Optimal. Leaf size=101 \[ -\frac{2 c \left (a e^2+3 c d^2\right )}{e^5 (d+e x)}+\frac{2 c d \left (a e^2+c d^2\right )}{e^5 (d+e x)^2}-\frac{\left (a e^2+c d^2\right )^2}{3 e^5 (d+e x)^3}-\frac{4 c^2 d \log (d+e x)}{e^5}+\frac{c^2 x}{e^4} \]
[Out]
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Rubi [A] time = 0.192998, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{2 c \left (a e^2+3 c d^2\right )}{e^5 (d+e x)}+\frac{2 c d \left (a e^2+c d^2\right )}{e^5 (d+e x)^2}-\frac{\left (a e^2+c d^2\right )^2}{3 e^5 (d+e x)^3}-\frac{4 c^2 d \log (d+e x)}{e^5}+\frac{c^2 x}{e^4} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)^2/(d + e*x)^4,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{4 c^{2} d \log{\left (d + e x \right )}}{e^{5}} + \frac{2 c d \left (a e^{2} + c d^{2}\right )}{e^{5} \left (d + e x\right )^{2}} - \frac{2 c \left (a e^{2} + 3 c d^{2}\right )}{e^{5} \left (d + e x\right )} + \frac{\int c^{2}\, dx}{e^{4}} - \frac{\left (a e^{2} + c d^{2}\right )^{2}}{3 e^{5} \left (d + e x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**2/(e*x+d)**4,x)
[Out]
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Mathematica [A] time = 0.111432, size = 110, normalized size = 1.09 \[ -\frac{a^2 e^4+2 a c e^2 \left (d^2+3 d e x+3 e^2 x^2\right )+c^2 \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )+12 c^2 d (d+e x)^3 \log (d+e x)}{3 e^5 (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)^2/(d + e*x)^4,x]
[Out]
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Maple [A] time = 0.013, size = 140, normalized size = 1.4 \[{\frac{{c}^{2}x}{{e}^{4}}}+2\,{\frac{acd}{{e}^{3} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{c}^{2}{d}^{3}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-4\,{\frac{{c}^{2}d\ln \left ( ex+d \right ) }{{e}^{5}}}-{\frac{{a}^{2}}{3\,e \left ( ex+d \right ) ^{3}}}-{\frac{2\,{d}^{2}ac}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{2}{d}^{4}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-2\,{\frac{ac}{{e}^{3} \left ( ex+d \right ) }}-6\,{\frac{{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^2/(e*x+d)^4,x)
[Out]
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Maxima [A] time = 0.70707, size = 176, normalized size = 1.74 \[ -\frac{13 \, c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4} + 6 \,{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 6 \,{\left (5 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} + \frac{c^{2} x}{e^{4}} - \frac{4 \, c^{2} d \log \left (e x + d\right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2/(e*x + d)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216752, size = 247, normalized size = 2.45 \[ \frac{3 \, c^{2} e^{4} x^{4} + 9 \, c^{2} d e^{3} x^{3} - 13 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - a^{2} e^{4} - 3 \,{\left (3 \, c^{2} d^{2} e^{2} + 2 \, a c e^{4}\right )} x^{2} - 3 \,{\left (9 \, c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x - 12 \,{\left (c^{2} d e^{3} x^{3} + 3 \, c^{2} d^{2} e^{2} x^{2} + 3 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \log \left (e x + d\right )}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2/(e*x + d)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.95338, size = 134, normalized size = 1.33 \[ - \frac{4 c^{2} d \log{\left (d + e x \right )}}{e^{5}} + \frac{c^{2} x}{e^{4}} - \frac{a^{2} e^{4} + 2 a c d^{2} e^{2} + 13 c^{2} d^{4} + x^{2} \left (6 a c e^{4} + 18 c^{2} d^{2} e^{2}\right ) + x \left (6 a c d e^{3} + 30 c^{2} d^{3} e\right )}{3 d^{3} e^{5} + 9 d^{2} e^{6} x + 9 d e^{7} x^{2} + 3 e^{8} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**2/(e*x+d)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.211458, size = 136, normalized size = 1.35 \[ -4 \, c^{2} d e^{\left (-5\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + c^{2} x e^{\left (-4\right )} - \frac{{\left (13 \, c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 6 \,{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + a^{2} e^{4} + 6 \,{\left (5 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2/(e*x + d)^4,x, algorithm="giac")
[Out]