Optimal. Leaf size=95 \[ -\frac {a^3 x \log (a+b x)}{b^4 c \sqrt {c x^2}}+\frac {a^2 x^2}{b^3 c \sqrt {c x^2}}-\frac {a x^3}{2 b^2 c \sqrt {c x^2}}+\frac {x^4}{3 b c \sqrt {c x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 95, 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^2}{b^3 c \sqrt {c x^2}}-\frac {a^3 x \log (a+b x)}{b^4 c \sqrt {c x^2}}-\frac {a x^3}{2 b^2 c \sqrt {c x^2}}+\frac {x^4}{3 b 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^6}{\left (c x^2\right )^{3/2} (a+b x)} \, dx &=\frac {x \int \frac {x^3}{a+b x} \, dx}{c \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}{c \sqrt {c x^2}}\\ &=\frac {a^2 x^2}{b^3 c \sqrt {c x^2}}-\frac {a x^3}{2 b^2 c \sqrt {c x^2}}+\frac {x^4}{3 b c \sqrt {c x^2}}-\frac {a^3 x \log (a+b x)}{b^4 c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 53, normalized size = 0.56 \begin {gather*} \frac {x^3 \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 \left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 59, normalized size = 0.62 \begin {gather*} \frac {\frac {6 a^2 x^4-3 a b x^5+2 b^2 x^6}{6 b^3}-\frac {a^3 x^3 \log (a+b x)}{b^4}}{\left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 54, normalized size = 0.57 \begin {gather*} \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^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.18, size = 86, normalized size = 0.91 \begin {gather*} \frac {\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 {6 \, 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}}}{6 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.55 \begin {gather*} -\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^{3}}{6 \left (c \,x^{2}\right )^{\frac {3}{2}} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.79, size = 162, normalized size = 1.71 \begin {gather*} \frac {x^{4}}{3 \, \sqrt {c x^{2}} b c} - \frac {a x^{3}}{2 \, \sqrt {c x^{2}} b^{2} c} + \frac {a^{2} x^{2}}{\sqrt {c x^{2}} b^{3} 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} c^{\frac {3}{2}}} + \frac {29 \, a^{3} x}{6 \, \sqrt {c x^{2}} b^{4} c} - \frac {a^{3} \log \left (b x\right )}{b^{4} c^{\frac {3}{2}}} - \frac {2 \, a^{4}}{\sqrt {c x^{2}} b^{5} c} + \frac {2 \, a^{4}}{b^{5} 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^6}{{\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right )} \,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^{6}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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