Optimal. Leaf size=45 \[ \frac {x^2}{b c \sqrt {c x^2}}-\frac {a x \log (a+b x)}{b^2 c \sqrt {c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 45, 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} \begin {gather*} \frac {x^2}{b c \sqrt {c x^2}}-\frac {a x \log (a+b x)}{b^2 c \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {x^4}{\left (c x^2\right )^{3/2} (a+b x)} \, dx &=\frac {x \int \frac {x}{a+b x} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx}{c \sqrt {c x^2}}\\ &=\frac {x^2}{b c \sqrt {c x^2}}-\frac {a x \log (a+b x)}{b^2 c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.64 \begin {gather*} \frac {x^3 (b x-a \log (a+b x))}{b^2 \left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 33, normalized size = 0.73 \begin {gather*} \frac {\frac {x^4}{b}-\frac {a x^3 \log (a+b x)}{b^2}}{\left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 30, normalized size = 0.67 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x - a \log \left (b x + a\right )\right )}}{b^{2} c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 55, normalized size = 1.22 \begin {gather*} \frac {\frac {a \log \left ({\left | -{\left (\sqrt {c} x - \sqrt {c x^{2}}\right )} b \sqrt {c} - 2 \, a c \right |}\right )}{b^{2} \sqrt {c}} + \frac {\sqrt {c x^{2}}}{b c}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 0.64 \begin {gather*} -\frac {\left (a \ln \left (b x +a \right )-b x \right ) x^{3}}{\left (c \,x^{2}\right )^{\frac {3}{2}} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.60, size = 116, normalized size = 2.58 \begin {gather*} \frac {x^{2}}{\sqrt {c x^{2}} b c} - \frac {\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^{2} c^{\frac {3}{2}}} + \frac {2 \, a x}{\sqrt {c x^{2}} b^{2} c} - \frac {a \log \left (b x\right )}{b^{2} c^{\frac {3}{2}}} - \frac {2 \, a^{2}}{\sqrt {c x^{2}} b^{3} c} + \frac {2 \, a^{2}}{b^{3} c^{\frac {3}{2}} 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 = -0.02 \begin {gather*} \int \frac {x^4}{{\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right )} \,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 {x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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