Optimal. Leaf size=163 \[ -\frac{c^2 d x \left (9 a e^2+10 c d^2\right )}{e^6}+\frac{3 c^2 x^2 \left (a e^2+2 c d^2\right )}{2 e^5}+\frac{6 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)}-\frac{\left (a e^2+c d^2\right )^3}{2 e^7 (d+e x)^2}+\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) \log (d+e x)}{e^7}-\frac{c^3 d x^3}{e^4}+\frac{c^3 x^4}{4 e^3} \]
[Out]
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Rubi [A] time = 0.365284, antiderivative size = 163, 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 \[ -\frac{c^2 d x \left (9 a e^2+10 c d^2\right )}{e^6}+\frac{3 c^2 x^2 \left (a e^2+2 c d^2\right )}{2 e^5}+\frac{6 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)}-\frac{\left (a e^2+c d^2\right )^3}{2 e^7 (d+e x)^2}+\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) \log (d+e x)}{e^7}-\frac{c^3 d x^3}{e^4}+\frac{c^3 x^4}{4 e^3} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)^3/(d + e*x)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{c^{3} d x^{3}}{e^{4}} + \frac{c^{3} x^{4}}{4 e^{3}} + \frac{3 c^{2} \left (a e^{2} + 2 c d^{2}\right ) \int x\, dx}{e^{5}} - \frac{c^{2} \left (9 a e^{2} + 10 c d^{2}\right ) \int d\, dx}{e^{6}} + \frac{6 c d \left (a e^{2} + c d^{2}\right )^{2}}{e^{7} \left (d + e x\right )} + \frac{3 c \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} - \frac{\left (a e^{2} + c d^{2}\right )^{3}}{2 e^{7} \left (d + e x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**3/(e*x+d)**3,x)
[Out]
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Mathematica [A] time = 0.14449, size = 198, normalized size = 1.21 \[ \frac{-2 a^3 e^6+12 c (d+e x)^2 \left (a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right ) \log (d+e x)+6 a^2 c d e^4 (3 d+4 e x)+6 a c^2 e^2 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+c^3 \left (22 d^6-16 d^5 e x-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-2 d e^5 x^5+e^6 x^6\right )}{4 e^7 (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)^3/(d + e*x)^3,x]
[Out]
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Maple [A] time = 0.013, size = 249, normalized size = 1.5 \[{\frac{{c}^{3}{x}^{4}}{4\,{e}^{3}}}-{\frac{{c}^{3}d{x}^{3}}{{e}^{4}}}+{\frac{3\,{c}^{2}{x}^{2}a}{2\,{e}^{3}}}+3\,{\frac{{x}^{2}{c}^{3}{d}^{2}}{{e}^{5}}}-9\,{\frac{a{c}^{2}dx}{{e}^{4}}}-10\,{\frac{{c}^{3}{d}^{3}x}{{e}^{6}}}-{\frac{{a}^{3}}{2\,e \left ( ex+d \right ) ^{2}}}-{\frac{3\,{a}^{2}c{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{d}^{4}a{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{3}{d}^{6}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}+3\,{\frac{c\ln \left ( ex+d \right ){a}^{2}}{{e}^{3}}}+18\,{\frac{{c}^{2}\ln \left ( ex+d \right ) a{d}^{2}}{{e}^{5}}}+15\,{\frac{{d}^{4}\ln \left ( ex+d \right ){c}^{3}}{{e}^{7}}}+6\,{\frac{{a}^{2}cd}{{e}^{3} \left ( ex+d \right ) }}+12\,{\frac{{d}^{3}a{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}+6\,{\frac{{c}^{3}{d}^{5}}{{e}^{7} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^3/(e*x+d)^3,x)
[Out]
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Maxima [A] time = 0.727918, size = 289, normalized size = 1.77 \[ \frac{11 \, c^{3} d^{6} + 21 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 12 \,{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac{c^{3} e^{3} x^{4} - 4 \, c^{3} d e^{2} x^{3} + 6 \,{\left (2 \, c^{3} d^{2} e + a c^{2} e^{3}\right )} x^{2} - 4 \,{\left (10 \, c^{3} d^{3} + 9 \, a c^{2} d e^{2}\right )} x}{4 \, e^{6}} + \frac{3 \,{\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/(e*x + d)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.207293, size = 425, normalized size = 2.61 \[ \frac{c^{3} e^{6} x^{6} - 2 \, c^{3} d e^{5} x^{5} + 22 \, c^{3} d^{6} + 42 \, a c^{2} d^{4} e^{2} + 18 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} +{\left (5 \, c^{3} d^{2} e^{4} + 6 \, a c^{2} e^{6}\right )} x^{4} - 4 \,{\left (5 \, c^{3} d^{3} e^{3} + 6 \, a c^{2} d e^{5}\right )} x^{3} - 2 \,{\left (34 \, c^{3} d^{4} e^{2} + 33 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 4 \,{\left (4 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3} - 6 \, a^{2} c d e^{5}\right )} x + 12 \,{\left (5 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} +{\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 2 \,{\left (5 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/(e*x + d)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.68147, size = 216, normalized size = 1.33 \[ - \frac{c^{3} d x^{3}}{e^{4}} + \frac{c^{3} x^{4}}{4 e^{3}} + \frac{3 c \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} + \frac{- a^{3} e^{6} + 9 a^{2} c d^{2} e^{4} + 21 a c^{2} d^{4} e^{2} + 11 c^{3} d^{6} + x \left (12 a^{2} c d e^{5} + 24 a c^{2} d^{3} e^{3} + 12 c^{3} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} + \frac{x^{2} \left (3 a c^{2} e^{2} + 6 c^{3} d^{2}\right )}{2 e^{5}} - \frac{x \left (9 a c^{2} d e^{2} + 10 c^{3} d^{3}\right )}{e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**3/(e*x+d)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.212469, size = 259, normalized size = 1.59 \[ 3 \,{\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{4} \,{\left (c^{3} x^{4} e^{9} - 4 \, c^{3} d x^{3} e^{8} + 12 \, c^{3} d^{2} x^{2} e^{7} - 40 \, c^{3} d^{3} x e^{6} + 6 \, a c^{2} x^{2} e^{9} - 36 \, a c^{2} d x e^{8}\right )} e^{\left (-12\right )} + \frac{{\left (11 \, c^{3} d^{6} + 21 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 12 \,{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} e^{\left (-7\right )}}{2 \,{\left (x e + d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/(e*x + d)^3,x, algorithm="giac")
[Out]