Optimal. Leaf size=78 \[ \frac{2 e^{3/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c e+d e x}}{\sqrt{e}}\right ),-1\right )}{3 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} \sqrt{c e+d e x}}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0587986, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081, Rules used = {692, 689, 221} \[ \frac{2 e^{3/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{3 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} \sqrt{c e+d e x}}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 692
Rule 689
Rule 221
Rubi steps
\begin{align*} \int \frac{(c e+d e x)^{3/2}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx &=-\frac{2 e \sqrt{c e+d e x} \sqrt{1-c^2-2 c d x-d^2 x^2}}{3 d}+\frac{1}{3} e^2 \int \frac{1}{\sqrt{c e+d e x} \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\\ &=-\frac{2 e \sqrt{c e+d e x} \sqrt{1-c^2-2 c d x-d^2 x^2}}{3 d}+\frac{(2 e) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^4}{e^2}}} \, dx,x,\sqrt{c e+d e x}\right )}{3 d}\\ &=-\frac{2 e \sqrt{c e+d e x} \sqrt{1-c^2-2 c d x-d^2 x^2}}{3 d}+\frac{2 e^{3/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d e x}}{\sqrt{e}}\right )\right |-1\right )}{3 d}\\ \end{align*}
Mathematica [C] time = 0.0200724, size = 54, normalized size = 0.69 \[ -\frac{2 e \sqrt{e (c+d x)} \left (\sqrt{1-(c+d x)^2}-\, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};(c+d x)^2\right )\right )}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.172, size = 308, normalized size = 4. \begin{align*} -{\frac{e}{6\,d \left ({x}^{3}{d}^{3}+3\,{x}^{2}c{d}^{2}+3\,x{c}^{2}d+{c}^{3}-dx-c \right ) }\sqrt{e \left ( dx+c \right ) }\sqrt{-{d}^{2}{x}^{2}-2\,cdx-{c}^{2}+1} \left ( 4\,{x}^{3}{d}^{3}+12\,{x}^{2}c{d}^{2}+\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}\sqrt{-2\,dx-2\,c+2}{\it EllipticF} \left ({\frac{1}{2}\sqrt{2\,dx+2\,c+2}},\sqrt{2} \right ) +3\,\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}\sqrt{-2\,dx-2\,c+2}{\it EllipticE} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) -\sqrt{2\,dx+2\,c+2}\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}{\it EllipticF} \left ({\frac{1}{2}\sqrt{-2\,dx-2\,c+2}},\sqrt{2} \right ) -3\,\sqrt{2\,dx+2\,c+2}\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) +12\,x{c}^{2}d+4\,{c}^{3}-4\,dx-4\,c \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d e x + c e\right )}^{\frac{3}{2}}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}{\left (d e x + c e\right )}^{\frac{3}{2}}}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \left (c + d x\right )\right )^{\frac{3}{2}}}{\sqrt{- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \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*} \int \frac{{\left (d e x + c e\right )}^{\frac{3}{2}}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]