Optimal. Leaf size=417 \[ -\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^3 \left (f+g x^2\right )}-\frac {f \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}-\frac {f \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac {b d e f^{3/2} n \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 g^{5/2} \left (d^2 g+e^2 f\right )}-\frac {b e^2 f^2 n \log \left (f+g x^2\right )}{4 g^3 \left (d^2 g+e^2 f\right )}+\frac {b e^2 f^2 n \log (d+e x)}{2 g^3 \left (d^2 g+e^2 f\right )}-\frac {b d^2 n \log (d+e x)}{2 e^2 g^2}-\frac {b f n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^3}-\frac {b f n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{g^3}+\frac {b d n x}{2 e g^2}-\frac {b n x^2}{4 g^2} \]
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Rubi [A] time = 0.49, antiderivative size = 417, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.482, Rules used = {266, 43, 2416, 2395, 2413, 706, 31, 635, 205, 260, 2394, 2393, 2391} \[ -\frac {b f n \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^3}-\frac {b f n \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right )}{g^3}-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^3 \left (f+g x^2\right )}-\frac {f \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}-\frac {f \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}-\frac {b e^2 f^2 n \log \left (f+g x^2\right )}{4 g^3 \left (d^2 g+e^2 f\right )}+\frac {b e^2 f^2 n \log (d+e x)}{2 g^3 \left (d^2 g+e^2 f\right )}+\frac {b d e f^{3/2} n \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 g^{5/2} \left (d^2 g+e^2 f\right )}-\frac {b d^2 n \log (d+e x)}{2 e^2 g^2}+\frac {b d n x}{2 e g^2}-\frac {b n x^2}{4 g^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 43
Rule 205
Rule 260
Rule 266
Rule 635
Rule 706
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2413
Rule 2416
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx &=\int \left (\frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {f^2 x \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 \left (f+g x^2\right )^2}-\frac {2 f x \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 \left (f+g x^2\right )}\right ) \, dx\\ &=\frac {\int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g^2}-\frac {(2 f) \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx}{g^2}+\frac {f^2 \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx}{g^2}\\ &=\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^3 \left (f+g x^2\right )}-\frac {(2 f) \int \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{2 \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {a+b \log \left (c (d+e x)^n\right )}{2 \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{g^2}+\frac {\left (b e f^2 n\right ) \int \frac {1}{(d+e x) \left (f+g x^2\right )} \, dx}{2 g^3}-\frac {(b e n) \int \frac {x^2}{d+e x} \, dx}{2 g^2}\\ &=\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^3 \left (f+g x^2\right )}+\frac {f \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}-\sqrt {g} x} \, dx}{g^{5/2}}-\frac {f \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}+\sqrt {g} x} \, dx}{g^{5/2}}-\frac {(b e n) \int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx}{2 g^2}+\frac {\left (b e f^2 n\right ) \int \frac {d g-e g x}{f+g x^2} \, dx}{2 g^3 \left (e^2 f+d^2 g\right )}+\frac {\left (b e^3 f^2 n\right ) \int \frac {1}{d+e x} \, dx}{2 g^3 \left (e^2 f+d^2 g\right )}\\ &=\frac {b d n x}{2 e g^2}-\frac {b n x^2}{4 g^2}-\frac {b d^2 n \log (d+e x)}{2 e^2 g^2}+\frac {b e^2 f^2 n \log (d+e x)}{2 g^3 \left (e^2 f+d^2 g\right )}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^3 \left (f+g x^2\right )}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^3}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^3}+\frac {(b e f n) \int \frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{d+e x} \, dx}{g^3}+\frac {(b e f n) \int \frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx}{g^3}+\frac {\left (b d e f^2 n\right ) \int \frac {1}{f+g x^2} \, dx}{2 g^2 \left (e^2 f+d^2 g\right )}-\frac {\left (b e^2 f^2 n\right ) \int \frac {x}{f+g x^2} \, dx}{2 g^2 \left (e^2 f+d^2 g\right )}\\ &=\frac {b d n x}{2 e g^2}-\frac {b n x^2}{4 g^2}+\frac {b d e f^{3/2} n \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 g^{5/2} \left (e^2 f+d^2 g\right )}-\frac {b d^2 n \log (d+e x)}{2 e^2 g^2}+\frac {b e^2 f^2 n \log (d+e x)}{2 g^3 \left (e^2 f+d^2 g\right )}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^3 \left (f+g x^2\right )}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^3}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^3}-\frac {b e^2 f^2 n \log \left (f+g x^2\right )}{4 g^3 \left (e^2 f+d^2 g\right )}+\frac {(b f n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{g^3}+\frac {(b f n) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{g^3}\\ &=\frac {b d n x}{2 e g^2}-\frac {b n x^2}{4 g^2}+\frac {b d e f^{3/2} n \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 g^{5/2} \left (e^2 f+d^2 g\right )}-\frac {b d^2 n \log (d+e x)}{2 e^2 g^2}+\frac {b e^2 f^2 n \log (d+e x)}{2 g^3 \left (e^2 f+d^2 g\right )}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^3 \left (f+g x^2\right )}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^3}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^3}-\frac {b e^2 f^2 n \log \left (f+g x^2\right )}{4 g^3 \left (e^2 f+d^2 g\right )}-\frac {b f n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^3}-\frac {b f n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^3}\\ \end {align*}
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Mathematica [C] time = 1.48, size = 530, normalized size = 1.27 \[ \frac {-\frac {2 f^2 \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )}{f+g x^2}-4 f \log \left (f+g x^2\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )+2 g x^2 \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )+b n \left (\frac {g \left (e x (2 d-e x)-2 \left (d^2-e^2 x^2\right ) \log (d+e x)\right )}{e^2}+\frac {f^{3/2} \left (i \sqrt {g} (d+e x) \log (d+e x)-e \left (\sqrt {f}+i \sqrt {g} x\right ) \log \left (-\sqrt {g} x+i \sqrt {f}\right )\right )}{\left (\sqrt {f}+i \sqrt {g} x\right ) \left (e \sqrt {f}-i d \sqrt {g}\right )}+\frac {i f^{3/2} \left (-\sqrt {g} (d+e x) \log (d+e x)+e \left (\sqrt {g} x+i \sqrt {f}\right ) \log \left (\sqrt {g} x+i \sqrt {f}\right )\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (e \sqrt {f}+i d \sqrt {g}\right )}-4 f \left (\text {Li}_2\left (-\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}-i d \sqrt {g}}\right )+\log (d+e x) \log \left (\frac {e \left (\sqrt {f}+i \sqrt {g} x\right )}{e \sqrt {f}-i d \sqrt {g}}\right )\right )-4 f \left (\text {Li}_2\left (\frac {i \sqrt {g} (d+e x)}{i \sqrt {g} d+e \sqrt {f}}\right )+\log (d+e x) \log \left (\frac {e \left (\sqrt {f}-i \sqrt {g} x\right )}{e \sqrt {f}+i d \sqrt {g}}\right )\right )\right )}{4 g^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{5} \log \left ({\left (e x + d\right )}^{n} c\right ) + a x^{5}}{g^{2} x^{4} + 2 \, f g x^{2} + f^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{5}}{{\left (g x^{2} + f\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 1008, normalized size = 2.42 \[ \text {result too large to display} \]
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 \[ -\frac {1}{2} \, a {\left (\frac {f^{2}}{g^{4} x^{2} + f g^{3}} - \frac {x^{2}}{g^{2}} + \frac {2 \, f \log \left (g x^{2} + f\right )}{g^{3}}\right )} + b \int \frac {x^{5} \log \left ({\left (e x + d\right )}^{n}\right ) + x^{5} \log \relax (c)}{g^{2} x^{4} + 2 \, f g x^{2} + f^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{{\left (g\,x^2+f\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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