Optimal. Leaf size=178 \[ -\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{16 \sqrt{2} d^{3/2} e}-\frac{c \sqrt{c d^2-c e^2 x^2}}{16 d e (d+e x)^{3/2}}+\frac{c \sqrt{c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}} \]
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Rubi [A] time = 0.0967206, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {663, 673, 661, 208} \[ -\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{16 \sqrt{2} d^{3/2} e}-\frac{c \sqrt{c d^2-c e^2 x^2}}{16 d e (d+e x)^{3/2}}+\frac{c \sqrt{c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}} \]
Antiderivative was successfully verified.
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Rule 663
Rule 673
Rule 661
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx &=-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}-\frac{1}{2} c \int \frac{\sqrt{c d^2-c e^2 x^2}}{(d+e x)^{7/2}} \, dx\\ &=\frac{c \sqrt{c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}+\frac{1}{8} c^2 \int \frac{1}{(d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}} \, dx\\ &=\frac{c \sqrt{c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac{c \sqrt{c d^2-c e^2 x^2}}{16 d e (d+e x)^{3/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}+\frac{c^2 \int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}} \, dx}{32 d}\\ &=\frac{c \sqrt{c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac{c \sqrt{c d^2-c e^2 x^2}}{16 d e (d+e x)^{3/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}+\frac{\left (c^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{-2 c d e^2+e^2 x^2} \, dx,x,\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{d+e x}}\right )}{16 d}\\ &=\frac{c \sqrt{c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac{c \sqrt{c d^2-c e^2 x^2}}{16 d e (d+e x)^{3/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}-\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{16 \sqrt{2} d^{3/2} e}\\ \end{align*}
Mathematica [A] time = 0.247028, size = 134, normalized size = 0.75 \[ \frac{\left (c \left (d^2-e^2 x^2\right )\right )^{3/2} \left (-\frac{2 \sqrt{d} \left (7 d^2-22 d e x+3 e^2 x^2\right )}{(d-e x) (d+e x)^{9/2}}-\frac{3 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\left (d^2-e^2 x^2\right )^{3/2}}\right )}{96 d^{3/2} e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.174, size = 259, normalized size = 1.5 \begin{align*} -{\frac{c}{96\,de}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( 3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ){x}^{3}c{e}^{3}+9\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ){x}^{2}cd{e}^{2}+9\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) xc{d}^{2}e+3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) c{d}^{3}+6\,{x}^{2}{e}^{2}\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}-44\,xde\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}+14\,\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}{d}^{2} \right ) \left ( ex+d \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{- \left ( ex-d \right ) c}}}{\frac{1}{\sqrt{cd}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25512, size = 965, normalized size = 5.42 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{2}}{\left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + 6 \, c d^{2} e^{2} x^{2} + 4 \, c d^{3} e x + c d^{4}\right )} \sqrt{\frac{c}{d}} \log \left (-\frac{c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 4 \, \sqrt{\frac{1}{2}} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d \sqrt{\frac{c}{d}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \,{\left (3 \, c e^{2} x^{2} - 22 \, c d e x + 7 \, c d^{2}\right )} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{96 \,{\left (d e^{5} x^{4} + 4 \, d^{2} e^{4} x^{3} + 6 \, d^{3} e^{3} x^{2} + 4 \, d^{4} e^{2} x + d^{5} e\right )}}, -\frac{3 \, \sqrt{\frac{1}{2}}{\left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + 6 \, c d^{2} e^{2} x^{2} + 4 \, c d^{3} e x + c d^{4}\right )} \sqrt{-\frac{c}{d}} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d \sqrt{-\frac{c}{d}}}{c e^{2} x^{2} - c d^{2}}\right ) +{\left (3 \, c e^{2} x^{2} - 22 \, c d e x + 7 \, c d^{2}\right )} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{48 \,{\left (d e^{5} x^{4} + 4 \, d^{2} e^{4} x^{3} + 6 \, d^{3} e^{3} x^{2} + 4 \, d^{4} e^{2} x + d^{5} e\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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