Optimal. Leaf size=73 \[ -\frac {a^2 x}{b^3 c \sqrt {c x^2} (a+b x)}-\frac {2 a x \log (a+b x)}{b^3 c \sqrt {c x^2}}+\frac {x^2}{b^2 c \sqrt {c x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 73, 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} \begin {gather*} -\frac {a^2 x}{b^3 c \sqrt {c x^2} (a+b x)}-\frac {2 a x \log (a+b x)}{b^3 c \sqrt {c x^2}}+\frac {x^2}{b^2 c \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {x^5}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx &=\frac {x \int \frac {x^2}{(a+b x)^2} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {1}{b^2}+\frac {a^2}{b^2 (a+b x)^2}-\frac {2 a}{b^2 (a+b x)}\right ) \, dx}{c \sqrt {c x^2}}\\ &=\frac {x^2}{b^2 c \sqrt {c x^2}}-\frac {a^2 x}{b^3 c \sqrt {c x^2} (a+b x)}-\frac {2 a x \log (a+b x)}{b^3 c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 54, normalized size = 0.74 \begin {gather*} \frac {x^3 \left (-a^2+a b x-2 a (a+b x) \log (a+b x)+b^2 x^2\right )}{b^3 \left (c x^2\right )^{3/2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 59, normalized size = 0.81 \begin {gather*} \frac {\frac {-a^2 x^3+a b x^4+b^2 x^5}{b^3 (a+b x)}-\frac {2 a x^3 \log (a+b x)}{b^3}}{\left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 63, normalized size = 0.86 \begin {gather*} \frac {{\left (b^{2} x^{2} + a b x - a^{2} - 2 \, {\left (a b x + a^{2}\right )} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{b^{4} c^{2} x^{2} + a b^{3} c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.26, size = 127, normalized size = 1.74 \begin {gather*} -\frac {\frac {2 \, a \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{3} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} + \frac {b x + a}{b^{3} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} - \frac {a^{2}}{{\left (b x + a\right )} b^{3} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )}}{c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 62, normalized size = 0.85 \begin {gather*} -\frac {\left (2 a b x \ln \left (b x +a \right )-b^{2} x^{2}+2 a^{2} \ln \left (b x +a \right )-a b x +a^{2}\right ) x^{3}}{\left (c \,x^{2}\right )^{\frac {3}{2}} \left (b x +a \right ) b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.61, size = 149, normalized size = 2.04 \begin {gather*} \frac {a^{3}}{\sqrt {c x^{2}} b^{5} c x + \sqrt {c x^{2}} a b^{4} c} + \frac {x^{2}}{\sqrt {c x^{2}} b^{2} c} - \frac {2 \, \left (-1\right )^{\frac {2 \, a c x}{b}} a \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{3} c^{\frac {3}{2}}} + \frac {2 \, a x}{\sqrt {c x^{2}} b^{3} c} - \frac {2 \, a \log \left (b x\right )}{b^{3} c^{\frac {3}{2}}} - \frac {5 \, a^{2}}{\sqrt {c x^{2}} b^{4} c} + \frac {4 \, a^{2}}{b^{4} c^{\frac {3}{2}} x} \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 = -0.01 \begin {gather*} \int \frac {x^5}{{\left (c\,x^2\right )}^{3/2}\,{\left (a+b\,x\right )}^2} \,d 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^{5}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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