Optimal. Leaf size=108 \[ -\frac {a+b \log \left (c x^n\right )}{e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {b n}{3 e^2 \sqrt {d+e x^2}}-\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 \sqrt {d} e^2} \]
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Rubi [A] time = 0.16, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {266, 43, 2350, 12, 446, 78, 63, 208} \[ -\frac {a+b \log \left (c x^n\right )}{e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {b n}{3 e^2 \sqrt {d+e x^2}}-\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 \sqrt {d} e^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 63
Rule 78
Rule 208
Rule 266
Rule 446
Rule 2350
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{e^2 \sqrt {d+e x^2}}-(b n) \int \frac {-2 d-3 e x^2}{3 e^2 x \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{e^2 \sqrt {d+e x^2}}-\frac {(b n) \int \frac {-2 d-3 e x^2}{x \left (d+e x^2\right )^{3/2}} \, dx}{3 e^2}\\ &=\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{e^2 \sqrt {d+e x^2}}-\frac {(b n) \operatorname {Subst}\left (\int \frac {-2 d-3 e x}{x (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e^2}\\ &=-\frac {b n}{3 e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{e^2 \sqrt {d+e x^2}}+\frac {(b n) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{3 e^2}\\ &=-\frac {b n}{3 e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{e^2 \sqrt {d+e x^2}}+\frac {(2 b n) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{3 e^3}\\ &=-\frac {b n}{3 e^2 \sqrt {d+e x^2}}-\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 \sqrt {d} e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{e^2 \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 137, normalized size = 1.27 \[ \frac {\frac {d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-\left (d+e x^2\right ) \left (3 a+3 b \log \left (c x^n\right )-3 b n \log (x)+b n\right )}{\left (d+e x^2\right )^{3/2}}-\frac {b n \log (x) \left (2 d+3 e x^2\right )}{\left (d+e x^2\right )^{3/2}}-\frac {2 b n \log \left (\sqrt {d} \sqrt {d+e x^2}+d\right )}{\sqrt {d}}+\frac {2 b n \log (x)}{\sqrt {d}}}{3 e^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 325, normalized size = 3.01 \[ \left [\frac {{\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \sqrt {d} \log \left (-\frac {e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) - {\left (b d^{2} n + 2 \, a d^{2} + {\left (b d e n + 3 \, a d e\right )} x^{2} + {\left (3 \, b d e x^{2} + 2 \, b d^{2}\right )} \log \relax (c) + {\left (3 \, b d e n x^{2} + 2 \, b d^{2} n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{3 \, {\left (d e^{4} x^{4} + 2 \, d^{2} e^{3} x^{2} + d^{3} e^{2}\right )}}, \frac {2 \, {\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d}}{\sqrt {e x^{2} + d}}\right ) - {\left (b d^{2} n + 2 \, a d^{2} + {\left (b d e n + 3 \, a d e\right )} x^{2} + {\left (3 \, b d e x^{2} + 2 \, b d^{2}\right )} \log \relax (c) + {\left (3 \, b d e n x^{2} + 2 \, b d^{2} n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{3 \, {\left (d e^{4} x^{4} + 2 \, d^{2} e^{3} x^{2} + d^{3} e^{2}\right )}}\right ] \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) x^{3}}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 137, normalized size = 1.27 \[ \frac {1}{3} \, b n {\left (\frac {\log \left (\frac {\sqrt {e x^{2} + d} - \sqrt {d}}{\sqrt {e x^{2} + d} + \sqrt {d}}\right )}{\sqrt {d} e^{2}} - \frac {1}{\sqrt {e x^{2} + d} e^{2}}\right )} - \frac {1}{3} \, b {\left (\frac {3 \, x^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e} + \frac {2 \, d}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{2}}\right )} \log \left (c x^{n}\right ) - \frac {1}{3} \, a {\left (\frac {3 \, x^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e} + \frac {2 \, d}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 57.19, size = 333, normalized size = 3.08 \[ a \left (\begin {cases} \frac {x^{4}}{4 d^{\frac {5}{2}}} & \text {for}\: e = 0 \\\frac {d}{3 e^{2} \left (d + e x^{2}\right )^{\frac {3}{2}}} - \frac {1}{e^{2} \sqrt {d + e x^{2}}} & \text {otherwise} \end {cases}\right ) - b n \left (\begin {cases} \frac {x^{4}}{16 d^{\frac {5}{2}}} & \text {for}\: e = 0 \\\frac {2 d^{4} \sqrt {1 + \frac {e x^{2}}{d}}}{6 d^{\frac {9}{2}} e^{2} + 6 d^{\frac {7}{2}} e^{3} x^{2}} + \frac {d^{4} \log {\left (\frac {e x^{2}}{d} \right )}}{6 d^{\frac {9}{2}} e^{2} + 6 d^{\frac {7}{2}} e^{3} x^{2}} - \frac {2 d^{4} \log {\left (\sqrt {1 + \frac {e x^{2}}{d}} + 1 \right )}}{6 d^{\frac {9}{2}} e^{2} + 6 d^{\frac {7}{2}} e^{3} x^{2}} + \frac {d^{3} x^{2} \log {\left (\frac {e x^{2}}{d} \right )}}{6 d^{\frac {9}{2}} e + 6 d^{\frac {7}{2}} e^{2} x^{2}} - \frac {2 d^{3} x^{2} \log {\left (\sqrt {1 + \frac {e x^{2}}{d}} + 1 \right )}}{6 d^{\frac {9}{2}} e + 6 d^{\frac {7}{2}} e^{2} x^{2}} + \frac {\operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {e} x} \right )}}{\sqrt {d} e^{2}} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} \frac {x^{4}}{4 d^{\frac {5}{2}}} & \text {for}\: e = 0 \\\frac {d}{3 e^{2} \left (d + e x^{2}\right )^{\frac {3}{2}}} - \frac {1}{e^{2} \sqrt {d + e x^{2}}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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