Optimal. Leaf size=133 \[ \frac{3 \sqrt{2} c^{3/2} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{e}-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{e (d+e x)^{5/2}} \]
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Rubi [A] time = 0.0789978, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {663, 665, 661, 208} \[ \frac{3 \sqrt{2} c^{3/2} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{e}-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{e (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 663
Rule 665
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)^{7/2}} \, dx &=-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{e (d+e x)^{5/2}}-\frac{1}{2} (3 c) \int \frac{\sqrt{c d^2-c e^2 x^2}}{(d+e x)^{3/2}} \, dx\\ &=-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{e (d+e x)^{5/2}}-\left (3 c^2 d\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}} \, dx\\ &=-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{e (d+e x)^{5/2}}-\left (6 c^2 d 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 )\\ &=-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{e (d+e x)^{5/2}}+\frac{3 \sqrt{2} c^{3/2} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{e}\\ \end{align*}
Mathematica [A] time = 0.203967, size = 107, normalized size = 0.8 \[ \frac{c \sqrt{c \left (d^2-e^2 x^2\right )} \left (\frac{3 \sqrt{2} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d^2-e^2 x^2}}-\frac{2 (2 d+e x)}{(d+e x)^{3/2}}\right )}{e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.171, size = 154, normalized size = 1.2 \begin{align*}{\frac{c}{e}\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 ) xcde+3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) c{d}^{2}-2\,xe\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}-4\,\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}d \right ) \left ( ex+d \right ) ^{-{\frac{3}{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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21309, size = 689, normalized size = 5.18 \begin{align*} \left [\frac{3 \, \sqrt{2}{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )} \sqrt{c d} \log \left (-\frac{c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} - 2 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{c d} \sqrt{e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 4 \, \sqrt{-c e^{2} x^{2} + c d^{2}}{\left (c e x + 2 \, c d\right )} \sqrt{e x + d}}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}}, \frac{3 \, \sqrt{2}{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )} \sqrt{-c d} \arctan \left (\frac{\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{-c d} \sqrt{e x + d}}{c e^{2} x^{2} - c d^{2}}\right ) - 2 \, \sqrt{-c e^{2} x^{2} + c d^{2}}{\left (c e x + 2 \, c d\right )} \sqrt{e x + d}}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{7}{2}}}\, dx \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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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