Optimal. Leaf size=213 \[ \frac {2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}+\frac {32 b d^{9/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{315 e^3}-\frac {32 b d^4 n \sqrt {d+e x}}{315 e^3}-\frac {32 b d^3 n (d+e x)^{3/2}}{945 e^3}-\frac {32 b d^2 n (d+e x)^{5/2}}{1575 e^3}+\frac {44 b d n (d+e x)^{7/2}}{441 e^3}-\frac {4 b n (d+e x)^{9/2}}{81 e^3} \]
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Rubi [A] time = 0.20, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {43, 2350, 12, 897, 1261, 208} \[ \frac {2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {32 b d^4 n \sqrt {d+e x}}{315 e^3}-\frac {32 b d^3 n (d+e x)^{3/2}}{945 e^3}-\frac {32 b d^2 n (d+e x)^{5/2}}{1575 e^3}+\frac {32 b d^{9/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{315 e^3}+\frac {44 b d n (d+e x)^{7/2}}{441 e^3}-\frac {4 b n (d+e x)^{9/2}}{81 e^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 208
Rule 897
Rule 1261
Rule 2350
Rubi steps
\begin {align*} \int x^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-(b n) \int \frac {2 (d+e x)^{5/2} \left (8 d^2-20 d e x+35 e^2 x^2\right )}{315 e^3 x} \, dx\\ &=\frac {2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {(2 b n) \int \frac {(d+e x)^{5/2} \left (8 d^2-20 d e x+35 e^2 x^2\right )}{x} \, dx}{315 e^3}\\ &=\frac {2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {(4 b n) \operatorname {Subst}\left (\int \frac {x^6 \left (63 d^2-90 d x^2+35 x^4\right )}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{315 e^4}\\ &=\frac {2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {(4 b n) \operatorname {Subst}\left (\int \left (8 d^4 e+8 d^3 e x^2+8 d^2 e x^4-55 d e x^6+35 e x^8+\frac {8 d^5}{-\frac {d}{e}+\frac {x^2}{e}}\right ) \, dx,x,\sqrt {d+e x}\right )}{315 e^4}\\ &=-\frac {32 b d^4 n \sqrt {d+e x}}{315 e^3}-\frac {32 b d^3 n (d+e x)^{3/2}}{945 e^3}-\frac {32 b d^2 n (d+e x)^{5/2}}{1575 e^3}+\frac {44 b d n (d+e x)^{7/2}}{441 e^3}-\frac {4 b n (d+e x)^{9/2}}{81 e^3}+\frac {2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {\left (32 b d^5 n\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{315 e^4}\\ &=-\frac {32 b d^4 n \sqrt {d+e x}}{315 e^3}-\frac {32 b d^3 n (d+e x)^{3/2}}{945 e^3}-\frac {32 b d^2 n (d+e x)^{5/2}}{1575 e^3}+\frac {44 b d n (d+e x)^{7/2}}{441 e^3}-\frac {4 b n (d+e x)^{9/2}}{81 e^3}+\frac {32 b d^{9/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{315 e^3}+\frac {2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 153, normalized size = 0.72 \[ \frac {2 \left (\sqrt {d+e x} \left (315 a \left (8 d^2-20 d e x+35 e^2 x^2\right ) (d+e x)^2+315 b \left (8 d^2-20 d e x+35 e^2 x^2\right ) (d+e x)^2 \log \left (c x^n\right )-2 b n \left (2614 d^4-677 d^3 e x+429 d^2 e^2 x^2+2425 d e^3 x^3+1225 e^4 x^4\right )\right )+5040 b d^{9/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{99225 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 496, normalized size = 2.33 \[ \left [\frac {2 \, {\left (2520 \, b d^{\frac {9}{2}} n \log \left (\frac {e x + 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) - {\left (5228 \, b d^{4} n - 2520 \, a d^{4} + 1225 \, {\left (2 \, b e^{4} n - 9 \, a e^{4}\right )} x^{4} + 50 \, {\left (97 \, b d e^{3} n - 315 \, a d e^{3}\right )} x^{3} + 3 \, {\left (286 \, b d^{2} e^{2} n - 315 \, a d^{2} e^{2}\right )} x^{2} - 2 \, {\left (677 \, b d^{3} e n - 630 \, a d^{3} e\right )} x - 315 \, {\left (35 \, b e^{4} x^{4} + 50 \, b d e^{3} x^{3} + 3 \, b d^{2} e^{2} x^{2} - 4 \, b d^{3} e x + 8 \, b d^{4}\right )} \log \relax (c) - 315 \, {\left (35 \, b e^{4} n x^{4} + 50 \, b d e^{3} n x^{3} + 3 \, b d^{2} e^{2} n x^{2} - 4 \, b d^{3} e n x + 8 \, b d^{4} n\right )} \log \relax (x)\right )} \sqrt {e x + d}\right )}}{99225 \, e^{3}}, -\frac {2 \, {\left (5040 \, b \sqrt {-d} d^{4} n \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) + {\left (5228 \, b d^{4} n - 2520 \, a d^{4} + 1225 \, {\left (2 \, b e^{4} n - 9 \, a e^{4}\right )} x^{4} + 50 \, {\left (97 \, b d e^{3} n - 315 \, a d e^{3}\right )} x^{3} + 3 \, {\left (286 \, b d^{2} e^{2} n - 315 \, a d^{2} e^{2}\right )} x^{2} - 2 \, {\left (677 \, b d^{3} e n - 630 \, a d^{3} e\right )} x - 315 \, {\left (35 \, b e^{4} x^{4} + 50 \, b d e^{3} x^{3} + 3 \, b d^{2} e^{2} x^{2} - 4 \, b d^{3} e x + 8 \, b d^{4}\right )} \log \relax (c) - 315 \, {\left (35 \, b e^{4} n x^{4} + 50 \, b d e^{3} n x^{3} + 3 \, b d^{2} e^{2} n x^{2} - 4 \, b d^{3} e n x + 8 \, b d^{4} n\right )} \log \relax (x)\right )} \sqrt {e x + d}\right )}}{99225 \, e^{3}}\right ] \]
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 \[ \int {\left (e x + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.45, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{\frac {3}{2}} \left (b \ln \left (c \,x^{n}\right )+a \right ) x^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.53, size = 196, normalized size = 0.92 \[ -\frac {4}{99225} \, {\left (\frac {1260 \, d^{\frac {9}{2}} \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{e^{3}} + \frac {1225 \, {\left (e x + d\right )}^{\frac {9}{2}} - 2475 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 504 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} + 840 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 2520 \, \sqrt {e x + d} d^{4}}{e^{3}}\right )} b n + \frac {2}{315} \, {\left (\frac {35 \, {\left (e x + d\right )}^{\frac {9}{2}}}{e^{3}} - \frac {90 \, {\left (e x + d\right )}^{\frac {7}{2}} d}{e^{3}} + \frac {63 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2}}{e^{3}}\right )} b \log \left (c x^{n}\right ) + \frac {2}{315} \, {\left (\frac {35 \, {\left (e x + d\right )}^{\frac {9}{2}}}{e^{3}} - \frac {90 \, {\left (e x + d\right )}^{\frac {7}{2}} d}{e^{3}} + \frac {63 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2}}{e^{3}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )\,{\left (d+e\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 105.83, size = 870, normalized size = 4.08 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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