3.1860 \(\int \frac{1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx\)

Optimal. Leaf size=73 \[ \frac{1}{(d+e x) \left (c d^2-a e^2\right )}+\frac{c d \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}-\frac{c d \log (d+e x)}{\left (c d^2-a e^2\right )^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.133147, antiderivative size = 73, 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 \[ \frac{1}{(d+e x) \left (c d^2-a e^2\right )}+\frac{c d \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}-\frac{c d \log (d+e x)}{\left (c d^2-a e^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 37.3464, size = 63, normalized size = 0.86 \[ - \frac{c d \log{\left (d + e x \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{c d \log{\left (a e + c d x \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{1}{\left (d + e x\right ) \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0565733, size = 66, normalized size = 0.9 \[ \frac{c d (d+e x) \log (a e+c d x)-a e^2+c d^2-c d (d+e x) \log (d+e x)}{(d+e x) \left (c d^2-a e^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.025, size = 75, normalized size = 1. \[ -{\frac{1}{ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( ex+d \right ) }}-{\frac{cd\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}}+{\frac{cd\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.716639, size = 144, normalized size = 1.97 \[ \frac{c d \log \left (c d x + a e\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac{c d \log \left (e x + d\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac{1}{c d^{3} - a d e^{2} +{\left (c d^{2} e - a e^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.209312, size = 147, normalized size = 2.01 \[ \frac{c d^{2} - a e^{2} +{\left (c d e x + c d^{2}\right )} \log \left (c d x + a e\right ) -{\left (c d e x + c d^{2}\right )} \log \left (e x + d\right )}{c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4} +{\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)),x, algorithm="fricas")

[Out]

(c*d^2 - a*e^2 + (c*d*e*x + c*d^2)*log(c*d*x + a*e) - (c*d*e*x + c*d^2)*log(e*x
+ d))/(c^2*d^5 - 2*a*c*d^3*e^2 + a^2*d*e^4 + (c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^
5)*x)

_______________________________________________________________________________________

Sympy [A]  time = 3.24132, size = 301, normalized size = 4.12 \[ - \frac{c d \log{\left (x + \frac{- \frac{a^{3} c d e^{6}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{3 a^{2} c^{2} d^{3} e^{4}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{3 a c^{3} d^{5} e^{2}}{\left (a e^{2} - c d^{2}\right )^{2}} + a c d e^{2} + \frac{c^{4} d^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} + c^{2} d^{3}}{2 c^{2} d^{2} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{c d \log{\left (x + \frac{\frac{a^{3} c d e^{6}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{3 a^{2} c^{2} d^{3} e^{4}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{3 a c^{3} d^{5} e^{2}}{\left (a e^{2} - c d^{2}\right )^{2}} + a c d e^{2} - \frac{c^{4} d^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} + c^{2} d^{3}}{2 c^{2} d^{2} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{1}{a d e^{2} - c d^{3} + x \left (a e^{3} - c d^{2} e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

-c*d*log(x + (-a**3*c*d*e**6/(a*e**2 - c*d**2)**2 + 3*a**2*c**2*d**3*e**4/(a*e**
2 - c*d**2)**2 - 3*a*c**3*d**5*e**2/(a*e**2 - c*d**2)**2 + a*c*d*e**2 + c**4*d**
7/(a*e**2 - c*d**2)**2 + c**2*d**3)/(2*c**2*d**2*e))/(a*e**2 - c*d**2)**2 + c*d*
log(x + (a**3*c*d*e**6/(a*e**2 - c*d**2)**2 - 3*a**2*c**2*d**3*e**4/(a*e**2 - c*
d**2)**2 + 3*a*c**3*d**5*e**2/(a*e**2 - c*d**2)**2 + a*c*d*e**2 - c**4*d**7/(a*e
**2 - c*d**2)**2 + c**2*d**3)/(2*c**2*d**2*e))/(a*e**2 - c*d**2)**2 - 1/(a*d*e**
2 - c*d**3 + x*(a*e**3 - c*d**2*e))

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError