Optimal. Leaf size=211 \[ \frac {d^3 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \log \left (c x^n\right )\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \log \left (c x^n\right )\right )}{f^7 (m+7)}-\frac {b d^3 n (f x)^{m+1}}{f (m+1)^2}-\frac {3 b d^2 e n (f x)^{m+3}}{f^3 (m+3)^2}-\frac {3 b d e^2 n (f x)^{m+5}}{f^5 (m+5)^2}-\frac {b e^3 n (f x)^{m+7}}{f^7 (m+7)^2} \]
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Rubi [A] time = 1.68, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {270, 2350, 14} \[ \frac {3 d^2 e (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}+\frac {d^3 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \log \left (c x^n\right )\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \log \left (c x^n\right )\right )}{f^7 (m+7)}-\frac {3 b d^2 e n (f x)^{m+3}}{f^3 (m+3)^2}-\frac {b d^3 n (f x)^{m+1}}{f (m+1)^2}-\frac {3 b d e^2 n (f x)^{m+5}}{f^5 (m+5)^2}-\frac {b e^3 n (f x)^{m+7}}{f^7 (m+7)^2} \]
Antiderivative was successfully verified.
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Rule 14
Rule 270
Rule 2350
Rubi steps
\begin {align*} \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \log \left (c x^n\right )\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \log \left (c x^n\right )\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \log \left (c x^n\right )\right )}{f^7 (7+m)}-(b n) \int (f x)^m \left (\frac {d^3}{1+m}+\frac {3 d^2 e x^2}{3+m}+\frac {3 d e^2 x^4}{5+m}+\frac {e^3 x^6}{7+m}\right ) \, dx\\ &=\frac {d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \log \left (c x^n\right )\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \log \left (c x^n\right )\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \log \left (c x^n\right )\right )}{f^7 (7+m)}-(b n) \int \left (\frac {d^3 (f x)^m}{1+m}+\frac {3 d^2 e (f x)^{2+m}}{f^2 (3+m)}+\frac {3 d e^2 (f x)^{4+m}}{f^4 (5+m)}+\frac {e^3 (f x)^{6+m}}{f^6 (7+m)}\right ) \, dx\\ &=-\frac {b d^3 n (f x)^{1+m}}{f (1+m)^2}-\frac {3 b d^2 e n (f x)^{3+m}}{f^3 (3+m)^2}-\frac {3 b d e^2 n (f x)^{5+m}}{f^5 (5+m)^2}-\frac {b e^3 n (f x)^{7+m}}{f^7 (7+m)^2}+\frac {d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \log \left (c x^n\right )\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \log \left (c x^n\right )\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \log \left (c x^n\right )\right )}{f^7 (7+m)}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 156, normalized size = 0.74 \[ x (f x)^m \left (\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{m+1}+\frac {3 d^2 e x^2 \left (a+b \log \left (c x^n\right )\right )}{m+3}+\frac {3 d e^2 x^4 \left (a+b \log \left (c x^n\right )\right )}{m+5}+\frac {e^3 x^6 \left (a+b \log \left (c x^n\right )\right )}{m+7}-\frac {b d^3 n}{(m+1)^2}-\frac {3 b d^2 e n x^2}{(m+3)^2}-\frac {3 b d e^2 n x^4}{(m+5)^2}-\frac {b e^3 n x^6}{(m+7)^2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 1222, normalized size = 5.79 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.62, size = 553, normalized size = 2.62 \[ \frac {b f^{6} f^{m} x^{7} x^{m} e^{3} \log \relax (c)}{f^{6} m + 7 \, f^{6}} + \frac {a f^{6} f^{m} x^{7} x^{m} e^{3}}{f^{6} m + 7 \, f^{6}} + \frac {3 \, b d f^{4} f^{m} x^{5} x^{m} e^{2} \log \relax (c)}{f^{4} m + 5 \, f^{4}} + \frac {3 \, a d f^{4} f^{m} x^{5} x^{m} e^{2}}{f^{4} m + 5 \, f^{4}} + \frac {b f^{m} m n x^{7} x^{m} e^{3} \log \relax (x)}{m^{2} + 14 \, m + 49} + \frac {7 \, b f^{m} n x^{7} x^{m} e^{3} \log \relax (x)}{m^{2} + 14 \, m + 49} + \frac {3 \, b d f^{m} m n x^{5} x^{m} e^{2} \log \relax (x)}{m^{2} + 10 \, m + 25} - \frac {b f^{m} n x^{7} x^{m} e^{3}}{m^{2} + 14 \, m + 49} + \frac {3 \, b d^{2} f^{2} f^{m} x^{3} x^{m} e \log \relax (c)}{f^{2} m + 3 \, f^{2}} + \frac {15 \, b d f^{m} n x^{5} x^{m} e^{2} \log \relax (x)}{m^{2} + 10 \, m + 25} + \frac {3 \, b d^{2} f^{m} m n x^{3} x^{m} e \log \relax (x)}{m^{2} + 6 \, m + 9} - \frac {3 \, b d f^{m} n x^{5} x^{m} e^{2}}{m^{2} + 10 \, m + 25} + \frac {3 \, a d^{2} f^{2} f^{m} x^{3} x^{m} e}{f^{2} m + 3 \, f^{2}} + \frac {9 \, b d^{2} f^{m} n x^{3} x^{m} e \log \relax (x)}{m^{2} + 6 \, m + 9} - \frac {3 \, b d^{2} f^{m} n x^{3} x^{m} e}{m^{2} + 6 \, m + 9} + \frac {b d^{3} f^{m} m n x x^{m} \log \relax (x)}{m^{2} + 2 \, m + 1} + \frac {b d^{3} f^{m} n x x^{m} \log \relax (x)}{m^{2} + 2 \, m + 1} - \frac {b d^{3} f^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac {\left (f x\right )^{m} b d^{3} x \log \relax (c)}{m + 1} + \frac {\left (f x\right )^{m} a d^{3} x}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.62, size = 5139, normalized size = 24.36 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 271, normalized size = 1.28 \[ \frac {b e^{3} f^{m} x^{7} x^{m} \log \left (c x^{n}\right )}{m + 7} + \frac {a e^{3} f^{m} x^{7} x^{m}}{m + 7} - \frac {b e^{3} f^{m} n x^{7} x^{m}}{{\left (m + 7\right )}^{2}} + \frac {3 \, b d e^{2} f^{m} x^{5} x^{m} \log \left (c x^{n}\right )}{m + 5} + \frac {3 \, a d e^{2} f^{m} x^{5} x^{m}}{m + 5} - \frac {3 \, b d e^{2} f^{m} n x^{5} x^{m}}{{\left (m + 5\right )}^{2}} + \frac {3 \, b d^{2} e f^{m} x^{3} x^{m} \log \left (c x^{n}\right )}{m + 3} + \frac {3 \, a d^{2} e f^{m} x^{3} x^{m}}{m + 3} - \frac {3 \, b d^{2} e f^{m} n x^{3} x^{m}}{{\left (m + 3\right )}^{2}} - \frac {b d^{3} f^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {\left (f x\right )^{m + 1} b d^{3} \log \left (c x^{n}\right )}{f {\left (m + 1\right )}} + \frac {\left (f x\right )^{m + 1} a d^{3}}{f {\left (m + 1\right )}} \]
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 \[ \int {\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
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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