Optimal. Leaf size=97 \[ -\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) \log (d+e x)}{e^4}-\frac{3 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)}+\frac{\left (c d^2-a e^2\right )^3}{2 e^4 (d+e x)^2}+\frac{c^3 d^3 x}{e^3} \]
[Out]
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Rubi [A] time = 0.179802, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) \log (d+e x)}{e^4}-\frac{3 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)}+\frac{\left (c d^2-a e^2\right )^3}{2 e^4 (d+e x)^2}+\frac{c^3 d^3 x}{e^3} \]
Antiderivative was successfully verified.
[In] Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^6,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{3 c^{2} d^{2} \left (a e^{2} - c d^{2}\right ) \log{\left (d + e x \right )}}{e^{4}} - \frac{3 c d \left (a e^{2} - c d^{2}\right )^{2}}{e^{4} \left (d + e x\right )} + \frac{d^{3} \int c^{3}\, dx}{e^{3}} - \frac{\left (a e^{2} - c d^{2}\right )^{3}}{2 e^{4} \left (d + e x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**6,x)
[Out]
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Mathematica [A] time = 0.0873365, size = 129, normalized size = 1.33 \[ \frac{-a^3 e^6-3 a^2 c d e^4 (d+2 e x)+3 a c^2 d^3 e^2 (3 d+4 e x)-6 c^2 d^2 (d+e x)^2 \left (c d^2-a e^2\right ) \log (d+e x)+c^3 d^3 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )}{2 e^4 (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^6,x]
[Out]
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Maple [A] time = 0.011, size = 167, normalized size = 1.7 \[{\frac{{c}^{3}{d}^{3}x}{{e}^{3}}}+3\,{\frac{{c}^{2}{d}^{2}\ln \left ( ex+d \right ) a}{{e}^{2}}}-3\,{\frac{{c}^{3}{d}^{4}\ln \left ( ex+d \right ) }{{e}^{4}}}-{\frac{{e}^{2}{a}^{3}}{2\, \left ( ex+d \right ) ^{2}}}+{\frac{3\,{a}^{2}c{d}^{2}}{2\, \left ( ex+d \right ) ^{2}}}-{\frac{3\,{c}^{2}{d}^{4}a}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{{c}^{3}{d}^{6}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-3\,{\frac{{a}^{2}cd}{ex+d}}+6\,{\frac{{c}^{2}{d}^{3}a}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{{c}^{3}{d}^{5}}{{e}^{4} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^6,x)
[Out]
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Maxima [A] time = 0.7313, size = 192, normalized size = 1.98 \[ \frac{c^{3} d^{3} x}{e^{3}} - \frac{5 \, c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \,{\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} - \frac{3 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3/(e*x + d)^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.239069, size = 282, normalized size = 2.91 \[ \frac{2 \, c^{3} d^{3} e^{3} x^{3} + 4 \, c^{3} d^{4} e^{2} x^{2} - 5 \, c^{3} d^{6} + 9 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} - 2 \,{\left (2 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x - 6 \,{\left (c^{3} d^{6} - a c^{2} d^{4} e^{2} +{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (c^{3} d^{5} e - a c^{2} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3/(e*x + d)^6,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.81302, size = 144, normalized size = 1.48 \[ \frac{c^{3} d^{3} x}{e^{3}} + \frac{3 c^{2} d^{2} \left (a e^{2} - c d^{2}\right ) \log{\left (d + e x \right )}}{e^{4}} - \frac{a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} - 9 a c^{2} d^{4} e^{2} + 5 c^{3} d^{6} + x \left (6 a^{2} c d e^{5} - 12 a c^{2} d^{3} e^{3} + 6 c^{3} d^{5} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**6,x)
[Out]
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GIAC/XCAS [A] time = 0.219514, size = 352, normalized size = 3.63 \[ c^{3} d^{3} x e^{\left (-3\right )} - 3 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) - \frac{{\left (5 \, c^{3} d^{9} - 9 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} + a^{3} d^{3} e^{6} + 6 \,{\left (c^{3} d^{5} e^{4} - 2 \, a c^{2} d^{3} e^{6} + a^{2} c d e^{8}\right )} x^{4} +{\left (23 \, c^{3} d^{6} e^{3} - 45 \, a c^{2} d^{4} e^{5} + 21 \, a^{2} c d^{2} e^{7} + a^{3} e^{9}\right )} x^{3} + 3 \,{\left (11 \, c^{3} d^{7} e^{2} - 21 \, a c^{2} d^{5} e^{4} + 9 \, a^{2} c d^{3} e^{6} + a^{3} d e^{8}\right )} x^{2} + 3 \,{\left (7 \, c^{3} d^{8} e - 13 \, a c^{2} d^{6} e^{3} + 5 \, a^{2} c d^{4} e^{5} + a^{3} d^{2} e^{7}\right )} x\right )} e^{\left (-4\right )}}{2 \,{\left (x e + d\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3/(e*x + d)^6,x, algorithm="giac")
[Out]