Optimal. Leaf size=56 \[ -\frac {a^2}{c \sqrt {c x^2}}+\frac {b^2 x^2}{c \sqrt {c x^2}}+\frac {2 a b x \log (x)}{c \sqrt {c x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 56, 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}{c \sqrt {c x^2}}+\frac {2 a b x \log (x)}{c \sqrt {c x^2}}+\frac {b^2 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 (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x^2} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (b^2+\frac {a^2}{x^2}+\frac {2 a b}{x}\right ) \, dx}{c \sqrt {c x^2}}\\ &=-\frac {a^2}{c \sqrt {c x^2}}+\frac {b^2 x^2}{c \sqrt {c x^2}}+\frac {2 a b x \log (x)}{c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 33, normalized size = 0.59 \begin {gather*} \frac {x^2 \left (-a^2+b^2 x^2+2 a b x \log (x)\right )}{\left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 32, normalized size = 0.57
method | result | size |
default | \(\frac {x^{2} \left (2 a b \ln \left (x \right ) x +x^{2} b^{2}-a^{2}\right )}{\left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(32\) |
risch | \(-\frac {a^{2}}{c \sqrt {c \,x^{2}}}+\frac {b^{2} x^{2}}{c \sqrt {c \,x^{2}}}+\frac {2 a b x \ln \left (x \right )}{c \sqrt {c \,x^{2}}}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 42, normalized size = 0.75 \begin {gather*} \frac {b^{2} x^{2}}{\sqrt {c x^{2}} c} + \frac {2 \, a b \log \left (x\right )}{c^{\frac {3}{2}}} - \frac {a^{2}}{\sqrt {c x^{2}} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 34, normalized size = 0.61 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x \log \left (x\right ) - a^{2}\right )} \sqrt {c x^{2}}}{c^{2} x^{2}} \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 \left (a + b x\right )^{2}}{\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 = 0.84, size = 46, normalized size = 0.82 \begin {gather*} \frac {\frac {b^{2} x}{\sqrt {c} \mathrm {sgn}\left (x\right )} + \frac {2 \, a b \log \left ({\left | x \right |}\right )}{\sqrt {c} \mathrm {sgn}\left (x\right )} - \frac {a^{2}}{\sqrt {c} x \mathrm {sgn}\left (x\right )}}{c} \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\,{\left (a+b\,x\right )}^2}{{\left (c\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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