Optimal. Leaf size=30 \[ \frac {a c \sqrt {c x^2} \log (x)}{x}+b c \sqrt {c x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 43} \begin {gather*} \frac {a c \sqrt {c x^2} \log (x)}{x}+b 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 {\left (c x^2\right )^{3/2} (a+b x)}{x^4} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int \frac {a+b x}{x} \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (b+\frac {a}{x}\right ) \, dx}{x}\\ &=b c \sqrt {c x^2}+\frac {a c \sqrt {c x^2} \log (x)}{x}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 0.70 \begin {gather*} \frac {\left (c x^2\right )^{3/2} (a \log (x)+b x)}{x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 23, normalized size = 0.77 \begin {gather*} \left (c x^2\right )^{3/2} \left (\frac {a \log (x)}{x^3}+\frac {b}{x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 21, normalized size = 0.70 \begin {gather*} \frac {{\left (b c x + a c \log \relax (x)\right )} \sqrt {c x^{2}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 17, normalized size = 0.57 \begin {gather*} {\left (b x \mathrm {sgn}\relax (x) + a \log \left ({\left | x \right |}\right ) \mathrm {sgn}\relax (x)\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 = 20, normalized size = 0.67 \begin {gather*} \frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (a \ln \relax (x )+b x \right )}{x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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 = -0.03 \begin {gather*} \int \frac {{\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right )}{x^4} \,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 {\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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