Optimal. Leaf size=178 \[ -\frac{7 e \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}}+\frac{7 e \sqrt{d+e x} \left (c d^2-a e^2\right )^2}{c^4 d^4}+\frac{7 e (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^3 d^3}-\frac{(d+e x)^{7/2}}{c d (a e+c d x)}+\frac{7 e (d+e x)^{5/2}}{5 c^2 d^2} \]
[Out]
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Rubi [A] time = 0.334248, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{7 e \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}}+\frac{7 e \sqrt{d+e x} \left (c d^2-a e^2\right )^2}{c^4 d^4}+\frac{7 e (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^3 d^3}-\frac{(d+e x)^{7/2}}{c d (a e+c d x)}+\frac{7 e (d+e x)^{5/2}}{5 c^2 d^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(11/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 75.6733, size = 162, normalized size = 0.91 \[ - \frac{\left (d + e x\right )^{\frac{7}{2}}}{c d \left (a e + c d x\right )} + \frac{7 e \left (d + e x\right )^{\frac{5}{2}}}{5 c^{2} d^{2}} - \frac{7 e \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )}{3 c^{3} d^{3}} + \frac{7 e \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{2}}{c^{4} d^{4}} - \frac{7 e \left (a e^{2} - c d^{2}\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{c^{\frac{9}{2}} d^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(11/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
[Out]
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Mathematica [A] time = 0.34401, size = 191, normalized size = 1.07 \[ \frac{\sqrt{d+e x} \left (105 a^3 e^6-35 a^2 c d e^4 (7 d-2 e x)+7 a c^2 d^2 e^2 \left (23 d^2-24 d e x-2 e^2 x^2\right )+c^3 d^3 \left (-15 d^3+116 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )}{15 c^4 d^4 (a e+c d x)}-\frac{7 e \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(11/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
[Out]
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Maple [B] time = 0.023, size = 457, normalized size = 2.6 \[{\frac{2\,e}{5\,{c}^{2}{d}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{4\,a{e}^{3}}{3\,{c}^{3}{d}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{4\,e}{3\,{c}^{2}d} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+6\,{\frac{{a}^{2}{e}^{5}\sqrt{ex+d}}{{c}^{4}{d}^{4}}}-12\,{\frac{a{e}^{3}\sqrt{ex+d}}{{c}^{3}{d}^{2}}}+6\,{\frac{e\sqrt{ex+d}}{{c}^{2}}}+{\frac{{a}^{3}{e}^{7}}{{c}^{4}{d}^{4} \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}-3\,{\frac{{a}^{2}{e}^{5}\sqrt{ex+d}}{{c}^{3}{d}^{2} \left ( cdex+a{e}^{2} \right ) }}+3\,{\frac{a{e}^{3}\sqrt{ex+d}}{{c}^{2} \left ( cdex+a{e}^{2} \right ) }}-{\frac{e{d}^{2}}{c \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}-7\,{\frac{{a}^{3}{e}^{7}}{{c}^{4}{d}^{4}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+21\,{\frac{{a}^{2}{e}^{5}}{{c}^{3}{d}^{2}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }-21\,{\frac{a{e}^{3}}{{c}^{2}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+7\,{\frac{e{d}^{2}}{c\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(11/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(11/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226425, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (a c^{2} d^{4} e^{2} - 2 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} +{\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \sqrt{\frac{c d^{2} - a e^{2}}{c d}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt{e x + d} c d \sqrt{\frac{c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \,{\left (6 \, c^{3} d^{3} e^{3} x^{3} - 15 \, c^{3} d^{6} + 161 \, a c^{2} d^{4} e^{2} - 245 \, a^{2} c d^{2} e^{4} + 105 \, a^{3} e^{6} + 2 \,{\left (16 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (58 \, c^{3} d^{5} e - 84 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{30 \,{\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}}, -\frac{105 \,{\left (a c^{2} d^{4} e^{2} - 2 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} +{\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \sqrt{-\frac{c d^{2} - a e^{2}}{c d}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d^{2} - a e^{2}}{c d}}}\right ) -{\left (6 \, c^{3} d^{3} e^{3} x^{3} - 15 \, c^{3} d^{6} + 161 \, a c^{2} d^{4} e^{2} - 245 \, a^{2} c d^{2} e^{4} + 105 \, a^{3} e^{6} + 2 \,{\left (16 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (58 \, c^{3} d^{5} e - 84 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(11/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(11/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(11/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="giac")
[Out]