\(\int \frac {(d+e x)^5}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\) [1877]

Optimal result

Integrand size = 35, antiderivative size = 105 \[ \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {e^2 \left (3 c d^2-2 a e^2\right ) x}{c^3 d^3}+\frac {e^3 x^2}{2 c^2 d^2}-\frac {\left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)}+\frac {3 e \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^4 d^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45} \[ \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {\left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)}+\frac {3 e \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^4 d^4}+\frac {e^2 x \left (3 c d^2-2 a e^2\right )}{c^3 d^3}+\frac {e^3 x^2}{2 c^2 d^2} \]

[In]

Int[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^3}{(a e+c d x)^2} \, dx \\ & = \int \left (\frac {3 c d^2 e^2-2 a e^4}{c^3 d^3}+\frac {e^3 x}{c^2 d^2}+\frac {\left (c d^2-a e^2\right )^3}{c^3 d^3 (a e+c d x)^2}+\frac {3 e \left (c d^2-a e^2\right )^2}{c^3 d^3 (a e+c d x)}\right ) \, dx \\ & = \frac {e^2 \left (3 c d^2-2 a e^2\right ) x}{c^3 d^3}+\frac {e^3 x^2}{2 c^2 d^2}-\frac {\left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)}+\frac {3 e \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^4 d^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {e^2 \left (-3 c d^2+2 a e^2\right ) x}{c^3 d^3}+\frac {e^3 x^2}{2 c^2 d^2}+\frac {-c^3 d^6+3 a c^2 d^4 e^2-3 a^2 c d^2 e^4+a^3 e^6}{c^4 d^4 (a e+c d x)}+\frac {3 \left (c^2 d^4 e-2 a c d^2 e^3+a^2 e^5\right ) \log (a e+c d x)}{c^4 d^4} \]

[In]

Integrate[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

-((e^2*(-3*c*d^2 + 2*a*e^2)*x)/(c^3*d^3)) + (e^3*x^2)/(2*c^2*d^2) + (-(c^3*d^6) + 3*a*c^2*d^4*e^2 - 3*a^2*c*d^
2*e^4 + a^3*e^6)/(c^4*d^4*(a*e + c*d*x)) + (3*(c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*Log[a*e + c*d*x])/(c^4*d^4
)

Maple [A] (verified)

Time = 2.74 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.30

method	result	size
default	\(-\frac {e^{2} \left (-\frac {1}{2} c d e \,x^{2}+2 a \,e^{2} x -3 c \,d^{2} x \right )}{c^{3} d^{3}}+\frac {3 e \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (c d x +a e \right )}{c^{4} d^{4}}-\frac {-e^{6} a^{3}+3 d^{2} e^{4} a^{2} c -3 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{c^{4} d^{4} \left (c d x +a e \right )}\)	\(137\)
risch	\(\frac {e^{3} x^{2}}{2 c^{2} d^{2}}-\frac {2 e^{4} a x}{c^{3} d^{3}}+\frac {3 e^{2} x}{c^{2} d}+\frac {e^{6} a^{3}}{c^{4} d^{4} \left (c d x +a e \right )}-\frac {3 e^{4} a^{2}}{c^{3} d^{2} \left (c d x +a e \right )}+\frac {3 e^{2} a}{c^{2} \left (c d x +a e \right )}-\frac {d^{2}}{c \left (c d x +a e \right )}+\frac {3 e^{5} \ln \left (c d x +a e \right ) a^{2}}{c^{4} d^{4}}-\frac {6 e^{3} \ln \left (c d x +a e \right ) a}{c^{3} d^{2}}+\frac {3 e \ln \left (c d x +a e \right )}{c^{2}}\)	\(184\)
norman	\(\frac {\frac {6 e^{6} a^{3}-9 d^{2} e^{4} a^{2} c -2 c^{3} d^{6}}{2 d^{3} c^{4}}+\frac {e^{4} x^{4}}{2 c d}+\frac {\left (6 a^{3} e^{8}-9 a^{2} c \,d^{2} e^{6}+3 a \,c^{2} d^{4} e^{4}-8 c^{3} d^{6} e^{2}\right ) x}{2 c^{4} d^{4} e}-\frac {e^{3} \left (3 e^{2} a -7 c \,d^{2}\right ) x^{3}}{2 c^{2} d^{2}}}{\left (c d x +a e \right ) \left (e x +d \right )}+\frac {3 e \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (c d x +a e \right )}{c^{4} d^{4}}\)	\(198\)
parallelrisch	\(\frac {x^{3} c^{3} d^{3} e^{3}+6 \ln \left (c d x +a e \right ) x \,a^{2} c d \,e^{5}-12 \ln \left (c d x +a e \right ) x a \,c^{2} d^{3} e^{3}+6 \ln \left (c d x +a e \right ) x \,c^{3} d^{5} e -3 x^{2} a \,c^{2} d^{2} e^{4}+6 x^{2} c^{3} d^{4} e^{2}+6 \ln \left (c d x +a e \right ) a^{3} e^{6}-12 \ln \left (c d x +a e \right ) a^{2} c \,d^{2} e^{4}+6 \ln \left (c d x +a e \right ) a \,c^{2} d^{4} e^{2}+6 e^{6} a^{3}-12 d^{2} e^{4} a^{2} c +6 d^{4} e^{2} c^{2} a -2 c^{3} d^{6}}{2 c^{4} d^{4} \left (c d x +a e \right )}\)	\(222\)

