Optimal. Leaf size=192 \[ \frac {2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {32 b d^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{105 e^3}-\frac {32 b d^3 n \sqrt {d+e x}}{105 e^3}-\frac {32 b d^2 n (d+e x)^{3/2}}{315 e^3}+\frac {36 b d n (d+e x)^{5/2}}{175 e^3}-\frac {4 b n (d+e x)^{7/2}}{49 e^3} \]
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Rubi [A] time = 0.18, antiderivative size = 192, 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} \[ \frac {2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}-\frac {32 b d^3 n \sqrt {d+e x}}{105 e^3}-\frac {32 b d^2 n (d+e x)^{3/2}}{315 e^3}+\frac {32 b d^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{105 e^3}+\frac {36 b d n (d+e x)^{5/2}}{175 e^3}-\frac {4 b n (d+e x)^{7/2}}{49 e^3} \]
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 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}-(b n) \int \frac {2 (d+e x)^{3/2} \left (8 d^2-12 d e x+15 e^2 x^2\right )}{105 e^3 x} \, dx\\ &=\frac {2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}-\frac {(2 b n) \int \frac {(d+e x)^{3/2} \left (8 d^2-12 d e x+15 e^2 x^2\right )}{x} \, dx}{105 e^3}\\ &=\frac {2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}-\frac {(4 b n) \operatorname {Subst}\left (\int \frac {x^4 \left (35 d^2-42 d x^2+15 x^4\right )}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{105 e^4}\\ &=\frac {2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}-\frac {(4 b n) \operatorname {Subst}\left (\int \left (8 d^3 e+8 d^2 e x^2-27 d e x^4+15 e x^6+\frac {8 d^4}{-\frac {d}{e}+\frac {x^2}{e}}\right ) \, dx,x,\sqrt {d+e x}\right )}{105 e^4}\\ &=-\frac {32 b d^3 n \sqrt {d+e x}}{105 e^3}-\frac {32 b d^2 n (d+e x)^{3/2}}{315 e^3}+\frac {36 b d n (d+e x)^{5/2}}{175 e^3}-\frac {4 b n (d+e x)^{7/2}}{49 e^3}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}-\frac {\left (32 b d^4 n\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{105 e^4}\\ &=-\frac {32 b d^3 n \sqrt {d+e x}}{105 e^3}-\frac {32 b d^2 n (d+e x)^{3/2}}{315 e^3}+\frac {36 b d n (d+e x)^{5/2}}{175 e^3}-\frac {4 b n (d+e x)^{7/2}}{49 e^3}+\frac {32 b d^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{105 e^3}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 151, normalized size = 0.79 \[ \frac {2 \sqrt {d+e x} \left (105 a \left (8 d^3-4 d^2 e x+3 d e^2 x^2+15 e^3 x^3\right )+105 b \left (8 d^3-4 d^2 e x+3 d e^2 x^2+15 e^3 x^3\right ) \log \left (c x^n\right )-2 b n \left (778 d^3-179 d^2 e x+108 d e^2 x^2+225 e^3 x^3\right )\right )+3360 b d^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{11025 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 396, normalized size = 2.06 \[ \left [\frac {2 \, {\left (840 \, b d^{\frac {7}{2}} n \log \left (\frac {e x + 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) - {\left (1556 \, b d^{3} n - 840 \, a d^{3} + 225 \, {\left (2 \, b e^{3} n - 7 \, a e^{3}\right )} x^{3} + 9 \, {\left (24 \, b d e^{2} n - 35 \, a d e^{2}\right )} x^{2} - 2 \, {\left (179 \, b d^{2} e n - 210 \, a d^{2} e\right )} x - 105 \, {\left (15 \, b e^{3} x^{3} + 3 \, b d e^{2} x^{2} - 4 \, b d^{2} e x + 8 \, b d^{3}\right )} \log \relax (c) - 105 \, {\left (15 \, b e^{3} n x^{3} + 3 \, b d e^{2} n x^{2} - 4 \, b d^{2} e n x + 8 \, b d^{3} n\right )} \log \relax (x)\right )} \sqrt {e x + d}\right )}}{11025 \, e^{3}}, -\frac {2 \, {\left (1680 \, b \sqrt {-d} d^{3} n \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) + {\left (1556 \, b d^{3} n - 840 \, a d^{3} + 225 \, {\left (2 \, b e^{3} n - 7 \, a e^{3}\right )} x^{3} + 9 \, {\left (24 \, b d e^{2} n - 35 \, a d e^{2}\right )} x^{2} - 2 \, {\left (179 \, b d^{2} e n - 210 \, a d^{2} e\right )} x - 105 \, {\left (15 \, b e^{3} x^{3} + 3 \, b d e^{2} x^{2} - 4 \, b d^{2} e x + 8 \, b d^{3}\right )} \log \relax (c) - 105 \, {\left (15 \, b e^{3} n x^{3} + 3 \, b d e^{2} n x^{2} - 4 \, b d^{2} e n x + 8 \, b d^{3} n\right )} \log \relax (x)\right )} \sqrt {e x + d}\right )}}{11025 \, e^{3}}\right ] \]
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 \[ \int \sqrt {e x + d} {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right ) \sqrt {e x +d}\, x^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 184, normalized size = 0.96 \[ -\frac {4}{11025} \, {\left (\frac {420 \, d^{\frac {7}{2}} \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{e^{3}} + \frac {225 \, {\left (e x + d\right )}^{\frac {7}{2}} - 567 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 280 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} + 840 \, \sqrt {e x + d} d^{3}}{e^{3}}\right )} b n + \frac {2}{105} \, b {\left (\frac {15 \, {\left (e x + d\right )}^{\frac {7}{2}}}{e^{3}} - \frac {42 \, {\left (e x + d\right )}^{\frac {5}{2}} d}{e^{3}} + \frac {35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2}}{e^{3}}\right )} \log \left (c x^{n}\right ) + \frac {2}{105} \, a {\left (\frac {15 \, {\left (e x + d\right )}^{\frac {7}{2}}}{e^{3}} - \frac {42 \, {\left (e x + d\right )}^{\frac {5}{2}} d}{e^{3}} + \frac {35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2}}{e^{3}}\right )} \]
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 \[ \int x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )\,\sqrt {d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.16, size = 364, normalized size = 1.90 \[ \frac {2 \left (\frac {a d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 a d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {a \left (d + e x\right )^{\frac {7}{2}}}{7} + b d^{2} \left (\frac {\left (d + e x\right )^{\frac {3}{2}} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{3} - \frac {2 n \left (\frac {d^{2} e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{\sqrt {- d}} + d e \sqrt {d + e x} + \frac {e \left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{3 e}\right ) - 2 b d \left (\frac {\left (d + e x\right )^{\frac {5}{2}} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{5} - \frac {2 n \left (\frac {d^{3} e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{\sqrt {- d}} + d^{2} e \sqrt {d + e x} + \frac {d e \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {e \left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{5 e}\right ) + b \left (\frac {\left (d + e x\right )^{\frac {7}{2}} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{7} - \frac {2 n \left (\frac {d^{4} e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{\sqrt {- d}} + d^{3} e \sqrt {d + e x} + \frac {d^{2} e \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {d e \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {e \left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{7 e}\right )\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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