Optimal. Leaf size=160 \[ -\frac{2 (d+e x)^{7/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{8 d (d+e x)^{5/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{64 d^2 (d+e x)^{3/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}+\frac{256 d^3 \sqrt{d+e x}}{5 c e \sqrt{c d^2-c e^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0726321, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {657, 649} \[ -\frac{2 (d+e x)^{7/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{8 d (d+e x)^{5/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{64 d^2 (d+e x)^{3/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}+\frac{256 d^3 \sqrt{d+e x}}{5 c e \sqrt{c d^2-c e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 657
Rule 649
Rubi steps
\begin{align*} \int \frac{(d+e x)^{9/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx &=-\frac{2 (d+e x)^{7/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}+\frac{1}{5} (12 d) \int \frac{(d+e x)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{8 d (d+e x)^{5/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{7/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}+\frac{1}{5} \left (32 d^2\right ) \int \frac{(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{64 d^2 (d+e x)^{3/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{8 d (d+e x)^{5/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{7/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}+\frac{1}{5} \left (128 d^3\right ) \int \frac{(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{256 d^3 \sqrt{d+e x}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{64 d^2 (d+e x)^{3/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{8 d (d+e x)^{5/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{7/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0834965, size = 66, normalized size = 0.41 \[ -\frac{2 \sqrt{d+e x} \left (43 d^2 e x-91 d^3+7 d e^2 x^2+e^3 x^3\right )}{5 c e \sqrt{c \left (d^2-e^2 x^2\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.043, size = 66, normalized size = 0.4 \begin{align*}{\frac{ \left ( -2\,ex+2\,d \right ) \left ( -{e}^{3}{x}^{3}-7\,d{e}^{2}{x}^{2}-43\,{d}^{2}xe+91\,{d}^{3} \right ) }{5\,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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.11729, size = 61, normalized size = 0.38 \begin{align*} -\frac{2 \,{\left (e^{3} x^{3} + 7 \, d e^{2} x^{2} + 43 \, d^{2} e x - 91 \, d^{3}\right )}}{5 \, \sqrt{-e x + d} c^{\frac{3}{2}} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.18276, size = 157, normalized size = 0.98 \begin{align*} \frac{2 \,{\left (e^{3} x^{3} + 7 \, d e^{2} x^{2} + 43 \, d^{2} e x - 91 \, d^{3}\right )} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{5 \,{\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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]