Optimal. Leaf size=170 \[ \frac{30 e^{11/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c e+d e x}}{\sqrt{e}}\right ),-1\right )}{77 d}-\frac{30 e^5 \sqrt{-c^2-2 c d x-d^2 x^2+1} \sqrt{c e+d e x}}{77 d}-\frac{18 e^3 \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{77 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{9/2}}{11 d} \]
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Rubi [A] time = 0.145616, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081, Rules used = {692, 689, 221} \[ -\frac{30 e^5 \sqrt{-c^2-2 c d x-d^2 x^2+1} \sqrt{c e+d e x}}{77 d}-\frac{18 e^3 \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{77 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{9/2}}{11 d}+\frac{30 e^{11/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{77 d} \]
Antiderivative was successfully verified.
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Rule 692
Rule 689
Rule 221
Rubi steps
\begin{align*} \int \frac{(c e+d e x)^{11/2}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx &=-\frac{2 e (c e+d e x)^{9/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{11 d}+\frac{1}{11} \left (9 e^2\right ) \int \frac{(c e+d e x)^{7/2}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\\ &=-\frac{18 e^3 (c e+d e x)^{5/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac{2 e (c e+d e x)^{9/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{11 d}+\frac{1}{77} \left (45 e^4\right ) \int \frac{(c e+d e x)^{3/2}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\\ &=-\frac{30 e^5 \sqrt{c e+d e x} \sqrt{1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac{18 e^3 (c e+d e x)^{5/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac{2 e (c e+d e x)^{9/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{11 d}+\frac{1}{77} \left (15 e^6\right ) \int \frac{1}{\sqrt{c e+d e x} \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\\ &=-\frac{30 e^5 \sqrt{c e+d e x} \sqrt{1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac{18 e^3 (c e+d e x)^{5/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac{2 e (c e+d e x)^{9/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{11 d}+\frac{\left (30 e^5\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^4}{e^2}}} \, dx,x,\sqrt{c e+d e x}\right )}{77 d}\\ &=-\frac{30 e^5 \sqrt{c e+d e x} \sqrt{1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac{18 e^3 (c e+d e x)^{5/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac{2 e (c e+d e x)^{9/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{11 d}+\frac{30 e^{11/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d e x}}{\sqrt{e}}\right )\right |-1\right )}{77 d}\\ \end{align*}
Mathematica [C] time = 0.101802, size = 125, normalized size = 0.74 \[ -\frac{2 e^5 \sqrt{e (c+d x)} \left (\sqrt{-c^2-2 c d x-d^2 x^2+1} \left (c^2 \left (42 d^2 x^2+9\right )+28 c^3 d x+7 c^4+2 c d x \left (14 d^2 x^2+9\right )+7 d^4 x^4+9 d^2 x^2+15\right )-15 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};(c+d x)^2\right )\right )}{77 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.384, size = 641, normalized size = 3.8 \begin{align*} -{\frac{{e}^{5}}{2310\,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 ( -900\,c+120\,{c}^{5}+924\,\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 ){c}^{3}-924\,\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 ) c+924\,\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 ) c-900\,dx+2940\,{x}^{6}c{d}^{6}+8820\,{x}^{5}{c}^{2}{d}^{5}+14700\,{x}^{4}{c}^{3}{d}^{4}+14700\,{x}^{3}{c}^{4}{d}^{3}+600\,{x}^{4}c{d}^{4}+8820\,{x}^{2}{c}^{5}{d}^{2}+1200\,{x}^{3}{c}^{2}{d}^{3}+2940\,x{c}^{6}d+1200\,{x}^{2}{c}^{3}{d}^{2}+600\,x{c}^{4}d+1080\,{x}^{2}c{d}^{2}+420\,{c}^{7}+65\,\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}\sqrt{-2\,dx-2\,c+2}{\it EllipticF} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) +1155\,\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 ) -385\,\sqrt{2\,dx+2\,c+2}\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}{\it EllipticF} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) -1155\,\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 ) +360\,{c}^{3}+1080\,x{c}^{2}d-924\,\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 ){c}^{3}+420\,{x}^{7}{d}^{7}+120\,{x}^{5}{d}^{5}+360\,{x}^{3}{d}^{3} \right ) } \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 (d e x + c e\right )}^{\frac{11}{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.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (d^{5} e^{5} x^{5} + 5 \, c d^{4} e^{5} x^{4} + 10 \, c^{2} d^{3} e^{5} x^{3} + 10 \, c^{3} d^{2} e^{5} x^{2} + 5 \, c^{4} d e^{5} x + c^{5} e^{5}\right )} \sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt{d e x + c e}}{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.
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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 (d e x + c e\right )}^{\frac{11}{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.
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