Optimal. Leaf size=112 \[ -\frac{3 e \sqrt{c d^2-a e^2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{5/2} d^{5/2}}-\frac{(d+e x)^{3/2}}{c d (a e+c d x)}+\frac{3 e \sqrt{d+e x}}{c^2 d^2} \]
[Out]
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Rubi [A] time = 0.185039, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{3 e \sqrt{c d^2-a e^2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{5/2} d^{5/2}}-\frac{(d+e x)^{3/2}}{c d (a e+c d x)}+\frac{3 e \sqrt{d+e x}}{c^2 d^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(7/2)/(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 = 49.1522, size = 99, normalized size = 0.88 \[ - \frac{\left (d + e x\right )^{\frac{3}{2}}}{c d \left (a e + c d x\right )} + \frac{3 e \sqrt{d + e x}}{c^{2} d^{2}} - \frac{3 e \sqrt{a e^{2} - c d^{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{c^{\frac{5}{2}} d^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(7/2)/(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 = 0.243372, size = 110, normalized size = 0.98 \[ \frac{\sqrt{d+e x} \left (3 a e^2-c d (d-2 e x)\right )}{c^2 d^2 (a e+c d x)}-\frac{3 e \sqrt{c d^2-a e^2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{5/2} d^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
[Out]
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Maple [A] time = 0.02, size = 183, normalized size = 1.6 \[ 2\,{\frac{e\sqrt{ex+d}}{{c}^{2}{d}^{2}}}+{\frac{a{e}^{3}}{{c}^{2}{d}^{2} \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}-{\frac{e}{c \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}-3\,{\frac{a{e}^{3}}{{c}^{2}{d}^{2}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+3\,{\frac{e}{c\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(7/2)/(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)^(7/2)/(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 = 0.224586, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (c d e x + a e^{2}\right )} \sqrt{\frac{c d^{2} - a e^{2}}{c d}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt{e x + d} c d \sqrt{\frac{c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \,{\left (2 \, c d e x - c d^{2} + 3 \, a e^{2}\right )} \sqrt{e x + d}}{2 \,{\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )}}, -\frac{3 \,{\left (c d e x + a e^{2}\right )} \sqrt{-\frac{c d^{2} - a e^{2}}{c d}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d^{2} - a e^{2}}{c d}}}\right ) -{\left (2 \, c d e x - c d^{2} + 3 \, a e^{2}\right )} \sqrt{e x + d}}{c^{3} d^{3} x + a c^{2} d^{2} e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(7/2)/(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)**(7/2)/(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)^(7/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="giac")
[Out]