Optimal. Leaf size=87 \[ \frac {b^2 \log (x) \left (c x^n\right )^{2/n}}{a^3 x^2}-\frac {b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^3 x^2}+\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a^2 x^2}-\frac {1}{2 a x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {368, 44} \begin {gather*} \frac {b^2 \log (x) \left (c x^n\right )^{2/n}}{a^3 x^2}-\frac {b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^3 x^2}+\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a^2 x^2}-\frac {1}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 368
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx &=\frac {\left (c x^n\right )^{2/n} \operatorname {Subst}\left (\int \frac {1}{x^3 (a+b x)} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{x^2}\\ &=\frac {\left (c x^n\right )^{2/n} \operatorname {Subst}\left (\int \left (\frac {1}{a x^3}-\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}-\frac {b^3}{a^3 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{x^2}\\ &=-\frac {1}{2 a x^2}+\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a^2 x^2}+\frac {b^2 \left (c x^n\right )^{2/n} \log (x)}{a^3 x^2}-\frac {b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^3 x^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 75, normalized size = 0.86 \begin {gather*} -\frac {a^2+2 b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )-2 a b \left (c x^n\right )^{\frac {1}{n}}-2 b^2 \log (x) \left (c x^n\right )^{2/n}}{2 a^3 x^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.99, size = 65, normalized size = 0.75 \begin {gather*} -\frac {2 \, b^{2} c^{\frac {2}{n}} x^{2} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right ) - 2 \, b^{2} c^{\frac {2}{n}} x^{2} \log \relax (x) - 2 \, a b c^{\left (\frac {1}{n}\right )} x + a^{2}}{2 \, a^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 302, normalized size = 3.47 \begin {gather*} \frac {b^{2} c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{n}} \ln \relax (x )}{a^{3} x^{2}}-\frac {b^{2} c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{n}} \ln \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right )}{a^{3} x^{2}}+\frac {b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}}{a^{2} x^{2}}-\frac {1}{2 a \,x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )} x^{3}}\,{d 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 {1}{x^3\,\left (a+b\,{\left (c\,x^n\right )}^{1/n}\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 {1}{x^{3} \left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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