Optimal. Leaf size=158 \[ \frac{c x \left (3 a^2 e^4+9 a c d^2 e^2+5 c^2 d^4\right )}{e^6}-\frac{c^2 d x^2 \left (3 a e^2+2 c d^2\right )}{e^5}+\frac{c^2 x^3 \left (a e^2+c d^2\right )}{e^4}-\frac{\left (a e^2+c d^2\right )^3}{e^7 (d+e x)}-\frac{6 c d \left (a e^2+c d^2\right )^2 \log (d+e x)}{e^7}-\frac{c^3 d x^4}{2 e^3}+\frac{c^3 x^5}{5 e^2} \]
[Out]
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Rubi [A] time = 0.335005, antiderivative size = 158, 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 \[ \frac{c x \left (3 a^2 e^4+9 a c d^2 e^2+5 c^2 d^4\right )}{e^6}-\frac{c^2 d x^2 \left (3 a e^2+2 c d^2\right )}{e^5}+\frac{c^2 x^3 \left (a e^2+c d^2\right )}{e^4}-\frac{\left (a e^2+c d^2\right )^3}{e^7 (d+e x)}-\frac{6 c d \left (a e^2+c d^2\right )^2 \log (d+e x)}{e^7}-\frac{c^3 d x^4}{2 e^3}+\frac{c^3 x^5}{5 e^2} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)^3/(d + e*x)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{c^{3} d x^{4}}{2 e^{3}} + \frac{c^{3} x^{5}}{5 e^{2}} - \frac{2 c^{2} d \left (3 a e^{2} + 2 c d^{2}\right ) \int x\, dx}{e^{5}} + \frac{c^{2} x^{3} \left (a e^{2} + c d^{2}\right )}{e^{4}} - \frac{6 c d \left (a e^{2} + c d^{2}\right )^{2} \log{\left (d + e x \right )}}{e^{7}} + \frac{\left (3 a^{2} e^{4} + 9 a c d^{2} e^{2} + 5 c^{2} d^{4}\right ) \int c\, dx}{e^{6}} - \frac{\left (a e^{2} + c d^{2}\right )^{3}}{e^{7} \left (d + e x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**3/(e*x+d)**2,x)
[Out]
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Mathematica [A] time = 0.158498, size = 193, normalized size = 1.22 \[ \frac{-10 a^3 e^6+30 a^2 c e^4 \left (-d^2+d e x+e^2 x^2\right )+10 a c^2 e^2 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )-60 c d (d+e x) \left (a e^2+c d^2\right )^2 \log (d+e x)+c^3 \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )}{10 e^7 (d+e x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)^3/(d + e*x)^2,x]
[Out]
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Maple [A] time = 0.013, size = 233, normalized size = 1.5 \[{\frac{{c}^{3}{x}^{5}}{5\,{e}^{2}}}-{\frac{{c}^{3}d{x}^{4}}{2\,{e}^{3}}}+{\frac{{c}^{2}{x}^{3}a}{{e}^{2}}}+{\frac{{x}^{3}{c}^{3}{d}^{2}}{{e}^{4}}}-3\,{\frac{{c}^{2}{x}^{2}ad}{{e}^{3}}}-2\,{\frac{{x}^{2}{c}^{3}{d}^{3}}{{e}^{5}}}+3\,{\frac{{a}^{2}cx}{{e}^{2}}}+9\,{\frac{{d}^{2}a{c}^{2}x}{{e}^{4}}}+5\,{\frac{{d}^{4}{c}^{3}x}{{e}^{6}}}-6\,{\frac{cd\ln \left ( ex+d \right ){a}^{2}}{{e}^{3}}}-12\,{\frac{{c}^{2}{d}^{3}\ln \left ( ex+d \right ) a}{{e}^{5}}}-6\,{\frac{{d}^{5}\ln \left ( ex+d \right ){c}^{3}}{{e}^{7}}}-{\frac{{a}^{3}}{e \left ( ex+d \right ) }}-3\,{\frac{{a}^{2}c{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}-3\,{\frac{{d}^{4}a{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}-{\frac{{c}^{3}{d}^{6}}{{e}^{7} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^3/(e*x+d)^2,x)
[Out]
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Maxima [A] time = 0.710285, size = 278, normalized size = 1.76 \[ -\frac{c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}}{e^{8} x + d e^{7}} + \frac{2 \, c^{3} e^{4} x^{5} - 5 \, c^{3} d e^{3} x^{4} + 10 \,{\left (c^{3} d^{2} e^{2} + a c^{2} e^{4}\right )} x^{3} - 10 \,{\left (2 \, c^{3} d^{3} e + 3 \, a c^{2} d e^{3}\right )} x^{2} + 10 \,{\left (5 \, c^{3} d^{4} + 9 \, a c^{2} d^{2} e^{2} + 3 \, a^{2} c e^{4}\right )} x}{10 \, e^{6}} - \frac{6 \,{\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205324, size = 367, normalized size = 2.32 \[ \frac{2 \, c^{3} e^{6} x^{6} - 3 \, c^{3} d e^{5} x^{5} - 10 \, c^{3} d^{6} - 30 \, a c^{2} d^{4} e^{2} - 30 \, a^{2} c d^{2} e^{4} - 10 \, a^{3} e^{6} + 5 \,{\left (c^{3} d^{2} e^{4} + 2 \, a c^{2} e^{6}\right )} x^{4} - 10 \,{\left (c^{3} d^{3} e^{3} + 2 \, a c^{2} d e^{5}\right )} x^{3} + 30 \,{\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 10 \,{\left (5 \, c^{3} d^{5} e + 9 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x - 60 \,{\left (c^{3} d^{6} + 2 \, a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} +{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \log \left (e x + d\right )}{10 \,{\left (e^{8} x + d e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.61871, size = 189, normalized size = 1.2 \[ - \frac{c^{3} d x^{4}}{2 e^{3}} + \frac{c^{3} x^{5}}{5 e^{2}} - \frac{6 c d \left (a e^{2} + c d^{2}\right )^{2} \log{\left (d + e x \right )}}{e^{7}} - \frac{a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}}{d e^{7} + e^{8} x} + \frac{x^{3} \left (a c^{2} e^{2} + c^{3} d^{2}\right )}{e^{4}} - \frac{x^{2} \left (3 a c^{2} d e^{2} + 2 c^{3} d^{3}\right )}{e^{5}} + \frac{x \left (3 a^{2} c e^{4} + 9 a c^{2} d^{2} e^{2} + 5 c^{3} d^{4}\right )}{e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**3/(e*x+d)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.213496, size = 351, normalized size = 2.22 \[ \frac{1}{10} \,{\left (2 \, c^{3} - \frac{15 \, c^{3} d}{x e + d} + \frac{10 \,{\left (5 \, c^{3} d^{2} e^{2} + a c^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{20 \,{\left (5 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac{30 \,{\left (5 \, c^{3} d^{4} e^{4} + 6 \, a c^{2} d^{2} e^{6} + a^{2} c e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}}\right )}{\left (x e + d\right )}^{5} e^{\left (-7\right )} + 6 \,{\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} e^{\left (-7\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{c^{3} d^{6} e^{5}}{x e + d} + \frac{3 \, a c^{2} d^{4} e^{7}}{x e + d} + \frac{3 \, a^{2} c d^{2} e^{9}}{x e + d} + \frac{a^{3} e^{11}}{x e + d}\right )} e^{\left (-12\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/(e*x + d)^2,x, algorithm="giac")
[Out]