Optimal. Leaf size=87 \[ \frac{a^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{5/2}}+\frac{a^2 e \log \left (a+c x^2\right )}{2 c^3}-\frac{a d x}{c^2}-\frac{a e x^2}{2 c^2}+\frac{d x^3}{3 c}+\frac{e x^4}{4 c} \]
[Out]
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Rubi [A] time = 0.141018, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{a^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{5/2}}+\frac{a^2 e \log \left (a+c x^2\right )}{2 c^3}-\frac{a d x}{c^2}-\frac{a e x^2}{2 c^2}+\frac{d x^3}{3 c}+\frac{e x^4}{4 c} \]
Antiderivative was successfully verified.
[In] Int[(x^4*(d + e*x))/(a + c*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a^{\frac{3}{2}} d \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{c^{\frac{5}{2}}} + \frac{a^{2} e \log{\left (a + c x^{2} \right )}}{2 c^{3}} - \frac{a e \int x\, dx}{c^{2}} + \frac{d x^{3}}{3 c} + \frac{e x^{4}}{4 c} - \frac{d \int a\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(e*x+d)/(c*x**2+a),x)
[Out]
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Mathematica [A] time = 0.0638324, size = 75, normalized size = 0.86 \[ \frac{12 a^{3/2} \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )+6 a^2 e \log \left (a+c x^2\right )+c x \left (c x^2 (4 d+3 e x)-6 a (2 d+e x)\right )}{12 c^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^4*(d + e*x))/(a + c*x^2),x]
[Out]
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Maple [A] time = 0.007, size = 77, normalized size = 0.9 \[{\frac{e{x}^{4}}{4\,c}}+{\frac{d{x}^{3}}{3\,c}}-{\frac{ae{x}^{2}}{2\,{c}^{2}}}-{\frac{adx}{{c}^{2}}}+{\frac{{a}^{2}e\ln \left ( c{x}^{2}+a \right ) }{2\,{c}^{3}}}+{\frac{{a}^{2}d}{{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(e*x+d)/(c*x^2+a),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),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.280359, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, c^{2} e x^{4} + 4 \, c^{2} d x^{3} - 6 \, a c e x^{2} + 6 \, a c d \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{2} + 2 \, c x \sqrt{-\frac{a}{c}} - a}{c x^{2} + a}\right ) - 12 \, a c d x + 6 \, a^{2} e \log \left (c x^{2} + a\right )}{12 \, c^{3}}, \frac{3 \, c^{2} e x^{4} + 4 \, c^{2} d x^{3} - 6 \, a c e x^{2} + 12 \, a c d \sqrt{\frac{a}{c}} \arctan \left (\frac{x}{\sqrt{\frac{a}{c}}}\right ) - 12 \, a c d x + 6 \, a^{2} e \log \left (c x^{2} + a\right )}{12 \, c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*x^4/(c*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.9265, size = 189, normalized size = 2.17 \[ - \frac{a d x}{c^{2}} - \frac{a e x^{2}}{2 c^{2}} + \left (\frac{a^{2} e}{2 c^{3}} - \frac{d \sqrt{- a^{3} c^{7}}}{2 c^{6}}\right ) \log{\left (x + \frac{- a^{2} e + 2 c^{3} \left (\frac{a^{2} e}{2 c^{3}} - \frac{d \sqrt{- a^{3} c^{7}}}{2 c^{6}}\right )}{a c d} \right )} + \left (\frac{a^{2} e}{2 c^{3}} + \frac{d \sqrt{- a^{3} c^{7}}}{2 c^{6}}\right ) \log{\left (x + \frac{- a^{2} e + 2 c^{3} \left (\frac{a^{2} e}{2 c^{3}} + \frac{d \sqrt{- a^{3} c^{7}}}{2 c^{6}}\right )}{a c d} \right )} + \frac{d x^{3}}{3 c} + \frac{e x^{4}}{4 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(e*x+d)/(c*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.272359, size = 115, normalized size = 1.32 \[ \frac{a^{2} d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c^{2}} + \frac{a^{2} e{\rm ln}\left (c x^{2} + a\right )}{2 \, c^{3}} + \frac{3 \, c^{3} x^{4} e + 4 \, c^{3} d x^{3} - 6 \, a c^{2} x^{2} e - 12 \, a c^{2} d x}{12 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*x^4/(c*x^2 + a),x, algorithm="giac")
[Out]