Optimal. Leaf size=41 \[ \frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 d}-\frac {4 b n (d x)^{7/2}}{49 d} \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2304} \[ \frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 d}-\frac {4 b n (d x)^{7/2}}{49 d} \]
Antiderivative was successfully verified.
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Rule 2304
Rubi steps
\begin {align*} \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {4 b n (d x)^{7/2}}{49 d}+\frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.71 \[ \frac {2}{49} x (d x)^{5/2} \left (7 a+7 b \log \left (c x^n\right )-2 b n\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 50, normalized size = 1.22 \[ \frac {2}{49} \, {\left (7 \, b d^{2} n x^{3} \log \relax (x) + 7 \, b d^{2} x^{3} \log \relax (c) - {\left (2 \, b d^{2} n - 7 \, a d^{2}\right )} x^{3}\right )} \sqrt {d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.52, size = 117, normalized size = 2.85 \[ \left (\frac {1}{7} i + \frac {1}{7}\right ) \, \sqrt {2} b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) \log \relax (x) - \left (\frac {1}{7} i - \frac {1}{7}\right ) \, \sqrt {2} b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \log \relax (x) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) - \left (\frac {2}{49} i + \frac {2}{49}\right ) \, \sqrt {2} b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) + \left (\frac {2}{49} i - \frac {2}{49}\right ) \, \sqrt {2} b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) + \frac {2}{7} \, b d^{\frac {5}{2}} x^{\frac {7}{2}} \log \relax (c) + \frac {2}{7} \, a d^{\frac {5}{2}} x^{\frac {7}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.15, size = 128, normalized size = 3.12 \[ \frac {2 b \,d^{3} x^{4} \ln \left (x^{n}\right )}{7 \sqrt {d x}}+\frac {\left (-7 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+7 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+7 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-7 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 b n +14 b \ln \relax (c )+14 a \right ) d^{3} x^{4}}{49 \sqrt {d x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 41, normalized size = 1.00 \[ -\frac {4 \, \left (d x\right )^{\frac {7}{2}} b n}{49 \, d} + \frac {2 \, \left (d x\right )^{\frac {7}{2}} b \log \left (c x^{n}\right )}{7 \, d} + \frac {2 \, \left (d x\right )^{\frac {7}{2}} a}{7 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (d\,x\right )}^{5/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
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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