Optimal. Leaf size=105 \[ \frac{3 c^2 d^2 \left (c d^2-a e^2\right )}{e^4 (d+e x)}-\frac{3 c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^2}+\frac{\left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^3}+\frac{c^3 d^3 \log (d+e x)}{e^4} \]
[Out]
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Rubi [A] time = 0.174293, antiderivative size = 105, 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 \[ \frac{3 c^2 d^2 \left (c d^2-a e^2\right )}{e^4 (d+e x)}-\frac{3 c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^2}+\frac{\left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^3}+\frac{c^3 d^3 \log (d+e x)}{e^4} \]
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)^7,x]
[Out]
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Rubi in Sympy [A] time = 43.274, size = 95, normalized size = 0.9 \[ \frac{c^{3} d^{3} \log{\left (d + e x \right )}}{e^{4}} - \frac{3 c^{2} d^{2} \left (a e^{2} - c d^{2}\right )}{e^{4} \left (d + e x\right )} - \frac{3 c d \left (a e^{2} - c d^{2}\right )^{2}}{2 e^{4} \left (d + e x\right )^{2}} - \frac{\left (a e^{2} - c d^{2}\right )^{3}}{3 e^{4} \left (d + e x\right )^{3}} \]
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)**7,x)
[Out]
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Mathematica [A] time = 0.0893828, size = 92, normalized size = 0.88 \[ \frac{\frac{\left (c d^2-a e^2\right ) \left (2 a^2 e^4+a c d e^2 (5 d+9 e x)+c^2 d^2 \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )}{(d+e x)^3}+6 c^3 d^3 \log (d+e x)}{6 e^4} \]
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)^7,x]
[Out]
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Maple [A] time = 0.01, size = 173, normalized size = 1.7 \[{\frac{{c}^{3}{d}^{3}\ln \left ( ex+d \right ) }{{e}^{4}}}-{\frac{{e}^{2}{a}^{3}}{3\, \left ( ex+d \right ) ^{3}}}+{\frac{{a}^{2}c{d}^{2}}{ \left ( ex+d \right ) ^{3}}}-{\frac{a{c}^{2}{d}^{4}}{{e}^{2} \left ( ex+d \right ) ^{3}}}+{\frac{{c}^{3}{d}^{6}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{3\,{a}^{2}cd}{2\, \left ( ex+d \right ) ^{2}}}+3\,{\frac{a{c}^{2}{d}^{3}}{{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{c}^{3}{d}^{5}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-3\,{\frac{a{c}^{2}{d}^{2}}{{e}^{2} \left ( ex+d \right ) }}+3\,{\frac{{c}^{3}{d}^{4}}{{e}^{4} \left ( ex+d \right ) }} \]
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)^7,x)
[Out]
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Maxima [A] time = 0.730567, size = 213, normalized size = 2.03 \[ \frac{c^{3} d^{3} \log \left (e x + d\right )}{e^{4}} + \frac{11 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} + 18 \,{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 9 \,{\left (3 \, c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \]
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)^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222632, size = 262, normalized size = 2.5 \[ \frac{11 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} + 18 \,{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 9 \,{\left (3 \, c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x + 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + 3 \, c^{3} d^{4} e^{2} x^{2} + 3 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \]
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)^7,x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.52519, size = 163, normalized size = 1.55 \[ \frac{c^{3} d^{3} \log{\left (d + e x \right )}}{e^{4}} - \frac{2 a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 6 a c^{2} d^{4} e^{2} - 11 c^{3} d^{6} + x^{2} \left (18 a c^{2} d^{2} e^{4} - 18 c^{3} d^{4} e^{2}\right ) + x \left (9 a^{2} c d e^{5} + 18 a c^{2} d^{3} e^{3} - 27 c^{3} d^{5} e\right )}{6 d^{3} e^{4} + 18 d^{2} e^{5} x + 18 d e^{6} x^{2} + 6 e^{7} x^{3}} \]
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)**7,x)
[Out]
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GIAC/XCAS [A] time = 0.214033, size = 365, normalized size = 3.48 \[ c^{3} d^{3} e^{\left (-4\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{{\left (11 \, c^{3} d^{9} - 6 \, a c^{2} d^{7} e^{2} - 3 \, a^{2} c d^{5} e^{4} - 2 \, a^{3} d^{3} e^{6} + 18 \,{\left (c^{3} d^{4} e^{5} - a c^{2} d^{2} e^{7}\right )} x^{5} + 9 \,{\left (9 \, c^{3} d^{5} e^{4} - 8 \, a c^{2} d^{3} e^{6} - a^{2} c d e^{8}\right )} x^{4} + 2 \,{\left (73 \, c^{3} d^{6} e^{3} - 57 \, a c^{2} d^{4} e^{5} - 15 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} x^{3} + 6 \,{\left (22 \, c^{3} d^{7} e^{2} - 15 \, a c^{2} d^{5} e^{4} - 6 \, a^{2} c d^{3} e^{6} - a^{3} d e^{8}\right )} x^{2} + 6 \,{\left (10 \, c^{3} d^{8} e - 6 \, a c^{2} d^{6} e^{3} - 3 \, a^{2} c d^{4} e^{5} - a^{3} d^{2} e^{7}\right )} x\right )} e^{\left (-4\right )}}{6 \,{\left (x e + d\right )}^{6}} \]
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)^7,x, algorithm="giac")
[Out]