Optimal. Leaf size=65 \[ \frac {b n}{2 x}-\frac {b d n}{e \sqrt {x}}+\frac {b d^2 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{e^2}-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{x} \]
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Rubi [A]
time = 0.04, antiderivative size = 65, 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, 45}
\begin {gather*} -\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{x}+\frac {b d^2 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{e^2}-\frac {b d n}{e \sqrt {x}}+\frac {b n}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2442
Rule 2504
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{x^2} \, dx &=-\left (2 \text {Subst}\left (\int x \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 )}{x}+(b e n) \text {Subst}\left (\int \frac {x^2}{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 )}{x}+(b e n) \text {Subst}\left (\int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=\frac {b n}{2 x}-\frac {b d n}{e \sqrt {x}}+\frac {b d^2 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{e^2}-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{x}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 68, normalized size = 1.05 \begin {gather*} -\frac {a}{x}+\frac {b n}{2 x}-\frac {b d n}{e \sqrt {x}}+\frac {b d^2 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{e^2}-\frac {b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 63, normalized size = 0.97
method | result | size |
derivativedivides | \(-\frac {a}{x}+\frac {b n}{2 x}-\frac {b \ln \left (c \,{\mathrm e}^{n \ln \left (d +\frac {e}{\sqrt {x}}\right )}\right )}{x}+\frac {b \,d^{2} n \ln \left (d +\frac {e}{\sqrt {x}}\right )}{e^{2}}-\frac {b d n}{e \sqrt {x}}\) | \(63\) |
default | \(-\frac {a}{x}+\frac {b n}{2 x}-\frac {b \ln \left (c \,{\mathrm e}^{n \ln \left (d +\frac {e}{\sqrt {x}}\right )}\right )}{x}+\frac {b \,d^{2} n \ln \left (d +\frac {e}{\sqrt {x}}\right )}{e^{2}}-\frac {b d n}{e \sqrt {x}}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 76, normalized size = 1.17 \begin {gather*} \frac {1}{2} \, {\left (2 \, d^{2} e^{\left (-3\right )} \log \left (d \sqrt {x} + e\right ) - d^{2} e^{\left (-3\right )} \log \left (x\right ) - \frac {{\left (2 \, d \sqrt {x} - e\right )} e^{\left (-2\right )}}{x}\right )} b n e - \frac {b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{x} - \frac {a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 67, normalized size = 1.03 \begin {gather*} -\frac {{\left (2 \, b d n \sqrt {x} e + 2 \, b e^{2} \log \left (c\right ) - {\left (b n - 2 \, a\right )} e^{2} - 2 \, {\left (b d^{2} n x - b n e^{2}\right )} \log \left (\frac {d x + \sqrt {x} e}{x}\right )\right )} e^{\left (-2\right )}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 360 vs.
\(2 (58) = 116\).
time = 51.83, size = 360, normalized size = 5.54 \begin {gather*} \begin {cases} - \frac {2 a d e^{2} x^{3}}{2 d e^{2} x^{4} + 2 e^{3} x^{\frac {7}{2}}} - \frac {2 a e^{3} x^{\frac {5}{2}}}{2 d e^{2} x^{4} + 2 e^{3} x^{\frac {7}{2}}} + \frac {2 b d^{3} x^{4} \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}}{2 d e^{2} x^{4} + 2 e^{3} x^{\frac {7}{2}}} - \frac {2 b d^{2} e n x^{\frac {7}{2}}}{2 d e^{2} x^{4} + 2 e^{3} x^{\frac {7}{2}}} + \frac {2 b d^{2} e x^{\frac {7}{2}} \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}}{2 d e^{2} x^{4} + 2 e^{3} x^{\frac {7}{2}}} - \frac {b d e^{2} n x^{3}}{2 d e^{2} x^{4} + 2 e^{3} x^{\frac {7}{2}}} - \frac {2 b d e^{2} x^{3} \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}}{2 d e^{2} x^{4} + 2 e^{3} x^{\frac {7}{2}}} + \frac {b e^{3} n x^{\frac {5}{2}}}{2 d e^{2} x^{4} + 2 e^{3} x^{\frac {7}{2}}} - \frac {2 b e^{3} x^{\frac {5}{2}} \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}}{2 d e^{2} x^{4} + 2 e^{3} x^{\frac {7}{2}}} & \text {for}\: e \neq 0 \\- \frac {a + b \log {\left (c d^{n} \right )}}{x} & \text {otherwise} \end {cases} \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. 162 vs.
\(2 (57) = 114\).
time = 3.45, size = 162, normalized size = 2.49 \begin {gather*} \frac {1}{2} \, {\left (\frac {4 \, {\left (d \sqrt {x} + e\right )} b d n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{\sqrt {x}} - \frac {2 \, {\left (d \sqrt {x} + e\right )}^{2} b n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x} - \frac {4 \, {\left (d \sqrt {x} + e\right )} b d n}{\sqrt {x}} + \frac {4 \, {\left (d \sqrt {x} + e\right )} b d \log \left (c\right )}{\sqrt {x}} + \frac {{\left (d \sqrt {x} + e\right )}^{2} b n}{x} - \frac {2 \, {\left (d \sqrt {x} + e\right )}^{2} b \log \left (c\right )}{x} + \frac {4 \, {\left (d \sqrt {x} + e\right )} a d}{\sqrt {x}} - \frac {2 \, {\left (d \sqrt {x} + e\right )}^{2} a}{x}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 60, normalized size = 0.92 \begin {gather*} \frac {b\,n}{2\,x}-\frac {a}{x}-\frac {b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{x}-\frac {b\,d\,n}{e\,\sqrt {x}}+\frac {b\,d^2\,n\,\ln \left (d+\frac {e}{\sqrt {x}}\right )}{e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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