Optimal. Leaf size=35 \[ \frac {b x^2}{c \sqrt {c x^2}}+\frac {a x \log (x)}{c \sqrt {c x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 35, 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 x \log (x)}{c \sqrt {c x^2}}+\frac {b x^2}{c \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 45
Rubi steps
\begin {align*} \int \frac {x^2 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {a+b x}{x} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (b+\frac {a}{x}\right ) \, dx}{c \sqrt {c x^2}}\\ &=\frac {b x^2}{c \sqrt {c x^2}}+\frac {a x \log (x)}{c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 21, normalized size = 0.60 \begin {gather*} \frac {x^3 (b x+a \log (x))}{\left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 20, normalized size = 0.57
method | result | size |
default | \(\frac {x^{3} \left (b x +a \ln \left (x \right )\right )}{\left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(20\) |
risch | \(\frac {b \,x^{2}}{c \sqrt {c \,x^{2}}}+\frac {a x \ln \left (x \right )}{c \sqrt {c \,x^{2}}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 23, normalized size = 0.66 \begin {gather*} \frac {b x^{2}}{\sqrt {c x^{2}} c} + \frac {a \log \left (x\right )}{c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.10, size = 22, normalized size = 0.63 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x + a \log \left (x\right )\right )}}{c^{2} 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^{2} \left (a + b x\right )}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.68, size = 27, normalized size = 0.77 \begin {gather*} \frac {\frac {b x}{\sqrt {c} \mathrm {sgn}\left (x\right )} + \frac {a \log \left ({\left | x \right |}\right )}{\sqrt {c} \mathrm {sgn}\left (x\right )}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.32, size = 30, normalized size = 0.86 \begin {gather*} \frac {b\,\left |x\right |}{c^{3/2}}+\frac {a\,\ln \left (x+\left |x\right |\right )}{c^{3/2}}-\frac {a\,x}{c^{3/2}\,\sqrt {x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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