[In]

int((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.95 \[ \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {c^{3} d^{3} e^{3} x^{3} - 2 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} + 2 \, a^{3} e^{6} + 3 \, {\left (2 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (3 \, a c^{2} d^{3} e^{3} - 2 \, a^{2} c d e^{5}\right )} x + 6 \, {\left (a c^{2} d^{4} e^{2} - 2 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \log \left (c d x + a e\right )}{2 \, {\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}} \]

[In]

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.25 \[ \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=x \left (- \frac {2 a e^{4}}{c^{3} d^{3}} + \frac {3 e^{2}}{c^{2} d}\right ) + \frac {a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} - c^{3} d^{6}}{a c^{4} d^{4} e + c^{5} d^{5} x} + \frac {e^{3} x^{2}}{2 c^{2} d^{2}} + \frac {3 e \left (a e^{2} - c d^{2}\right )^{2} \log {\left (a e + c d x \right )}}{c^{4} d^{4}} \]

[In]

integrate((e*x+d)**5/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.36 \[ \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}}{c^{5} d^{5} x + a c^{4} d^{4} e} + \frac {c d e^{3} x^{2} + 2 \, {\left (3 \, c d^{2} e^{2} - 2 \, a e^{4}\right )} x}{2 \, c^{3} d^{3}} + \frac {3 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} \log \left (c d x + a e\right )}{c^{4} d^{4}} \]

[In]

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.39 \[ \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {3 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{4} d^{4}} + \frac {c^{2} d^{2} e^{3} x^{2} + 6 \, c^{2} d^{3} e^{2} x - 4 \, a c d e^{4} x}{2 \, c^{4} d^{4}} - \frac {c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}}{{\left (c d x + a e\right )} c^{4} d^{4}} \]

[In]

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.45 \[ \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=x\,\left (\frac {3\,e^2}{c^2\,d}-\frac {2\,a\,e^4}{c^3\,d^3}\right )+\frac {e^3\,x^2}{2\,c^2\,d^2}+\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (3\,a^2\,e^5-6\,a\,c\,d^2\,e^3+3\,c^2\,d^4\,e\right )}{c^4\,d^4}+\frac {a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6}{c\,d\,\left (x\,c^4\,d^4+a\,e\,c^3\,d^3\right )} \]

[In]

int((d + e*x)^5/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2,x)

[Out]

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