Optimal. Leaf size=112 \[ -\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}+\frac {b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d}-\frac {b e n \sqrt {d+e x^2}}{3 d x}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 d x^3} \]
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Rubi [A] time = 0.10, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2335, 277, 217, 206} \[ -\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}+\frac {b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d}-\frac {b e n \sqrt {d+e x^2}}{3 d x}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 d x^3} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 277
Rule 2335
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx &=-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}+\frac {(b n) \int \frac {\left (d+e x^2\right )^{3/2}}{x^4} \, dx}{3 d}\\ &=-\frac {b n \left (d+e x^2\right )^{3/2}}{9 d x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}+\frac {(b e n) \int \frac {\sqrt {d+e x^2}}{x^2} \, dx}{3 d}\\ &=-\frac {b e n \sqrt {d+e x^2}}{3 d x}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 d x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}+\frac {\left (b e^2 n\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{3 d}\\ &=-\frac {b e n \sqrt {d+e x^2}}{3 d x}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 d x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}+\frac {\left (b e^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{3 d}\\ &=-\frac {b e n \sqrt {d+e x^2}}{3 d x}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 d x^3}+\frac {b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 99, normalized size = 0.88 \[ -\frac {\sqrt {d+e x^2} \left (3 a \left (d+e x^2\right )+b n \left (d+4 e x^2\right )\right )+3 b \left (d+e x^2\right )^{3/2} \log \left (c x^n\right )-3 b e^{3/2} n x^3 \log \left (\sqrt {e} \sqrt {d+e x^2}+e x\right )}{9 d x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 212, normalized size = 1.89 \[ \left [\frac {3 \, b e^{\frac {3}{2}} n x^{3} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - 2 \, {\left (b d n + {\left (4 \, b e n + 3 \, a e\right )} x^{2} + 3 \, a d + 3 \, {\left (b e x^{2} + b d\right )} \log \relax (c) + 3 \, {\left (b e n x^{2} + b d n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{18 \, d x^{3}}, -\frac {3 \, b \sqrt {-e} e n x^{3} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (b d n + {\left (4 \, b e n + 3 \, a e\right )} x^{2} + 3 \, a d + 3 \, {\left (b e x^{2} + b d\right )} \log \relax (c) + 3 \, {\left (b e n x^{2} + b d n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{9 \, d x^{3}}\right ] \]
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 \[ \int \frac {\sqrt {e x^{2} + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \sqrt {e \,x^{2}+d}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 118, normalized size = 1.05 \[ \frac {{\left (\frac {3 \, \sqrt {e x^{2} + d} e^{2} x}{d} + 3 \, e^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right ) - \frac {2 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} e}{d x} - \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}}}{d x^{3}}\right )} b n}{9 \, d} - \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} b \log \left (c x^{n}\right )}{3 \, d x^{3}} - \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} a}{3 \, d x^{3}} \]
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 \[ \int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \sqrt {d + e x^{2}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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