Optimal. Leaf size=49 \[ \frac {a x}{b^2 c \sqrt {c x^2} (a+b x)}+\frac {x \log (a+b x)}{b^2 c \sqrt {c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 49, 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 {a x}{b^2 c \sqrt {c x^2} (a+b x)}+\frac {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)^2} \, dx &=\frac {x \int \frac {x}{(a+b x)^2} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx}{c \sqrt {c x^2}}\\ &=\frac {a x}{b^2 c \sqrt {c x^2} (a+b x)}+\frac {x \log (a+b x)}{b^2 c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 0.76 \begin {gather*} \frac {x^3 ((a+b x) \log (a+b x)+a)}{b^2 \left (c x^2\right )^{3/2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 39, normalized size = 0.80 \begin {gather*} \frac {\frac {a x^3}{b^2 (a+b x)}+\frac {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 = 1.12, size = 44, normalized size = 0.90 \begin {gather*} \frac {\sqrt {c x^{2}} {\left ({\left (b x + a\right )} \log \left (b x + a\right ) + a\right )}}{b^{3} c^{2} x^{2} + a b^{2} c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 89, normalized size = 1.82 \begin {gather*} \frac {\frac {\log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{2} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} - \frac {a}{{\left (b x + a\right )} b^{2} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )}}{c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 41, normalized size = 0.84 \begin {gather*} \frac {\left (b x \ln \left (b x +a \right )+a \ln \left (b x +a \right )+a \right ) x^{3}}{\left (c \,x^{2}\right )^{\frac {3}{2}} \left (b x +a \right ) b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.56, size = 108, normalized size = 2.20 \begin {gather*} -\frac {a^{2}}{\sqrt {c x^{2}} b^{4} c x + \sqrt {c x^{2}} a b^{3} c} + \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{2} c^{\frac {3}{2}}} + \frac {\log \left (b x\right )}{b^{2} c^{\frac {3}{2}}} + \frac {3 \, a}{\sqrt {c x^{2}} b^{3} c} - \frac {2 \, a}{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 )}^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 {x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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