Optimal. Leaf size=75 \[ \frac{3 d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}+\frac{3 d x}{8 a^2 \left (a+c x^2\right )}-\frac{a e-c d x}{4 a c \left (a+c x^2\right )^2} \]
[Out]
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Rubi [A] time = 0.0646081, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}+\frac{3 d x}{8 a^2 \left (a+c x^2\right )}-\frac{a e-c d x}{4 a c \left (a+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(a + c*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 9.00467, size = 66, normalized size = 0.88 \[ - \frac{a e - c d x}{4 a c \left (a + c x^{2}\right )^{2}} + \frac{3 d x}{8 a^{2} \left (a + c x^{2}\right )} + \frac{3 d \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/(c*x**2+a)**3,x)
[Out]
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Mathematica [A] time = 0.0926997, size = 71, normalized size = 0.95 \[ \frac{\frac{\sqrt{a} \left (-2 a^2 e+5 a c d x+3 c^2 d x^3\right )}{\left (a+c x^2\right )^2}+3 \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(a + c*x^2)^3,x]
[Out]
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Maple [A] time = 0.005, size = 65, normalized size = 0.9 \[{\frac{2\,cdx-2\,ae}{8\,ac \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{3\,dx}{8\,{a}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{3\,d}{8\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/(c*x^2+a)^3,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)/(c*x^2 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213286, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (c^{3} d x^{4} + 2 \, a c^{2} d x^{2} + a^{2} c d\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) + 2 \,{\left (3 \, c^{2} d x^{3} + 5 \, a c d x - 2 \, a^{2} e\right )} \sqrt{-a c}}{16 \,{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt{-a c}}, \frac{3 \,{\left (c^{3} d x^{4} + 2 \, a c^{2} d x^{2} + a^{2} c d\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (3 \, c^{2} d x^{3} + 5 \, a c d x - 2 \, a^{2} e\right )} \sqrt{a c}}{8 \,{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt{a c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^2 + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.34197, size = 124, normalized size = 1.65 \[ d \left (- \frac{3 \sqrt{- \frac{1}{a^{5} c}} \log{\left (- a^{3} \sqrt{- \frac{1}{a^{5} c}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a^{5} c}} \log{\left (a^{3} \sqrt{- \frac{1}{a^{5} c}} + x \right )}}{16}\right ) + \frac{- 2 a^{2} e + 5 a c d x + 3 c^{2} d x^{3}}{8 a^{4} c + 16 a^{3} c^{2} x^{2} + 8 a^{2} c^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/(c*x**2+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.211324, size = 82, normalized size = 1.09 \[ \frac{3 \, d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2}} + \frac{3 \, c^{2} d x^{3} + 5 \, a c d x - 2 \, a^{2} e}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^2 + a)^3,x, algorithm="giac")
[Out]