Optimal. Leaf size=276 \[ \frac {3 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^2}{4 e^3}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^3}-\frac {3 b^2 d^2 n^2}{e^2 x^{2/3}}+\frac {b^2 d^3 n^2 \log ^2\left (d+\frac {e}{x^{2/3}}\right )}{2 e^3}+\frac {3 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2} \]
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Rubi [A]
time = 0.21, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2445,
2458, 45, 2372, 12, 14, 2338} \begin {gather*} -\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}+\frac {3 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}+\frac {b^2 d^3 n^2 \log ^2\left (d+\frac {e}{x^{2/3}}\right )}{2 e^3}-\frac {3 b^2 d^2 n^2}{e^2 x^{2/3}}+\frac {3 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^2}{4 e^3}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 45
Rule 2338
Rule 2372
Rule 2445
Rule 2458
Rule 2504
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx &=-\left (\frac {3}{2} \text {Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\frac {1}{x^{2/3}}\right )\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}+(b e n) \text {Subst}\left (\int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\frac {1}{x^{2/3}}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}+(b n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+\frac {e}{x^{2/3}}\right )\\ &=\frac {1}{6} b n \left (\frac {18 d^2 \left (d+\frac {e}{x^{2/3}}\right )}{e^3}-\frac {9 d \left (d+\frac {e}{x^{2/3}}\right )^2}{e^3}+\frac {2 \left (d+\frac {e}{x^{2/3}}\right )^3}{e^3}-\frac {6 d^3 \log \left (d+\frac {e}{x^{2/3}}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}-\left (b^2 n^2\right ) \text {Subst}\left (\int \frac {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{6 e^3 x} \, dx,x,d+\frac {e}{x^{2/3}}\right )\\ &=\frac {1}{6} b n \left (\frac {18 d^2 \left (d+\frac {e}{x^{2/3}}\right )}{e^3}-\frac {9 d \left (d+\frac {e}{x^{2/3}}\right )^2}{e^3}+\frac {2 \left (d+\frac {e}{x^{2/3}}\right )^3}{e^3}-\frac {6 d^3 \log \left (d+\frac {e}{x^{2/3}}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}-\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \frac {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{x} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{6 e^3}\\ &=\frac {1}{6} b n \left (\frac {18 d^2 \left (d+\frac {e}{x^{2/3}}\right )}{e^3}-\frac {9 d \left (d+\frac {e}{x^{2/3}}\right )^2}{e^3}+\frac {2 \left (d+\frac {e}{x^{2/3}}\right )^3}{e^3}-\frac {6 d^3 \log \left (d+\frac {e}{x^{2/3}}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}-\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \left (18 d^2-9 d x+2 x^2-\frac {6 d^3 \log (x)}{x}\right ) \, dx,x,d+\frac {e}{x^{2/3}}\right )}{6 e^3}\\ &=\frac {3 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^2}{4 e^3}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^3}-\frac {3 b^2 d^2 n^2}{e^2 x^{2/3}}+\frac {1}{6} b n \left (\frac {18 d^2 \left (d+\frac {e}{x^{2/3}}\right )}{e^3}-\frac {9 d \left (d+\frac {e}{x^{2/3}}\right )^2}{e^3}+\frac {2 \left (d+\frac {e}{x^{2/3}}\right )^3}{e^3}-\frac {6 d^3 \log \left (d+\frac {e}{x^{2/3}}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}+\frac {\left (b^2 d^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{e^3}\\ &=\frac {3 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^2}{4 e^3}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^3}-\frac {3 b^2 d^2 n^2}{e^2 x^{2/3}}+\frac {b^2 d^3 n^2 \log ^2\left (d+\frac {e}{x^{2/3}}\right )}{2 e^3}+\frac {1}{6} b n \left (\frac {18 d^2 \left (d+\frac {e}{x^{2/3}}\right )}{e^3}-\frac {9 d \left (d+\frac {e}{x^{2/3}}\right )^2}{e^3}+\frac {2 \left (d+\frac {e}{x^{2/3}}\right )^3}{e^3}-\frac {6 d^3 \log \left (d+\frac {e}{x^{2/3}}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.36, size = 691, normalized size = 2.50 \begin {gather*} \frac {-18 e^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+b n \left (9 b d n x^{2/3} \left (e \left (e-2 d x^{2/3}\right )+2 d^2 x^{4/3} \log \left (d+\frac {e}{x^{2/3}}\right )\right )-2 b n \left (e \left (2 e^2-3 d e x^{2/3}+6 d^2 x^{4/3}\right )-6 d^3 x^2 \log \left (d+\frac {e}{x^{2/3}}\right )\right )+12 e^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )-18 d e^2 x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )+36 d^2 x^{4/3} \left (e (a-b n)+b \left (e+d x^{2/3}\right ) \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )-36 d^3 x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )-36 