Optimal. Leaf size=217 \[ -\frac{3 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{256 \sqrt{2} d^{5/2} e}-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{256 d^2 e (d+e x)^{3/2}}-\frac{c \sqrt{c d^2-c e^2 x^2}}{64 d e (d+e x)^{5/2}}+\frac{c \sqrt{c d^2-c e^2 x^2}}{8 e (d+e x)^{7/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}} \]
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Rubi [A] time = 0.134371, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {663, 673, 661, 208} \[ -\frac{3 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{256 \sqrt{2} d^{5/2} e}-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{256 d^2 e (d+e x)^{3/2}}-\frac{c \sqrt{c d^2-c e^2 x^2}}{64 d e (d+e x)^{5/2}}+\frac{c \sqrt{c d^2-c e^2 x^2}}{8 e (d+e x)^{7/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}} \]
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)^{13/2}} \, dx &=-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}-\frac{1}{8} (3 c) \int \frac{\sqrt{c d^2-c e^2 x^2}}{(d+e x)^{9/2}} \, dx\\ &=\frac{c \sqrt{c d^2-c e^2 x^2}}{8 e (d+e x)^{7/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}+\frac{1}{16} c^2 \int \frac{1}{(d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}} \, dx\\ &=\frac{c \sqrt{c d^2-c e^2 x^2}}{8 e (d+e x)^{7/2}}-\frac{c \sqrt{c d^2-c e^2 x^2}}{64 d e (d+e x)^{5/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}+\frac{\left (3 c^2\right ) \int \frac{1}{(d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}} \, dx}{128 d}\\ &=\frac{c \sqrt{c d^2-c e^2 x^2}}{8 e (d+e x)^{7/2}}-\frac{c \sqrt{c d^2-c e^2 x^2}}{64 d e (d+e x)^{5/2}}-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{256 d^2 e (d+e x)^{3/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}+\frac{\left (3 c^2\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}} \, dx}{512 d^2}\\ &=\frac{c \sqrt{c d^2-c e^2 x^2}}{8 e (d+e x)^{7/2}}-\frac{c \sqrt{c d^2-c e^2 x^2}}{64 d e (d+e x)^{5/2}}-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{256 d^2 e (d+e x)^{3/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}+\frac{\left (3 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 )}{256 d^2}\\ &=\frac{c \sqrt{c d^2-c e^2 x^2}}{8 e (d+e x)^{7/2}}-\frac{c \sqrt{c d^2-c e^2 x^2}}{64 d e (d+e x)^{5/2}}-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{256 d^2 e (d+e x)^{3/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}-\frac{3 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 )}{256 \sqrt{2} d^{5/2} e}\\ \end{align*}
Mathematica [A] time = 0.272458, size = 145, normalized size = 0.67 \[ \frac{\left (c \left (d^2-e^2 x^2\right )\right )^{3/2} \left (-\frac{2 \sqrt{d} \left (-79 d^2 e x+39 d^3+13 d e^2 x^2+3 e^3 x^3\right )}{(d-e x) (d+e x)^{11/2}}-\frac{3 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\left (d^2-e^2 x^2\right )^{3/2}}\right )}{512 d^{5/2} e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.176, size = 325, normalized size = 1.5 \begin{align*} -{\frac{c}{512\,e{d}^{2}}\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}^{4}c{e}^{4}+12\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ){x}^{3}cd{e}^{3}+18\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ){x}^{2}c{d}^{2}{e}^{2}+12\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) xc{d}^{3}e+6\,{x}^{3}{e}^{3}\sqrt{cd}\sqrt{- \left ( ex-d \right ) c}+3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) c{d}^{4}+26\,{x}^{2}d{e}^{2}\sqrt{cd}\sqrt{- \left ( ex-d \right ) c}-158\,x{d}^{2}e\sqrt{cd}\sqrt{- \left ( ex-d \right ) c}+78\,\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}{d}^{3} \right ) \left ( ex+d \right ) ^{-{\frac{9}{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{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28808, size = 1130, normalized size = 5.21 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{2}}{\left (c e^{5} x^{5} + 5 \, c d e^{4} x^{4} + 10 \, c d^{2} e^{3} x^{3} + 10 \, c d^{3} e^{2} x^{2} + 5 \, c d^{4} e x + c d^{5}\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^{3} x^{3} + 13 \, c d e^{2} x^{2} - 79 \, c d^{2} e x + 39 \, c d^{3}\right )} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{512 \,{\left (d^{2} e^{6} x^{5} + 5 \, d^{3} e^{5} x^{4} + 10 \, d^{4} e^{4} x^{3} + 10 \, d^{5} e^{3} x^{2} + 5 \, d^{6} e^{2} x + d^{7} e\right )}}, -\frac{3 \, \sqrt{\frac{1}{2}}{\left (c e^{5} x^{5} + 5 \, c d e^{4} x^{4} + 10 \, c d^{2} e^{3} x^{3} + 10 \, c d^{3} e^{2} x^{2} + 5 \, c d^{4} e x + c d^{5}\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^{3} x^{3} + 13 \, c d e^{2} x^{2} - 79 \, c d^{2} e x + 39 \, c d^{3}\right )} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{256 \,{\left (d^{2} e^{6} x^{5} + 5 \, d^{3} e^{5} x^{4} + 10 \, d^{4} e^{4} x^{3} + 10 \, d^{5} e^{3} x^{2} + 5 \, d^{6} e^{2} x + d^{7} 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{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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