Optimal. Leaf size=77 \[ -\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x}-\frac {2 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{e^{3/2}}-\frac {2 b d n}{e \sqrt [3]{x}}+\frac {2 b n}{3 x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2455, 263, 341, 325, 205} \[ -\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x}-\frac {2 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{e^{3/2}}-\frac {2 b d n}{e \sqrt [3]{x}}+\frac {2 b n}{3 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 263
Rule 325
Rule 341
Rule 2455
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^2} \, dx &=-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x}-\frac {1}{3} (2 b e n) \int \frac {1}{\left (d+\frac {e}{x^{2/3}}\right ) x^{8/3}} \, dx\\ &=-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x}-\frac {1}{3} (2 b e n) \int \frac {1}{\left (e+d x^{2/3}\right ) x^2} \, dx\\ &=-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x}-(2 b e n) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {2 b n}{3 x}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x}+(2 b d n) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {2 b n}{3 x}-\frac {2 b d n}{e \sqrt [3]{x}}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x}-\frac {\left (2 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=\frac {2 b n}{3 x}-\frac {2 b d n}{e \sqrt [3]{x}}-\frac {2 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{e^{3/2}}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 80, normalized size = 1.04 \[ -\frac {a}{x}-\frac {b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x}+\frac {2 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {e}}{\sqrt {d} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac {2 b d n}{e \sqrt [3]{x}}+\frac {2 b n}{3 x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 235, normalized size = 3.05 \[ \left [\frac {3 \, b d n x \sqrt {-\frac {d}{e}} \log \left (\frac {d^{3} x^{2} + 2 \, d e^{2} x \sqrt {-\frac {d}{e}} - e^{3} - 2 \, {\left (d^{2} e x \sqrt {-\frac {d}{e}} - d e^{2}\right )} x^{\frac {2}{3}} - 2 \, {\left (d^{2} e x + e^{3} \sqrt {-\frac {d}{e}}\right )} x^{\frac {1}{3}}}{d^{3} x^{2} + e^{3}}\right ) - 3 \, b e n \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right ) - 6 \, b d n x^{\frac {2}{3}} + 2 \, b e n - 3 \, b e \log \relax (c) - 3 \, a e}{3 \, e x}, -\frac {6 \, b d n x \sqrt {\frac {d}{e}} \arctan \left (x^{\frac {1}{3}} \sqrt {\frac {d}{e}}\right ) + 3 \, b e n \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right ) + 6 \, b d n x^{\frac {2}{3}} - 2 \, b e n + 3 \, b e \log \relax (c) + 3 \, a e}{3 \, e x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.46, size = 73, normalized size = 0.95 \[ -\frac {1}{3} \, {\left (2 \, {\left (3 \, d^{\frac {3}{2}} \arctan \left (\sqrt {d} x^{\frac {1}{3}} e^{\left (-\frac {1}{2}\right )}\right ) e^{\left (-\frac {5}{2}\right )} + \frac {{\left (3 \, d x^{\frac {2}{3}} - e\right )} e^{\left (-2\right )}}{x}\right )} e + \frac {3 \, \log \left (d + \frac {e}{x^{\frac {2}{3}}}\right )}{x}\right )} b n - \frac {b \log \relax (c)}{x} - \frac {a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )+a}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.99, size = 72, normalized size = 0.94 \[ -\frac {2}{3} \, b e n {\left (\frac {3 \, d^{2} \arctan \left (\frac {d x^{\frac {1}{3}}}{\sqrt {d e}}\right )}{\sqrt {d e} e^{2}} + \frac {3 \, d x^{\frac {2}{3}} - e}{e^{2} x}\right )} - \frac {b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )}{x} - \frac {a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________