Optimal. Leaf size=54 \[ -\frac {a^2 \sqrt {c x^2}}{2 x^3}-\frac {2 a b \sqrt {c x^2}}{x^2}+\frac {b^2 \sqrt {c x^2} \log (x)}{x} \]
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Rubi [A]
time = 0.01, antiderivative size = 54, 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, 45}
\begin {gather*} -\frac {a^2 \sqrt {c x^2}}{2 x^3}-\frac {2 a b \sqrt {c x^2}}{x^2}+\frac {b^2 \sqrt {c x^2} \log (x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 45
Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2} (a+b x)^2}{x^4} \, dx &=\frac {\sqrt {c x^2} \int \frac {(a+b x)^2}{x^3} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (\frac {a^2}{x^3}+\frac {2 a b}{x^2}+\frac {b^2}{x}\right ) \, dx}{x}\\ &=-\frac {a^2 \sqrt {c x^2}}{2 x^3}-\frac {2 a b \sqrt {c x^2}}{x^2}+\frac {b^2 \sqrt {c x^2} \log (x)}{x}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 36, normalized size = 0.67 \begin {gather*} \frac {\sqrt {c x^2} \left (-a (a+4 b x)+2 b^2 x^2 \log (x)\right )}{2 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 34, normalized size = 0.63
method | result | size |
default | \(\frac {\sqrt {c \,x^{2}}\, \left (2 b^{2} \ln \left (x \right ) x^{2}-4 a b x -a^{2}\right )}{2 x^{3}}\) | \(34\) |
risch | \(\frac {\sqrt {c \,x^{2}}\, \left (-\frac {1}{2} a^{2}-2 a b x \right )}{x^{3}}+\frac {b^{2} \ln \left (x \right ) \sqrt {c \,x^{2}}}{x}\) | \(40\) |
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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Fricas [A]
time = 0.68, size = 33, normalized size = 0.61 \begin {gather*} \frac {{\left (2 \, b^{2} x^{2} \log \left (x\right ) - 4 \, a b x - a^{2}\right )} \sqrt {c x^{2}}}{2 \, x^{3}} \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^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.58, size = 35, normalized size = 0.65 \begin {gather*} \frac {1}{2} \, {\left (2 \, b^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) - \frac {4 \, a b x \mathrm {sgn}\left (x\right ) + a^{2} \mathrm {sgn}\left (x\right )}{x^{2}}\right )} \sqrt {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 {\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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