Optimal. Leaf size=108 \[ -\frac {b n}{3 e^2 \sqrt {d+e x^2}}-\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 \sqrt {d} e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{e^2 \sqrt {d+e x^2}} \]
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Rubi [A]
time = 0.12, 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 = {272, 45, 2392,
12, 457, 79, 65, 214} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 65
Rule 79
Rule 214
Rule 272
Rule 457
Rule 2392
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) \text {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) \text {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) \text {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 = 0.17, size = 137, normalized size = 1.27 \begin {gather*} \frac {\frac {2 b n \log (x)}{\sqrt {d}}-\frac {b n \left (2 d+3 e x^2\right ) \log (x)}{\left (d+e x^2\right )^{3/2}}+\frac {d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )-\left (d+e x^2\right ) \left (3 a+b n-3 b n \log (x)+3 b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}}-\frac {2 b n \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )}{\sqrt {d}}}{3 e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 138, normalized size = 1.28 \begin {gather*} \frac {1}{3} \, {\left (\frac {e^{\left (-2\right )} \log \left (\frac {\sqrt {x^{2} e + d} - \sqrt {d}}{\sqrt {x^{2} e + d} + \sqrt {d}}\right )}{\sqrt {d}} - \frac {e^{\left (-2\right )}}{\sqrt {x^{2} e + d}}\right )} b n - \frac {1}{3} \, {\left (\frac {3 \, x^{2} e^{\left (-1\right )}}{{\left (x^{2} e + d\right )}^{\frac {3}{2}}} + \frac {2 \, d e^{\left (-2\right )}}{{\left (x^{2} e + d\right )}^{\frac {3}{2}}}\right )} b \log \left (c x^{n}\right ) - \frac {1}{3} \, {\left (\frac {3 \, x^{2} e^{\left (-1\right )}}{{\left (x^{2} e + d\right )}^{\frac {3}{2}}} + \frac {2 \, d e^{\left (-2\right )}}{{\left (x^{2} e + d\right )}^{\frac {3}{2}}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 328, normalized size = 3.04 \begin {gather*} \left [\frac {{\left (b n x^{4} e^{2} + 2 \, b d n x^{2} e + b d^{2} n\right )} \sqrt {d} \log \left (-\frac {x^{2} e - 2 \, \sqrt {x^{2} e + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) - {\left (b d^{2} n + {\left (b d n + 3 \, a d\right )} x^{2} e + 2 \, a d^{2} + {\left (3 \, b d x^{2} e + 2 \, b d^{2}\right )} \log \left (c\right ) + {\left (3 \, b d n x^{2} e + 2 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}}{3 \, {\left (d x^{4} e^{4} + 2 \, d^{2} x^{2} e^{3} + d^{3} e^{2}\right )}}, \frac {2 \, {\left (b n x^{4} e^{2} + 2 \, b d n x^{2} e + b d^{2} n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d}}{\sqrt {x^{2} e + d}}\right ) - {\left (b d^{2} n + {\left (b d n + 3 \, a d\right )} x^{2} e + 2 \, a d^{2} + {\left (3 \, b d x^{2} e + 2 \, b d^{2}\right )} \log \left (c\right ) + {\left (3 \, b d n x^{2} e + 2 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}}{3 \, {\left (d x^{4} e^{4} + 2 \, d^{2} x^{2} e^{3} + d^{3} e^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 30.87, size = 333, normalized size = 3.08 \begin {gather*} 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 )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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