Optimal. Leaf size=139 \[ \frac {b e^5 n \sqrt {x}}{3 d^5}-\frac {b e^4 n x}{6 d^4}+\frac {b e^3 n x^{3/2}}{9 d^3}-\frac {b e^2 n x^2}{12 d^2}+\frac {b e n x^{5/2}}{15 d}-\frac {b e^6 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{3 d^6}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {b e^6 n \log (x)}{6 d^6} \]
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Rubi [A]
time = 0.07, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2504, 2442, 46}
\begin {gather*} \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {b e^6 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{3 d^6}-\frac {b e^6 n \log (x)}{6 d^6}+\frac {b e^5 n \sqrt {x}}{3 d^5}-\frac {b e^4 n x}{6 d^4}+\frac {b e^3 n x^{3/2}}{9 d^3}-\frac {b e^2 n x^2}{12 d^2}+\frac {b e n x^{5/2}}{15 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 2442
Rule 2504
Rubi steps
\begin {align*} \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx &=-\left (2 \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^7} \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {1}{3} (b e n) \text {Subst}\left (\int \frac {1}{x^6 (d+e x)} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {1}{3} (b e n) \text {Subst}\left (\int \left (\frac {1}{d x^6}-\frac {e}{d^2 x^5}+\frac {e^2}{d^3 x^4}-\frac {e^3}{d^4 x^3}+\frac {e^4}{d^5 x^2}-\frac {e^5}{d^6 x}+\frac {e^6}{d^6 (d+e x)}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=\frac {b e^5 n \sqrt {x}}{3 d^5}-\frac {b e^4 n x}{6 d^4}+\frac {b e^3 n x^{3/2}}{9 d^3}-\frac {b e^2 n x^2}{12 d^2}+\frac {b e n x^{5/2}}{15 d}-\frac {b e^6 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{3 d^6}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {b e^6 n \log (x)}{6 d^6}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 130, normalized size = 0.94 \begin {gather*} \frac {a x^3}{3}+\frac {1}{3} b x^3 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-\frac {1}{3} b e n \left (-\frac {e^4 \sqrt {x}}{d^5}+\frac {e^3 x}{2 d^4}-\frac {e^2 x^{3/2}}{3 d^3}+\frac {e x^2}{4 d^2}-\frac {x^{5/2}}{5 d}+\frac {e^5 \log \left (d+\frac {e}{\sqrt {x}}\right )}{d^6}+\frac {e^5 \log (x)}{2 d^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 95, normalized size = 0.68 \begin {gather*} \frac {1}{3} \, b x^{3} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + \frac {1}{3} \, a x^{3} + \frac {1}{180} \, b n {\left (\frac {12 \, d^{4} x^{\frac {5}{2}} - 15 \, d^{3} x^{2} e + 20 \, d^{2} x^{\frac {3}{2}} e^{2} - 30 \, d x e^{3} + 60 \, \sqrt {x} e^{4}}{d^{5}} - \frac {60 \, e^{5} \log \left (d \sqrt {x} + e\right )}{d^{6}}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 152, normalized size = 1.09 \begin {gather*} \frac {60 \, b d^{6} x^{3} \log \left (c\right ) + 60 \, a d^{6} x^{3} - 15 \, b d^{4} n x^{2} e^{2} - 60 \, b d^{6} n \log \left (\sqrt {x}\right ) - 30 \, b d^{2} n x e^{4} + 60 \, {\left (b d^{6} n - b n e^{6}\right )} \log \left (d \sqrt {x} + e\right ) + 60 \, {\left (b d^{6} n x^{3} - b d^{6} n\right )} \log \left (\frac {d x + \sqrt {x} e}{x}\right ) + 4 \, {\left (3 \, b d^{5} n x^{2} e + 5 \, b d^{3} n x e^{3} + 15 \, b d n e^{5}\right )} \sqrt {x}}{180 \, d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 21.15, size = 134, normalized size = 0.96 \begin {gather*} \frac {a x^{3}}{3} + b \left (\frac {e n \left (\frac {2 x^{\frac {5}{2}}}{5 d} - \frac {e x^{2}}{2 d^{2}} + \frac {2 e^{2} x^{\frac {3}{2}}}{3 d^{3}} - \frac {e^{3} x}{d^{4}} + \frac {2 e^{4} \sqrt {x}}{d^{5}} - \frac {2 e^{6} \left (\begin {cases} \frac {1}{d \sqrt {x}} & \text {for}\: e = 0 \\\frac {\log {\left (d + \frac {e}{\sqrt {x}} \right )}}{e} & \text {otherwise} \end {cases}\right )}{d^{6}} + \frac {2 e^{5} \log {\left (\frac {1}{\sqrt {x}} \right )}}{d^{6}}\right )}{6} + \frac {x^{3} \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 236 vs.
\(2 (110) = 220\).
time = 3.09, size = 236, normalized size = 1.70 \begin {gather*} \frac {1}{3} \, b x^{3} \log \left (c\right ) + \frac {1}{3} \, a x^{3} - \frac {1}{180} \, {\left ({\left (\frac {60 \, \log \left (\frac {{\left | d \sqrt {x} + e \right |}}{\sqrt {{\left | x \right |}}}\right )}{d^{6}} - \frac {60 \, \log \left ({\left | -d + \frac {d \sqrt {x} + e}{\sqrt {x}} \right |}\right )}{d^{6}} + \frac {137 \, d^{5} - \frac {385 \, {\left (d \sqrt {x} + e\right )} d^{4}}{\sqrt {x}} + \frac {470 \, {\left (d \sqrt {x} + e\right )}^{2} d^{3}}{x} - \frac {270 \, {\left (d \sqrt {x} + e\right )}^{3} d^{2}}{x^{\frac {3}{2}}} + \frac {60 \, {\left (d \sqrt {x} + e\right )}^{4} d}{x^{2}}}{{\left (d - \frac {d \sqrt {x} + e}{\sqrt {x}}\right )}^{5} d^{6}}\right )} e^{7} - \frac {60 \, e^{7} \log \left ({\left (d e^{\left (-1\right )} - \frac {{\left (d \sqrt {x} + e\right )} e^{\left (-1\right )}}{\sqrt {x}}\right )} {\left (\frac {d}{d e^{\left (-1\right )} - \frac {{\left (d \sqrt {x} + e\right )} e^{\left (-1\right )}}{\sqrt {x}}} - e\right )}\right )}{{\left (d - \frac {d \sqrt {x} + e}{\sqrt {x}}\right )}^{6}}\right )} b n e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.69, size = 106, normalized size = 0.76 \begin {gather*} \frac {a\,x^3}{3}+\frac {b\,\left (60\,d^6\,x^3\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )-120\,e^6\,n\,\mathrm {atanh}\left (\frac {2\,e}{d\,\sqrt {x}}+1\right )-15\,d^4\,e^2\,n\,x^2+20\,d^3\,e^3\,n\,x^{3/2}-30\,d^2\,e^4\,n\,x+60\,d\,e^5\,n\,\sqrt {x}+12\,d^5\,e\,n\,x^{5/2}\right )}{180\,d^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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