Optimal. Leaf size=83 \[ -\frac {a^3 x \log (a+b x)}{b^4 \sqrt {c x^2}}+\frac {a^2 x^2}{b^3 \sqrt {c x^2}}-\frac {a x^3}{2 b^2 \sqrt {c x^2}}+\frac {x^4}{3 b \sqrt {c x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 83, 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, 43} \[ \frac {a^2 x^2}{b^3 \sqrt {c x^2}}-\frac {a^3 x \log (a+b x)}{b^4 \sqrt {c x^2}}-\frac {a x^3}{2 b^2 \sqrt {c x^2}}+\frac {x^4}{3 b \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {c x^2} (a+b x)} \, dx &=\frac {x \int \frac {x^3}{a+b x} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a^2}{b^3}-\frac {a x}{b^2}+\frac {x^2}{b}-\frac {a^3}{b^3 (a+b x)}\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {a^2 x^2}{b^3 \sqrt {c x^2}}-\frac {a x^3}{2 b^2 \sqrt {c x^2}}+\frac {x^4}{3 b \sqrt {c x^2}}-\frac {a^3 x \log (a+b x)}{b^4 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 51, normalized size = 0.61 \[ \frac {x \left (b x \left (6 a^2-3 a b x+2 b^2 x^2\right )-6 a^3 \log (a+b x)\right )}{6 b^4 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 54, normalized size = 0.65 \[ \frac {{\left (2 \, b^{3} x^{3} - 3 \, a b^{2} x^{2} + 6 \, a^{2} b x - 6 \, a^{3} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{6 \, b^{4} c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.15, size = 81, normalized size = 0.98 \[ \frac {1}{6} \, \sqrt {c x^{2}} {\left (x {\left (\frac {2 \, x}{b c} - \frac {3 \, a}{b^{2} c}\right )} + \frac {6 \, a^{2}}{b^{3} c}\right )} + \frac {a^{3} \log \left ({\left | -{\left (\sqrt {c} x - \sqrt {c x^{2}}\right )} b \sqrt {c} - 2 \, a c \right |}\right )}{b^{4} \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 50, normalized size = 0.60 \[ -\frac {\left (-2 b^{3} x^{3}+3 a \,b^{2} x^{2}+6 a^{3} \ln \left (b x +a \right )-6 a^{2} b x \right ) x}{6 \sqrt {c \,x^{2}}\, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.52, size = 142, normalized size = 1.71 \[ \frac {\sqrt {c x^{2}} x^{2}}{3 \, b c} - \frac {7 \, a x^{2}}{6 \, b^{2} \sqrt {c}} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a^{3} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{4} \sqrt {c}} + \frac {2 \, \sqrt {c x^{2}} a x}{3 \, b^{2} c} - \frac {14 \, a^{2} x}{3 \, b^{3} \sqrt {c}} - \frac {a^{3} \log \left (b x\right )}{b^{4} \sqrt {c}} + \frac {17 \, \sqrt {c x^{2}} a^{2}}{3 \, b^{3} c} - \frac {7 \, a^{3}}{2 \, b^{4} \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^4}{\sqrt {c\,x^2}\,\left (a+b\,x\right )} \,d x \]
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 \[ \int \frac {x^{4}}{\sqrt {c x^{2}} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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