Optimal. Leaf size=101 \[ \frac {a^2 x^4 \left (c x^n\right )^{-3/n}}{b^3}-\frac {a x^4 \left (c x^n\right )^{-2/n}}{2 b^2}+\frac {x^4 \left (c x^n\right )^{-1/n}}{3 b}-\frac {a^3 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^4} \]
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Rubi [A]
time = 0.04, antiderivative size = 101, 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 = {375, 45}
\begin {gather*} -\frac {a^3 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^4}+\frac {a^2 x^4 \left (c x^n\right )^{-3/n}}{b^3}-\frac {a x^4 \left (c x^n\right )^{-2/n}}{2 b^2}+\frac {x^4 \left (c x^n\right )^{-1/n}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 375
Rubi steps
\begin {align*} \int \frac {x^3}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx &=\left (x^4 \left (c x^n\right )^{-4/n}\right ) \text {Subst}\left (\int \frac {x^3}{a+b x} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\left (x^4 \left (c x^n\right )^{-4/n}\right ) \text {Subst}\left (\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,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\frac {a^2 x^4 \left (c x^n\right )^{-3/n}}{b^3}-\frac {a x^4 \left (c x^n\right )^{-2/n}}{2 b^2}+\frac {x^4 \left (c x^n\right )^{-1/n}}{3 b}-\frac {a^3 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^4}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 87, normalized size = 0.86 \begin {gather*} \frac {x^4 \left (c x^n\right )^{-4/n} \left (b \left (c x^n\right )^{\frac {1}{n}} \left (6 a^2-3 a b \left (c x^n\right )^{\frac {1}{n}}+2 b^2 \left (c x^n\right )^{2/n}\right )-6 a^3 \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )\right )}{6 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.50, size = 387, normalized size = 3.83
method | result | size |
risch | \(\frac {x^{4} c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} {\mathrm e}^{-\frac {i \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (\mathrm {csgn}\left (i c \,x^{n}\right )-\mathrm {csgn}\left (i x^{n}\right )\right )}{2 n}}}{3 b}-\frac {a \,x^{4} c^{-\frac {2}{n}} \left (x^{n}\right )^{-\frac {2}{n}} {\mathrm e}^{-\frac {i \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (\mathrm {csgn}\left (i c \,x^{n}\right )-\mathrm {csgn}\left (i x^{n}\right )\right )}{n}}}{2 b^{2}}+\frac {\left (x^{n}\right )^{-\frac {3}{n}} c^{-\frac {3}{n}} x^{4} {\mathrm e}^{-\frac {3 i \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (\mathrm {csgn}\left (i c \,x^{n}\right )-\mathrm {csgn}\left (i x^{n}\right )\right )}{2 n}} a^{2}}{b^{3}}-\frac {\ln \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (\mathrm {csgn}\left (i c \,x^{n}\right )-\mathrm {csgn}\left (i x^{n}\right )\right )}{2 n}}+a \right ) c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} a^{3} \left (x^{n}\right )^{-\frac {3}{n}} c^{-\frac {3}{n}} x^{4} {\mathrm e}^{-\frac {2 i \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (\mathrm {csgn}\left (i c \,x^{n}\right )-\mathrm {csgn}\left (i x^{n}\right )\right )}{n}}}{b^{4}}\) | \(387\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 74, normalized size = 0.73 \begin {gather*} \frac {2 \, b^{3} c^{\frac {3}{n}} x^{3} - 3 \, a b^{2} c^{\frac {2}{n}} x^{2} + 6 \, a^{2} b c^{\left (\frac {1}{n}\right )} x - 6 \, a^{3} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right )}{6 \, b^{4} c^{\frac {4}{n}}} \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^{3}}{a + b \left (c x^{n}\right )^{\frac {1}{n}}}\, dx \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*} \text {could not integrate} \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^3}{a+b\,{\left (c\,x^n\right )}^{1/n}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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