Optimal. Leaf size=104 \[ -\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{2 x^2}+\frac {b d^4 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 e^4}-\frac {b d^3 n}{2 e^3 \sqrt {x}}+\frac {b d^2 n}{4 e^2 x}-\frac {b d n}{6 e x^{3/2}}+\frac {b n}{8 x^2} \]
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Rubi [A] time = 0.08, antiderivative size = 104, 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 = {2454, 2395, 43} \[ -\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{2 x^2}-\frac {b d^3 n}{2 e^3 \sqrt {x}}+\frac {b d^2 n}{4 e^2 x}+\frac {b d^4 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 e^4}-\frac {b d n}{6 e x^{3/2}}+\frac {b n}{8 x^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2395
Rule 2454
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{x^3} \, dx &=-\left (2 \operatorname {Subst}\left (\int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{2 x^2}+\frac {1}{2} (b e n) \operatorname {Subst}\left (\int \frac {x^4}{d+e x} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{2 x^2}+\frac {1}{2} (b e n) \operatorname {Subst}\left (\int \left (-\frac {d^3}{e^4}+\frac {d^2 x}{e^3}-\frac {d x^2}{e^2}+\frac {x^3}{e}+\frac {d^4}{e^4 (d+e x)}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=\frac {b n}{8 x^2}-\frac {b d n}{6 e x^{3/2}}+\frac {b d^2 n}{4 e^2 x}-\frac {b d^3 n}{2 e^3 \sqrt {x}}+\frac {b d^4 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 e^4}-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 109, normalized size = 1.05 \[ -\frac {a}{2 x^2}-\frac {b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{2 x^2}+\frac {b d^4 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 e^4}-\frac {b d^3 n}{2 e^3 \sqrt {x}}+\frac {b d^2 n}{4 e^2 x}-\frac {b d n}{6 e x^{3/2}}+\frac {b n}{8 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 96, normalized size = 0.92 \[ \frac {6 \, b d^{2} e^{2} n x + 3 \, b e^{4} n - 12 \, b e^{4} \log \relax (c) - 12 \, a e^{4} + 12 \, {\left (b d^{4} n x^{2} - b e^{4} n\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right ) - 4 \, {\left (3 \, b d^{3} e n x + b d e^{3} n\right )} \sqrt {x}}{24 \, e^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 349, normalized size = 3.36 \[ \frac {1}{24} \, {\left (\frac {48 \, {\left (d \sqrt {x} + e\right )} b d^{3} n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{\sqrt {x}} - \frac {72 \, {\left (d \sqrt {x} + e\right )}^{2} b d^{2} n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x} - \frac {48 \, {\left (d \sqrt {x} + e\right )} b d^{3} n}{\sqrt {x}} + \frac {48 \, {\left (d \sqrt {x} + e\right )} b d^{3} \log \relax (c)}{\sqrt {x}} + \frac {48 \, {\left (d \sqrt {x} + e\right )}^{3} b d n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{\frac {3}{2}}} + \frac {36 \, {\left (d \sqrt {x} + e\right )}^{2} b d^{2} n}{x} - \frac {72 \, {\left (d \sqrt {x} + e\right )}^{2} b d^{2} \log \relax (c)}{x} - \frac {12 \, {\left (d \sqrt {x} + e\right )}^{4} b n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{2}} - \frac {16 \, {\left (d \sqrt {x} + e\right )}^{3} b d n}{x^{\frac {3}{2}}} + \frac {48 \, {\left (d \sqrt {x} + e\right )} a d^{3}}{\sqrt {x}} + \frac {48 \, {\left (d \sqrt {x} + e\right )}^{3} b d \log \relax (c)}{x^{\frac {3}{2}}} + \frac {3 \, {\left (d \sqrt {x} + e\right )}^{4} b n}{x^{2}} - \frac {72 \, {\left (d \sqrt {x} + e\right )}^{2} a d^{2}}{x} - \frac {12 \, {\left (d \sqrt {x} + e\right )}^{4} b \log \relax (c)}{x^{2}} + \frac {48 \, {\left (d \sqrt {x} + e\right )}^{3} a d}{x^{\frac {3}{2}}} - \frac {12 \, {\left (d \sqrt {x} + e\right )}^{4} a}{x^{2}}\right )} e^{\left (-4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )+a}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 95, normalized size = 0.91 \[ \frac {1}{24} \, b e n {\left (\frac {12 \, d^{4} \log \left (d \sqrt {x} + e\right )}{e^{5}} - \frac {6 \, d^{4} \log \relax (x)}{e^{5}} - \frac {12 \, d^{3} x^{\frac {3}{2}} - 6 \, d^{2} e x + 4 \, d e^{2} \sqrt {x} - 3 \, e^{3}}{e^{4} x^{2}}\right )} - \frac {b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{2 \, x^{2}} - \frac {a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 87, normalized size = 0.84 \[ \frac {b\,n}{8\,x^2}-\frac {a}{2\,x^2}-\frac {b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{2\,x^2}-\frac {b\,d\,n}{6\,e\,x^{3/2}}+\frac {b\,d^4\,n\,\ln \left (d+\frac {e}{\sqrt {x}}\right )}{2\,e^4}+\frac {b\,d^2\,n}{4\,e^2\,x}-\frac {b\,d^3\,n}{2\,e^3\,\sqrt {x}} \]
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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