Optimal. Leaf size=58 \[ -\frac {a \sqrt {c x^2}}{b^2}+\frac {x \sqrt {c x^2}}{2 b}+\frac {a^2 \sqrt {c x^2} \log (a+b x)}{b^3 x} \]
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Rubi [A]
time = 0.01, 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, 45}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 45
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 \begin {gather*} \frac {c x \left (b x (-2 a+b x)+2 a^2 \log (a+b x)\right )}{2 b^3 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 40, normalized size = 0.69
method | result | size |
default | \(\frac {\sqrt {c \,x^{2}}\, \left (x^{2} b^{2}+2 a^{2} \ln \left (b x +a \right )-2 a b x \right )}{2 b^{3} x}\) | \(40\) |
risch | \(\frac {\sqrt {c \,x^{2}}\, \left (\frac {1}{2} x^{2} b -a x \right )}{x \,b^{2}}+\frac {a^{2} \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b^{3} x}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 91, normalized size = 1.57 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 39, normalized size = 0.67 \begin {gather*} \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} \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 \sqrt {c x^{2}}}{a + b x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.04, size = 54, normalized size = 0.93 \begin {gather*} \frac {1}{2} \, \sqrt {c} {\left (\frac {2 \, a^{2} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{3}} - \frac {2 \, a^{2} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{3}} + \frac {b x^{2} \mathrm {sgn}\left (x\right ) - 2 \, a x \mathrm {sgn}\left (x\right )}{b^{2}}\right )} \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\,\sqrt {c\,x^2}}{a+b\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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