Optimal. Leaf size=49 \[ -\frac {a^2 \sqrt {c x^2}}{x^2}+\frac {2 a b \sqrt {c x^2} \log (x)}{x}+b^2 \sqrt {c x^2} \]
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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^2 \sqrt {c x^2}}{x^2}+\frac {2 a b \sqrt {c x^2} \log (x)}{x}+b^2 \sqrt {c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2} (a+b x)^2}{x^3} \, dx &=\frac {\sqrt {c x^2} \int \frac {(a+b x)^2}{x^2} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (b^2+\frac {a^2}{x^2}+\frac {2 a b}{x}\right ) \, dx}{x}\\ &=b^2 \sqrt {c x^2}-\frac {a^2 \sqrt {c x^2}}{x^2}+\frac {2 a b \sqrt {c x^2} \log (x)}{x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.63 \begin {gather*} \frac {c \left (-a^2+2 a b x \log (x)+b^2 x^2\right )}{\sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 37, normalized size = 0.76 \begin {gather*} \sqrt {c x^2} \left (\frac {b^2 x^2-a^2}{x^2}+\frac {2 a b \log (x)}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 31, normalized size = 0.63 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x \log \relax (x) - a^{2}\right )} \sqrt {c x^{2}}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 31, normalized size = 0.63 \begin {gather*} {\left (b^{2} x \mathrm {sgn}\relax (x) + 2 \, a b \log \left ({\left | x \right |}\right ) \mathrm {sgn}\relax (x) - \frac {a^{2} \mathrm {sgn}\relax (x)}{x}\right )} \sqrt {c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 32, normalized size = 0.65 \begin {gather*} \frac {\sqrt {c \,x^{2}}\, \left (2 a b x \ln \relax (x )+b^{2} x^{2}-a^{2}\right )}{x^{2}} \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 = -0.02 \begin {gather*} \int \frac {\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2}{x^3} \,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 {\sqrt {c x^{2}} \left (a + b x\right )^{2}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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