Optimal. Leaf size=81 \[ \frac {(f x)^{m+1} \log \left (c \left (d+e x^2\right )^p\right )}{f (m+1)}-\frac {2 e p (f x)^{m+3} \, _2F_1\left (1,\frac {m+3}{2};\frac {m+5}{2};-\frac {e x^2}{d}\right )}{d f^3 (m+1) (m+3)} \]
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Rubi [A] time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2455, 16, 364} \[ \frac {(f x)^{m+1} \log \left (c \left (d+e x^2\right )^p\right )}{f (m+1)}-\frac {2 e p (f x)^{m+3} \, _2F_1\left (1,\frac {m+3}{2};\frac {m+5}{2};-\frac {e x^2}{d}\right )}{d f^3 (m+1) (m+3)} \]
Antiderivative was successfully verified.
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Rule 16
Rule 364
Rule 2455
Rubi steps
\begin {align*} \int (f x)^m \log \left (c \left (d+e x^2\right )^p\right ) \, dx &=\frac {(f x)^{1+m} \log \left (c \left (d+e x^2\right )^p\right )}{f (1+m)}-\frac {(2 e p) \int \frac {x (f x)^{1+m}}{d+e x^2} \, dx}{f (1+m)}\\ &=\frac {(f x)^{1+m} \log \left (c \left (d+e x^2\right )^p\right )}{f (1+m)}-\frac {(2 e p) \int \frac {(f x)^{2+m}}{d+e x^2} \, dx}{f^2 (1+m)}\\ &=-\frac {2 e p (f x)^{3+m} \, _2F_1\left (1,\frac {3+m}{2};\frac {5+m}{2};-\frac {e x^2}{d}\right )}{d f^3 (1+m) (3+m)}+\frac {(f x)^{1+m} \log \left (c \left (d+e x^2\right )^p\right )}{f (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 70, normalized size = 0.86 \[ \frac {x (f x)^m \left (d (m+3) \log \left (c \left (d+e x^2\right )^p\right )-2 e p x^2 \, _2F_1\left (1,\frac {m+3}{2};\frac {m+5}{2};-\frac {e x^2}{d}\right )\right )}{d (m+1) (m+3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ), x\right ) \]
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 \[ \int \left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.12, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\, dx \]
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 \[ \frac {f^{m} p x x^{m} \log \left (e x^{2} + d\right )}{m + 1} + \int \frac {{\left (d f^{m} {\left (m + 1\right )} \log \relax (c) + {\left (e f^{m} {\left (m + 1\right )} \log \relax (c) - 2 \, e f^{m} p\right )} x^{2}\right )} x^{m}}{e {\left (m + 1\right )} x^{2} + d {\left (m + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,{\left (f\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 95.99, size = 359, normalized size = 4.43 \[ - 2 e p \left (\begin {cases} \frac {0^{m} \sqrt {- \frac {d}{e^{3}}} \log {\left (- e \sqrt {- \frac {d}{e^{3}}} + x \right )}}{2} - \frac {0^{m} \sqrt {- \frac {d}{e^{3}}} \log {\left (e \sqrt {- \frac {d}{e^{3}}} + x \right )}}{2} + \frac {0^{m} x}{e} & \text {for}\: \left (f = 0 \wedge m \neq -1\right ) \vee f = 0 \\\frac {f f^{m} m x^{3} x^{m} \Phi \left (\frac {e x^{2} e^{i \pi }}{d}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 d f m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 4 d f \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {3 f f^{m} x^{3} x^{m} \Phi \left (\frac {e x^{2} e^{i \pi }}{d}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 d f m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 4 d f \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} & \text {for}\: m > -\infty \wedge m < \infty \wedge m \neq -1 \\- \frac {\begin {cases} \log {\relax (d )} \log {\relax (x )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (d )} \log {\left (\frac {1}{x} \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (d )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (d )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {otherwise} \end {cases}}{2 e f} + \frac {\log {\left (f x \right )} \log {\left (d + e x^{2} \right )}}{2 e f} & \text {otherwise} \end {cases}\right ) + \left (\begin {cases} 0^{m} x & \text {for}\: f = 0 \\\frac {\begin {cases} \frac {\left (f x\right )^{m + 1}}{m + 1} & \text {for}\: m \neq -1 \\\log {\left (f x \right )} & \text {otherwise} \end {cases}}{f} & \text {otherwise} \end {cases}\right ) \log {\left (c \left (d + e x^{2}\right )^{p} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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