Optimal. Leaf size=86 \[ \frac {a^3 x}{b^4 \sqrt {c x^2} (a+b x)}+\frac {3 a^2 x \log (a+b x)}{b^4 \sqrt {c x^2}}-\frac {2 a x^2}{b^3 \sqrt {c x^2}}+\frac {x^3}{2 b^2 \sqrt {c x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 86, 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} \[ \frac {a^3 x}{b^4 \sqrt {c x^2} (a+b x)}+\frac {3 a^2 x \log (a+b x)}{b^4 \sqrt {c x^2}}-\frac {2 a x^2}{b^3 \sqrt {c x^2}}+\frac {x^3}{2 b^2 \sqrt {c x^2}} \]
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)^2} \, dx &=\frac {x \int \frac {x^3}{(a+b x)^2} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (-\frac {2 a}{b^3}+\frac {x}{b^2}-\frac {a^3}{b^3 (a+b x)^2}+\frac {3 a^2}{b^3 (a+b x)}\right ) \, dx}{\sqrt {c x^2}}\\ &=-\frac {2 a x^2}{b^3 \sqrt {c x^2}}+\frac {x^3}{2 b^2 \sqrt {c x^2}}+\frac {a^3 x}{b^4 \sqrt {c x^2} (a+b x)}+\frac {3 a^2 x \log (a+b x)}{b^4 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 69, normalized size = 0.80 \[ \frac {x \left (2 a^3-4 a^2 b x+6 a^2 (a+b x) \log (a+b x)-3 a b^2 x^2+b^3 x^3\right )}{2 b^4 \sqrt {c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 74, normalized size = 0.86 \[ \frac {{\left (b^{3} x^{3} - 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 2 \, a^{3} + 6 \, {\left (a^{2} b x + a^{3}\right )} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{2 \, {\left (b^{5} c x^{2} + a b^{4} c x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 143, normalized size = 1.66 \[ \frac {\frac {{\left (b x + a\right )}^{2} {\left (\frac {6 \, a}{b x + a} - 1\right )}}{b^{4} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} + \frac {6 \, a^{2} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{4} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} - \frac {2 \, a^{3}}{{\left (b x + a\right )} b^{4} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )}}{2 \, \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 74, normalized size = 0.86 \[ \frac {\left (b^{3} x^{3}+6 a^{2} b x \ln \left (b x +a \right )-3 a \,b^{2} x^{2}+6 a^{3} \ln \left (b x +a \right )-4 a^{2} b x +2 a^{3}\right ) x}{2 \sqrt {c \,x^{2}}\, \left (b x +a \right ) b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.55, size = 129, normalized size = 1.50 \[ -\frac {\sqrt {c x^{2}} a^{2}}{b^{4} c x + a b^{3} c} + \frac {x^{2}}{2 \, b^{2} \sqrt {c}} + \frac {3 \, \left (-1\right )^{\frac {2 \, a c x}{b}} a^{2} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{4} \sqrt {c}} + \frac {2 \, a x}{b^{3} \sqrt {c}} + \frac {3 \, a^{2} \log \left (b x\right )}{b^{4} \sqrt {c}} - \frac {4 \, \sqrt {c x^{2}} a}{b^{3} c} + \frac {3 \, a^{2}}{2 \, b^{4} \sqrt {c}} \]
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 \[ \int \frac {x^4}{\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2} \,d x \]
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 \[ \int \frac {x^{4}}{\sqrt {c x^{2}} \left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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