[ \frac {{\left (b^{2} x^{2} + a b x - a^{2} - 2 \, {\left (a b x + a^{2}\right )} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{b^{4} c x^{2} + a b^{3} c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.13, size = 127, normalized size = 1.98 \[ -\frac {\frac {2 \, a \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{3} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} + \frac {b x + a}{b^{3} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} - \frac {a^{2}}{{\left (b x + a\right )} b^{3} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )}}{\sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 60, normalized size = 0.94 \[ -\frac {\left (2 a b x \ln \left (b x +a \right )-b^{2} x^{2}+2 a^{2} \ln \left (b x +a \right )-a b x +a^{2}\right ) x}{\sqrt {c \,x^{2}}\, \left (b x +a \right ) b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 88, normalized size = 1.38 \[ \frac {\sqrt {c x^{2}} a}{b^{3} c x + a b^{2} c} - \frac {2 \, \left (-1\right )^{\frac {2 \, a c x}{b}} a \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{3} \sqrt {c}} - \frac {2 \, a \log \left (b x\right )}{b^{3} \sqrt {c}} + \frac {\sqrt {c x^{2}}}{b^{2} c} \]
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 \[ \int \frac {x^3}{\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2} \,d x \]
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 \[ \int \frac {x^{3}}{\sqrt {c x^{2}} \left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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