Optimal. Leaf size=58 \[ \frac {a^2 \sqrt {c x^2} \log (a+b x)}{b^3 x}-\frac {a \sqrt {c x^2}}{b^2}+\frac {x \sqrt {c x^2}}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 58, 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} \[ \frac {a^2 \sqrt {c x^2} \log (a+b x)}{b^3 x}-\frac {a \sqrt {c x^2}}{b^2}+\frac {x \sqrt {c x^2}}{2 b} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {x \sqrt {c x^2}}{a+b x} \, dx &=\frac {\sqrt {c x^2} \int \frac {x^2}{a+b x} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx}{x}\\ &=-\frac {a \sqrt {c x^2}}{b^2}+\frac {x \sqrt {c x^2}}{2 b}+\frac {a^2 \sqrt {c x^2} \log (a+b x)}{b^3 x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 0.69 \[ \frac {c x \left (2 a^2 \log (a+b x)+b x (b x-2 a)\right )}{2 b^3 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 39, normalized size = 0.67 \[ \frac {{\left (b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{2 \, b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 54, normalized size = 0.93 \[ \frac {1}{2} \, \sqrt {c} {\left (\frac {2 \, a^{2} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{3}} - \frac {2 \, a^{2} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{3}} + \frac {b x^{2} \mathrm {sgn}\relax (x) - 2 \, a x \mathrm {sgn}\relax (x)}{b^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 40, normalized size = 0.69 \[ \frac {\sqrt {c \,x^{2}}\, \left (b^{2} x^{2}+2 a^{2} \ln \left (b x +a \right )-2 a b x \right )}{2 b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.49, size = 91, normalized size = 1.57 \[ \frac {\left (-1\right )^{\frac {2 \, c x}{b}} a^{2} \sqrt {c} \log \left (\frac {2 \, c x}{b}\right )}{b^{3}} + \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a^{2} \sqrt {c} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{3}} + \frac {\sqrt {c x^{2}} x}{2 \, b} - \frac {\sqrt {c x^{2}} a}{b^{2}} \]
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 \[ \int \frac {x\,\sqrt {c\,x^2}}{a+b\,x} \,d x \]
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 \[ \int \frac {x \sqrt {c x^{2}}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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