Optimal. Leaf size=85 \[ -\frac{3 \sqrt{a} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 c^{5/2}}-\frac{a e \log \left (a+c x^2\right )}{c^3}-\frac{x^3 (d+e x)}{2 c \left (a+c x^2\right )}+\frac{3 d x}{2 c^2}+\frac{e x^2}{c^2} \]
[Out]
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Rubi [A] time = 0.184867, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{3 \sqrt{a} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 c^{5/2}}-\frac{a e \log \left (a+c x^2\right )}{c^3}-\frac{x^3 (d+e x)}{2 c \left (a+c x^2\right )}+\frac{3 d x}{2 c^2}+\frac{e x^2}{c^2} \]
Antiderivative was successfully verified.
[In] Int[(x^4*(d + e*x))/(a + c*x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{3 \sqrt{a} d \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 c^{\frac{5}{2}}} - \frac{a e \log{\left (a + c x^{2} \right )}}{c^{3}} - \frac{x^{3} \left (2 d + 2 e x\right )}{4 c \left (a + c x^{2}\right )} + \frac{3 d x}{2 c^{2}} + \frac{2 e \int x\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(e*x+d)/(c*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.116992, size = 77, normalized size = 0.91 \[ \frac{\frac{a (c d x-a e)}{a+c x^2}-3 \sqrt{a} \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )-2 a e \log \left (a+c x^2\right )+2 c d x+c e x^2}{2 c^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^4*(d + e*x))/(a + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.011, size = 88, normalized size = 1. \[{\frac{e{x}^{2}}{2\,{c}^{2}}}+{\frac{dx}{{c}^{2}}}+{\frac{axd}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{{a}^{2}e}{2\,{c}^{3} \left ( c{x}^{2}+a \right ) }}-{\frac{ae\ln \left ( c{x}^{2}+a \right ) }{{c}^{3}}}-{\frac{3\,ad}{2\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(e*x+d)/(c*x^2+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((e*x + d)*x^4/(c*x^2 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.280443, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, c^{2} e x^{4} + 4 \, c^{2} d x^{3} + 2 \, a c e x^{2} + 6 \, a c d x - 2 \, a^{2} e + 3 \,{\left (c^{2} d x^{2} + a c d\right )} \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{a}{c}} - a}{c x^{2} + a}\right ) - 4 \,{\left (a c e x^{2} + a^{2} e\right )} \log \left (c x^{2} + a\right )}{4 \,{\left (c^{4} x^{2} + a c^{3}\right )}}, \frac{c^{2} e x^{4} + 2 \, c^{2} d x^{3} + a c e x^{2} + 3 \, a c d x - a^{2} e - 3 \,{\left (c^{2} d x^{2} + a c d\right )} \sqrt{\frac{a}{c}} \arctan \left (\frac{x}{\sqrt{\frac{a}{c}}}\right ) - 2 \,{\left (a c e x^{2} + a^{2} e\right )} \log \left (c x^{2} + a\right )}{2 \,{\left (c^{4} x^{2} + a c^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*x^4/(c*x^2 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.08785, size = 189, normalized size = 2.22 \[ \left (- \frac{a e}{c^{3}} - \frac{3 d \sqrt{- a c^{7}}}{4 c^{6}}\right ) \log{\left (x + \frac{- 4 a e - 4 c^{3} \left (- \frac{a e}{c^{3}} - \frac{3 d \sqrt{- a c^{7}}}{4 c^{6}}\right )}{3 c d} \right )} + \left (- \frac{a e}{c^{3}} + \frac{3 d \sqrt{- a c^{7}}}{4 c^{6}}\right ) \log{\left (x + \frac{- 4 a e - 4 c^{3} \left (- \frac{a e}{c^{3}} + \frac{3 d \sqrt{- a c^{7}}}{4 c^{6}}\right )}{3 c d} \right )} + \frac{- a^{2} e + a c d x}{2 a c^{3} + 2 c^{4} x^{2}} + \frac{d x}{c^{2}} + \frac{e x^{2}}{2 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(e*x+d)/(c*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.273762, size = 117, normalized size = 1.38 \[ -\frac{3 \, a d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} c^{2}} - \frac{a e{\rm ln}\left (c x^{2} + a\right )}{c^{3}} + \frac{c^{2} x^{2} e + 2 \, c^{2} d x}{2 \, c^{4}} + \frac{a c d x - a^{2} e}{2 \,{\left (c x^{2} + a\right )} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*x^4/(c*x^2 + a)^2,x, algorithm="giac")
[Out]