d^3 x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )-36 d^3 x^2 \left (\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac {e}{d x^{2/3}}\right )+b n \text {Li}_2\left (1+\frac {e}{d x^{2/3}}\right )\right )+18 b d^3 n x^2 \left (\log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )+2 \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )\right )+18 b d^3 n x^2 \left (\log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )-4 \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )+2 \text {Li}_2\left (\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \text {Li}_2\left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )\right )}{36 e^3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )\right )^{2}}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 299, normalized size = 1.08 \begin {gather*} -\frac {1}{6} \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (d x^{\frac {2}{3}} + e\right ) - 6 \, d^{3} e^{\left (-4\right )} \log \left (x^{\frac {2}{3}}\right ) - \frac {{\left (6 \, d^{2} x^{\frac {4}{3}} - 3 \, d x^{\frac {2}{3}} e + 2 \, e^{2}\right )} e^{\left (-3\right )}}{x^{2}}\right )} a b n e - \frac {1}{36} \, {\left (6 \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (d x^{\frac {2}{3}} + e\right ) - 6 \, d^{3} e^{\left (-4\right )} \log \left (x^{\frac {2}{3}}\right ) - \frac {{\left (6 \, d^{2} x^{\frac {4}{3}} - 3 \, d x^{\frac {2}{3}} e + 2 \, e^{2}\right )} e^{\left (-3\right )}}{x^{2}}\right )} n e \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) - \frac {{\left (18 \, d^{3} x^{2} \log \left (d x^{\frac {2}{3}} + e\right )^{2} + 8 \, d^{3} x^{2} \log \left (x\right )^{2} - 44 \, d^{3} x^{2} \log \left (x\right ) - 66 \, d^{2} x^{\frac {4}{3}} e + 15 \, d x^{\frac {2}{3}} e^{2} - 6 \, {\left (4 \, d^{3} x^{2} \log \left (x\right ) - 11 \, d^{3} x^{2}\right )} \log \left (d x^{\frac {2}{3}} + e\right ) - 4 \, e^{3}\right )} n^{2} e^{\left (-3\right )}}{x^{2}}\right )} b^{2} - \frac {b^{2} \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )^{2}}{2 \, x^{2}} - \frac {a b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )}{x^{2}} - \frac {a^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 294, normalized size = 1.07 \begin {gather*} -\frac {{\left (18 \, b^{2} e^{3} \log \left (c\right )^{2} - 12 \, {\left (b^{2} n - 3 \, a b\right )} e^{3} \log \left (c\right ) + 18 \, {\left (b^{2} d^{3} n^{2} x^{2} + b^{2} n^{2} e^{3}\right )} \log \left (\frac {d x + x^{\frac {1}{3}} e}{x}\right )^{2} + 2 \, {\left (2 \, b^{2} n^{2} - 6 \, a b n + 9 \, a^{2}\right )} e^{3} - 6 \, {\left (6 \, b^{2} d^{2} n^{2} x^{\frac {4}{3}} e - 3 \, b^{2} d n^{2} x^{\frac {2}{3}} e^{2} + {\left (11 \, b^{2} d^{3} n^{2} - 6 \, a b d^{3} n\right )} x^{2} + 2 \, {\left (b^{2} n^{2} - 3 \, a b n\right )} e^{3} - 6 \, {\left (b^{2} d^{3} n x^{2} + b^{2} n e^{3}\right )} \log \left (c\right )\right )} \log \left (\frac {d x + x^{\frac {1}{3}} e}{x}\right ) + 3 \, {\left (6 \, b^{2} d n e^{2} \log \left (c\right ) - {\left (5 \, b^{2} d n^{2} - 6 \, a b d n\right )} e^{2}\right )} x^{\frac {2}{3}} - 6 \, {\left (6 \, b^{2} d^{2} n x e \log \left (c\right ) - {\left (11 \, b^{2} d^{2} n^{2} - 6 \, a b d^{2} n\right )} x e\right )} x^{\frac {1}{3}}\right )} e^{\left (-3\right )}}{36 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 [B]
time = 0.57, size = 302, normalized size = 1.09 \begin {gather*} \frac {\frac {d\,\left (\frac {3\,a^2}{2}-a\,b\,n+\frac {b^2\,n^2}{3}\right )}{2\,e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{4\,e}}{x^{4/3}}-{\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}^2\,\left (\frac {b^2}{2\,x^2}+\frac {b^2\,d^3}{2\,e^3}\right )-\frac {\frac {a^2}{2}-\frac {a\,b\,n}{3}+\frac {b^2\,n^2}{9}}{x^2}-\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\,\left (\frac {b\,\left (3\,a-b\,n\right )}{3\,x^2}-\frac {\frac {b\,d\,\left (3\,a-b\,n\right )}{2\,e}-\frac {3\,a\,b\,d}{2\,e}}{x^{4/3}}+\frac {d\,\left (\frac {b\,d\,\left (3\,a-b\,n\right )}{e}-\frac {3\,a\,b\,d}{e}\right )}{e\,x^{2/3}}\right )-\frac {\frac {d\,\left (\frac {d\,\left (\frac {3\,a^2}{2}-a\,b\,n+\frac {b^2\,n^2}{3}\right )}{e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{2\,e}\right )}{e}+\frac {b^2\,d^2\,n^2}{e^2}}{x^{2/3}}+\frac {\ln \left (d+\frac {e}{x^{2/3}}\right )\,\left (11\,b^2\,d^3\,n^2-6\,a\,b\,d^3\,n\right )}{6\,e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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