3.1856 \(\int \frac{(d+e x)^3}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\)

Optimal. Leaf size=69 \[ \frac{\left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^3 d^3}+\frac{e x \left (c d^2-a e^2\right )}{c^2 d^2}+\frac{(d+e x)^2}{2 c d} \]

[Out]

(e*(c*d^2 - a*e^2)*x)/(c^2*d^2) + (d + e*x)^2/(2*c*d) + ((c*d^2 - a*e^2)^2*Log[a
*e + c*d*x])/(c^3*d^3)

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Rubi [A]  time = 0.0890186, antiderivative size = 69, 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 \[ \frac{\left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^3 d^3}+\frac{e x \left (c d^2-a e^2\right )}{c^2 d^2}+\frac{(d+e x)^2}{2 c d} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

(e*(c*d^2 - a*e^2)*x)/(c^2*d^2) + (d + e*x)^2/(2*c*d) + ((c*d^2 - a*e^2)^2*Log[a
*e + c*d*x])/(c^3*d^3)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\left (d + e x\right )^{2}}{2 c d} - \frac{\left (a e^{2} - c d^{2}\right ) \int e\, dx}{c^{2} d^{2}} + \frac{\left (a e^{2} - c d^{2}\right )^{2} \log{\left (a e + c d x \right )}}{c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

(d + e*x)**2/(2*c*d) - (a*e**2 - c*d**2)*Integral(e, x)/(c**2*d**2) + (a*e**2 -
c*d**2)**2*log(a*e + c*d*x)/(c**3*d**3)

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Mathematica [A]  time = 0.0423942, size = 58, normalized size = 0.84 \[ \frac{2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)+c d e x \left (c d (4 d+e x)-2 a e^2\right )}{2 c^3 d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

(c*d*e*x*(-2*a*e^2 + c*d*(4*d + e*x)) + 2*(c*d^2 - a*e^2)^2*Log[a*e + c*d*x])/(2
*c^3*d^3)

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Maple [A]  time = 0.005, size = 93, normalized size = 1.4 \[{\frac{{e}^{2}{x}^{2}}{2\,cd}}-{\frac{a{e}^{3}x}{{c}^{2}{d}^{2}}}+2\,{\frac{ex}{c}}+{\frac{\ln \left ( cdx+ae \right ){a}^{2}{e}^{4}}{{c}^{3}{d}^{3}}}-2\,{\frac{\ln \left ( cdx+ae \right ) a{e}^{2}}{{c}^{2}d}}+{\frac{d\ln \left ( cdx+ae \right ) }{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2),x)

[Out]

1/2*e^2/c/d*x^2-e^3/c^2/d^2*a*x+2*e/c*x+1/c^3/d^3*ln(c*d*x+a*e)*a^2*e^4-2/c^2/d*
ln(c*d*x+a*e)*a*e^2+1/c*d*ln(c*d*x+a*e)

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Maxima [A]  time = 0.721702, size = 104, normalized size = 1.51 \[ \frac{c d e^{2} x^{2} + 2 \,{\left (2 \, c d^{2} e - a e^{3}\right )} x}{2 \, c^{2} d^{2}} + \frac{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (c d x + a e\right )}{c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="maxima")

[Out]

1/2*(c*d*e^2*x^2 + 2*(2*c*d^2*e - a*e^3)*x)/(c^2*d^2) + (c^2*d^4 - 2*a*c*d^2*e^2
+ a^2*e^4)*log(c*d*x + a*e)/(c^3*d^3)

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Fricas [A]  time = 0.222694, size = 107, normalized size = 1.55 \[ \frac{c^{2} d^{2} e^{2} x^{2} + 2 \,{\left (2 \, c^{2} d^{3} e - a c d e^{3}\right )} x + 2 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (c d x + a e\right )}{2 \, c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="fricas")

[Out]

1/2*(c^2*d^2*e^2*x^2 + 2*(2*c^2*d^3*e - a*c*d*e^3)*x + 2*(c^2*d^4 - 2*a*c*d^2*e^
2 + a^2*e^4)*log(c*d*x + a*e))/(c^3*d^3)

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Sympy [A]  time = 1.67113, size = 61, normalized size = 0.88 \[ \frac{e^{2} x^{2}}{2 c d} - \frac{x \left (a e^{3} - 2 c d^{2} e\right )}{c^{2} d^{2}} + \frac{\left (a e^{2} - c d^{2}\right )^{2} \log{\left (a e + c d x \right )}}{c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

e**2*x**2/(2*c*d) - x*(a*e**3 - 2*c*d**2*e)/(c**2*d**2) + (a*e**2 - c*d**2)**2*l
og(a*e + c*d*x)/(c**3*d**3)

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GIAC/XCAS [A]  time = 0.219085, size = 282, normalized size = 4.09 \[ \frac{{\left (c d x^{2} e^{4} + 4 \, c d^{2} x e^{3} - 2 \, a x e^{5}\right )} e^{\left (-2\right )}}{2 \, c^{2} d^{2}} + \frac{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\rm ln}\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{3} d^{3}} + \frac{{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \arctan \left (\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}} c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="giac")

[Out]

1/2*(c*d*x^2*e^4 + 4*c*d^2*x*e^3 - 2*a*x*e^5)*e^(-2)/(c^2*d^2) + 1/2*(c^2*d^4 -
2*a*c*d^2*e^2 + a^2*e^4)*ln(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)/(c^3*d^3) + (
c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*arctan((2*c*d*x*e + c*d^2
+ a*e^2)/sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4))/(sqrt(-c^2*d^4 + 2*a*c*d^2*e
^2 - a^2*e^4)*c^3*d^3)