Optimal. Leaf size=150 \[ \frac{\sqrt{a} e \left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 c^{3/2} \left (a e^2+c d^2\right )^2}+\frac{a \left (d-e x^2\right )}{4 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{d^3 \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2}-\frac{d^3 \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )^2} \]
[Out]
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Rubi [A] time = 0.485226, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\sqrt{a} e \left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 c^{3/2} \left (a e^2+c d^2\right )^2}+\frac{a \left (d-e x^2\right )}{4 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{d^3 \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2}-\frac{d^3 \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[x^7/((d + e*x^2)*(a + c*x^4)^2),x]
[Out]
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Rubi in Sympy [A] time = 70.3934, size = 170, normalized size = 1.13 \[ - \frac{\sqrt{a} e \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 c^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )} + \frac{\sqrt{a} e \left (a e^{2} + 2 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 c^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{a \left (d - e x^{2}\right )}{4 c \left (a + c x^{4}\right ) \left (a e^{2} + c d^{2}\right )} + \frac{d^{3} \log{\left (a + c x^{4} \right )}}{4 \left (a e^{2} + c d^{2}\right )^{2}} - \frac{d^{3} \log{\left (d + e x^{2} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**7/(e*x**2+d)/(c*x**4+a)**2,x)
[Out]
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Mathematica [A] time = 0.190637, size = 142, normalized size = 0.95 \[ \frac{\sqrt{a} e \left (a+c x^4\right ) \left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )+\sqrt{c} \left (-2 c d^3 \left (a+c x^4\right ) \log \left (d+e x^2\right )+c d^3 \left (a+c x^4\right ) \log \left (a+c x^4\right )+a \left (d-e x^2\right ) \left (a e^2+c d^2\right )\right )}{4 c^{3/2} \left (a+c x^4\right ) \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[x^7/((d + e*x^2)*(a + c*x^4)^2),x]
[Out]
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Maple [A] time = 0.023, size = 262, normalized size = 1.8 \[ -{\frac{{a}^{2}{e}^{3}{x}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) c}}-{\frac{ae{x}^{2}{d}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{d{a}^{2}{e}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) c}}+{\frac{a{d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{{d}^{3}\ln \left ( \left ( c{x}^{4}+a \right ) c \right ) }{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{{a}^{2}{e}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}c}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{d}^{2}ea}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{{d}^{3}\ln \left ( e{x}^{2}+d \right ) }{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^7/(e*x^2+d)/(c*x^4+a)^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(x^7/((c*x^4 + a)^2*(e*x^2 + d)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 7.11036, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, a c d^{3} + 2 \, a^{2} d e^{2} - 2 \,{\left (a c d^{2} e + a^{2} e^{3}\right )} x^{2} +{\left (3 \, a c d^{2} e + a^{2} e^{3} +{\left (3 \, c^{2} d^{2} e + a c e^{3}\right )} x^{4}\right )} \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{4} + 2 \, c x^{2} \sqrt{-\frac{a}{c}} - a}{c x^{4} + a}\right ) + 2 \,{\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (c x^{4} + a\right ) - 4 \,{\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (e x^{2} + d\right )}{8 \,{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4} +{\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{4}\right )}}, \frac{a c d^{3} + a^{2} d e^{2} -{\left (a c d^{2} e + a^{2} e^{3}\right )} x^{2} +{\left (3 \, a c d^{2} e + a^{2} e^{3} +{\left (3 \, c^{2} d^{2} e + a c e^{3}\right )} x^{4}\right )} \sqrt{\frac{a}{c}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{a}{c}}}\right ) +{\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (c x^{4} + a\right ) - 2 \,{\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (e x^{2} + d\right )}{4 \,{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4} +{\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^7/((c*x^4 + a)^2*(e*x^2 + d)),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(x**7/(e*x**2+d)/(c*x**4+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.274248, size = 301, normalized size = 2.01 \[ -\frac{d^{3} e{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}} + \frac{d^{3}{\rm ln}\left (c x^{4} + a\right )}{4 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac{{\left (3 \, a c d^{2} e + a^{2} e^{3}\right )} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \,{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} \sqrt{a c}} - \frac{c^{2} d^{3} x^{4} + a c d^{2} x^{2} e + a^{2} x^{2} e^{3} - a^{2} d e^{2}}{4 \,{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}{\left (c x^{4} + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^7/((c*x^4 + a)^2*(e*x^2 + d)),x, algorithm="giac")
[Out]