Optimal. Leaf size=107 \[ \frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {b e^4 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 d^4}-\frac {b e^4 n \log (x)}{4 d^4}+\frac {b e^3 n \sqrt {x}}{2 d^3}-\frac {b e^2 n x}{4 d^2}+\frac {b e n x^{3/2}}{6 d} \]
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Rubi [A] time = 0.07, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2454, 2395, 44} \[ \frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )+\frac {b e^3 n \sqrt {x}}{2 d^3}-\frac {b e^2 n x}{4 d^2}-\frac {b e^4 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 d^4}-\frac {b e^4 n \log (x)}{4 d^4}+\frac {b e n x^{3/2}}{6 d} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2395
Rule 2454
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^5} \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {1}{2} (b e n) \operatorname {Subst}\left (\int \frac {1}{x^4 (d+e x)} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {1}{2} (b e n) \operatorname {Subst}\left (\int \left (\frac {1}{d x^4}-\frac {e}{d^2 x^3}+\frac {e^2}{d^3 x^2}-\frac {e^3}{d^4 x}+\frac {e^4}{d^4 (d+e x)}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=\frac {b e^3 n \sqrt {x}}{2 d^3}-\frac {b e^2 n x}{4 d^2}+\frac {b e n x^{3/2}}{6 d}-\frac {b e^4 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 d^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {b e^4 n \log (x)}{4 d^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 102, normalized size = 0.95 \[ \frac {a x^2}{2}+\frac {1}{2} b x^2 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-\frac {1}{2} b e n \left (\frac {e^3 \log \left (d+\frac {e}{\sqrt {x}}\right )}{d^4}+\frac {e^3 \log (x)}{2 d^4}-\frac {e^2 \sqrt {x}}{d^3}+\frac {e x}{2 d^2}-\frac {x^{3/2}}{3 d}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 126, normalized size = 1.18 \[ \frac {6 \, b d^{4} x^{2} \log \relax (c) - 3 \, b d^{2} e^{2} n x + 6 \, a d^{4} x^{2} - 6 \, b d^{4} n \log \left (\sqrt {x}\right ) + 6 \, {\left (b d^{4} - b e^{4}\right )} n \log \left (d \sqrt {x} + e\right ) + 6 \, {\left (b d^{4} n x^{2} - b d^{4} n\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right ) + 2 \, {\left (b d^{3} e n x + 3 \, b d e^{3} n\right )} \sqrt {x}}{12 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 81, normalized size = 0.76 \[ \frac {1}{2} \, b x^{2} \log \relax (c) + \frac {1}{12} \, {\left (6 \, x^{2} \log \left (d + \frac {e}{\sqrt {x}}\right ) + {\left (\frac {2 \, d^{2} x^{\frac {3}{2}} - 3 \, d x e + 6 \, \sqrt {x} e^{2}}{d^{3}} - \frac {6 \, e^{3} \log \left ({\left | d \sqrt {x} + e \right |}\right )}{d^{4}}\right )} e\right )} b n + \frac {1}{2} \, a x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )+a \right ) x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 74, normalized size = 0.69 \[ -\frac {1}{12} \, b e n {\left (\frac {6 \, e^{3} \log \left (d \sqrt {x} + e\right )}{d^{4}} - \frac {2 \, d^{2} x^{\frac {3}{2}} - 3 \, d e x + 6 \, e^{2} \sqrt {x}}{d^{3}}\right )} + \frac {1}{2} \, b x^{2} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + \frac {1}{2} \, a x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 86, normalized size = 0.80 \[ \frac {x^{3/2}\,\left (\frac {b\,e\,n}{3\,d}-\frac {b\,e^2\,n}{2\,d^2\,\sqrt {x}}+\frac {b\,e^3\,n}{d^3\,x}\right )}{2}+\frac {a\,x^2}{2}+\frac {b\,x^2\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{2}-\frac {b\,e^4\,n\,\mathrm {atanh}\left (\frac {2\,e}{d\,\sqrt {x}}+1\right )}{d^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.26, size = 88, normalized size = 0.82 \[ \frac {a x^{2}}{2} + b \left (\frac {e n \left (\frac {2 x^{\frac {3}{2}}}{3 d} - \frac {e x}{d^{2}} - \frac {2 e^{3} \left (\begin {cases} \frac {\sqrt {x}}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d \sqrt {x} + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{3}} + \frac {2 e^{2} \sqrt {x}}{d^{3}}\right )}{4} + \frac {x^{2} \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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