Optimal. Leaf size=119 \[ -\frac{2 (d+e x)^{5/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}-\frac{16 d (d+e x)^{3/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}+\frac{64 d^2 \sqrt{d+e x}}{3 c e \sqrt{c d^2-c e^2 x^2}} \]
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Rubi [A] time = 0.0486483, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {657, 649} \[ -\frac{2 (d+e x)^{5/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}-\frac{16 d (d+e x)^{3/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}+\frac{64 d^2 \sqrt{d+e x}}{3 c e \sqrt{c d^2-c e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 657
Rule 649
Rubi steps
\begin{align*} \int \frac{(d+e x)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx &=-\frac{2 (d+e x)^{5/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}+\frac{1}{3} (8 d) \int \frac{(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{16 d (d+e x)^{3/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{5/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}+\frac{1}{3} \left (32 d^2\right ) \int \frac{(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{64 d^2 \sqrt{d+e x}}{3 c e \sqrt{c d^2-c e^2 x^2}}-\frac{16 d (d+e x)^{3/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{5/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0589753, size = 55, normalized size = 0.46 \[ -\frac{2 \sqrt{d+e x} \left (-23 d^2+10 d e x+e^2 x^2\right )}{3 c e \sqrt{c \left (d^2-e^2 x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 55, normalized size = 0.5 \begin{align*}{\frac{ \left ( -2\,ex+2\,d \right ) \left ( -{e}^{2}{x}^{2}-10\,dxe+23\,{d}^{2} \right ) }{3\,e} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13826, size = 58, normalized size = 0.49 \begin{align*} -\frac{2 \,{\left (\sqrt{c} e^{2} x^{2} + 10 \, \sqrt{c} d e x - 23 \, \sqrt{c} d^{2}\right )}}{3 \, \sqrt{-e x + d} c^{2} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11554, size = 135, normalized size = 1.13 \begin{align*} \frac{2 \, \sqrt{-c e^{2} x^{2} + c d^{2}}{\left (e^{2} x^{2} + 10 \, d e x - 23 \, d^{2}\right )} \sqrt{e x + d}}{3 \,{\left (c^{2} e^{3} x^{2} - c^{2} 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 (d + e x\right )^{\frac{7}{2}}}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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