Optimal. Leaf size=102 \[ -\frac {b d n \sqrt {d+e x^2}}{3 e}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e}+\frac {b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e} \]
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Rubi [A]
time = 0.06, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2376, 272, 52,
65, 214} \begin {gather*} \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac {b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e}-\frac {b d n \sqrt {d+e x^2}}{3 e}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 214
Rule 272
Rule 2376
Rubi steps
\begin {align*} \int x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {(b n) \int \frac {\left (d+e x^2\right )^{3/2}}{x} \, dx}{3 e}\\ &=\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {(b n) \text {Subst}\left (\int \frac {(d+e x)^{3/2}}{x} \, dx,x,x^2\right )}{6 e}\\ &=-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {(b d n) \text {Subst}\left (\int \frac {\sqrt {d+e x}}{x} \, dx,x,x^2\right )}{6 e}\\ &=-\frac {b d n \sqrt {d+e x^2}}{3 e}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {\left (b d^2 n\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{6 e}\\ &=-\frac {b d n \sqrt {d+e x^2}}{3 e}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {\left (b d^2 n\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{3 e^2}\\ &=-\frac {b d n \sqrt {d+e x^2}}{3 e}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e}+\frac {b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 136, normalized size = 1.33 \begin {gather*} \frac {3 a d \sqrt {d+e x^2}-4 b d n \sqrt {d+e x^2}+3 a e x^2 \sqrt {d+e x^2}-b e n x^2 \sqrt {d+e x^2}-3 b d^{3/2} n \log (x)+3 b \left (d+e x^2\right )^{3/2} \log \left (c x^n\right )+3 b d^{3/2} n \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )}{9 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {e \,x^{2}+d}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 85, normalized size = 0.83 \begin {gather*} \frac {1}{3} \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} b e^{\left (-1\right )} \log \left (c x^{n}\right ) + \frac {1}{9} \, {\left (3 \, d^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {\sqrt {d} e^{\left (-\frac {1}{2}\right )}}{{\left | x \right |}}\right ) - {\left (x^{2} e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x^{2} e + d} d\right )} b n e^{\left (-1\right )} + \frac {1}{3} \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} a e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 209, normalized size = 2.05 \begin {gather*} \left [\frac {1}{18} \, {\left (3 \, b d^{\frac {3}{2}} n \log \left (-\frac {x^{2} e + 2 \, \sqrt {x^{2} e + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) - 2 \, {\left ({\left (b n - 3 \, a\right )} x^{2} e + 4 \, b d n - 3 \, a d - 3 \, {\left (b x^{2} e + b d\right )} \log \left (c\right ) - 3 \, {\left (b n x^{2} e + b d n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-1\right )}, -\frac {1}{9} \, {\left (3 \, b \sqrt {-d} d n \arctan \left (\frac {\sqrt {-d}}{\sqrt {x^{2} e + d}}\right ) + {\left ({\left (b n - 3 \, a\right )} x^{2} e + 4 \, b d n - 3 \, a d - 3 \, {\left (b x^{2} e + b d\right )} \log \left (c\right ) - 3 \, {\left (b n x^{2} e + b d n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-1\right )}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 12.32, size = 155, normalized size = 1.52 \begin {gather*} a \left (\begin {cases} \frac {\sqrt {d} x^{2}}{2} & \text {for}\: e = 0 \\\frac {\left (d + e x^{2}\right )^{\frac {3}{2}}}{3 e} & \text {otherwise} \end {cases}\right ) - b n \left (\begin {cases} \frac {\sqrt {d} x^{2}}{4} & \text {for}\: e = 0 \\\frac {4 d^{\frac {3}{2}} \sqrt {1 + \frac {e x^{2}}{d}}}{9 e} + \frac {d^{\frac {3}{2}} \log {\left (\frac {e x^{2}}{d} \right )}}{6 e} - \frac {d^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {e x^{2}}{d}} + 1 \right )}}{3 e} + \frac {\sqrt {d} x^{2} \sqrt {1 + \frac {e x^{2}}{d}}}{9} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} \frac {\sqrt {d} x^{2}}{2} & \text {for}\: e = 0 \\\frac {\left (d + e x^{2}\right )^{\frac {3}{2}}}{3 e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.38, size = 145, normalized size = 1.42 \begin {gather*} \frac {1}{3} \, \sqrt {x^{2} e + d} b x^{2} \log \left (c\right ) + \frac {1}{3} \, \sqrt {x^{2} e + d} b d e^{\left (-1\right )} \log \left (c\right ) + \frac {1}{3} \, \sqrt {x^{2} e + d} a x^{2} + \frac {1}{3} \, \sqrt {x^{2} e + d} a d e^{\left (-1\right )} + \frac {1}{9} \, {\left (3 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} e^{\left (-1\right )} \log \left (x\right ) - {\left (\frac {3 \, d^{2} \arctan \left (\frac {\sqrt {x^{2} e + d}}{\sqrt {-d}}\right )}{\sqrt {-d}} + {\left (x^{2} e + d\right )}^{\frac {3}{2}} + 3 \, \sqrt {x^{2} e + d} d\right )} e^{\left (-1\right )}\right )} b n \end {gather*}
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 \begin {gather*} \int x\,\sqrt {e\,x^2+d}\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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