\(\int \frac {(d+e x)^4}{a d e+(c d^2+a e^2) x+c d e x^2} \, dx\) [1866]

Optimal result

Integrand size = 35, antiderivative size = 100 \[ \int \frac {(d+e x)^4}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {e \left (c d^2-a e^2\right )^2 x}{c^3 d^3}+\frac {\left (c d^2-a e^2\right ) (d+e x)^2}{2 c^2 d^2}+\frac {(d+e x)^3}{3 c d}+\frac {\left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^4 d^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 100, 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} \[ \int \frac {(d+e x)^4}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^4 d^4}+\frac {e x \left (c d^2-a e^2\right )^2}{c^3 d^3}+\frac {(d+e x)^2 \left (c d^2-a e^2\right )}{2 c^2 d^2}+\frac {(d+e x)^3}{3 c d} \]

[In]

Int[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^4}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {c d e x \left (6 a^2 e^4-3 a c d e^2 (6 d+e x)+c^2 d^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+6 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{6 c^4 d^4} \]

[In]

Integrate[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

(c*d*e*x*(6*a^2*e^4 - 3*a*c*d*e^2*(6*d + e*x) + c^2*d^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) + 6*(c*d^2 - a*e^2)^3*
Log[a*e + c*d*x])/(6*c^4*d^4)

Maple [A] (verified)

Time = 2.70 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.32

method	result	size
default	\(\frac {e \left (\frac {1}{3} x^{3} c^{2} d^{2} e^{2}-\frac {1}{2} x^{2} a c d \,e^{3}+\frac {3}{2} x^{2} c^{2} d^{3} e +a^{2} e^{4} x -3 a c \,d^{2} e^{2} x +3 c^{2} d^{4} x \right )}{c^{3} d^{3}}+\frac {\left (-e^{6} a^{3}+3 d^{2} e^{4} a^{2} c -3 d^{4} e^{2} c^{2} a +c^{3} d^{6}\right ) \ln \left (c d x +a e \right )}{c^{4} d^{4}}\)	\(132\)
norman	\(\frac {e \left (a^{2} e^{4}-3 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right ) x}{c^{3} d^{3}}+\frac {e^{3} x^{3}}{3 c d}-\frac {e^{2} \left (e^{2} a -3 c \,d^{2}\right ) x^{2}}{2 c^{2} d^{2}}-\frac {\left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right ) \ln \left (c d x +a e \right )}{c^{4} d^{4}}\)	\(134\)
risch	\(\frac {e^{3} x^{3}}{3 c d}-\frac {e^{4} x^{2} a}{2 c^{2} d^{2}}+\frac {3 e^{2} x^{2}}{2 c}+\frac {e^{5} a^{2} x}{c^{3} d^{3}}-\frac {3 e^{3} a x}{c^{2} d}+\frac {3 e d x}{c}-\frac {\ln \left (c d x +a e \right ) e^{6} a^{3}}{c^{4} d^{4}}+\frac {3 \ln \left (c d x +a e \right ) e^{4} a^{2}}{c^{3} d^{2}}-\frac {3 \ln \left (c d x +a e \right ) e^{2} a}{c^{2}}+\frac {d^{2} \ln \left (c d x +a e \right )}{c}\)	\(157\)
parallelrisch	\(-\frac {-2 x^{3} c^{3} d^{3} e^{3}+3 x^{2} a \,c^{2} d^{2} e^{4}-9 x^{2} c^{3} d^{4} e^{2}+6 \ln \left (c d x +a e \right ) a^{3} e^{6}-18 \ln \left (c d x +a e \right ) a^{2} c \,d^{2} e^{4}+18 \ln \left (c d x +a e \right ) a \,c^{2} d^{4} e^{2}-6 \ln \left (c d x +a e \right ) c^{3} d^{6}-6 x \,a^{2} c d \,e^{5}+18 x a \,c^{2} d^{3} e^{3}-18 x \,c^{3} d^{5} e}{6 c^{4} d^{4}}\)	\(163\)

[In]

int((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.37 \[ \int \frac {(d+e x)^4}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, c^{3} d^{3} e^{3} x^{3} + 3 \, {\left (3 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 6 \, {\left (3 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x + 6 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left (c d x + a e\right )}{6 \, c^{4} d^{4}} \]

[In]

integrate((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^4}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=x^{2} \left (- \frac {a e^{4}}{2 c^{2} d^{2}} + \frac {3 e^{2}}{2 c}\right ) + x \left (\frac {a^{2} e^{5}}{c^{3} d^{3}} - \frac {3 a e^{3}}{c^{2} d} + \frac {3 d e}{c}\right ) + \frac {e^{3} x^{3}}{3 c d} - \frac {\left (a e^{2} - c d^{2}\right )^{3} \log {\left (a e + c d x \right )}}{c^{4} d^{4}} \]

[In]

integrate((e*x+d)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x)^4}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, c^{2} d^{2} e^{3} x^{3} + 3 \, {\left (3 \, c^{2} d^{3} e^{2} - a c d e^{4}\right )} x^{2} + 6 \, {\left (3 \, c^{2} d^{4} e - 3 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x}{6 \, c^{3} d^{3}} + \frac {{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left (c d x + a e\right )}{c^{4} d^{4}} \]

[In]

integrate((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.36 \[ \int \frac {(d+e x)^4}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, c^{2} d^{2} e^{3} x^{3} + 9 \, c^{2} d^{3} e^{2} x^{2} - 3 \, a c d e^{4} x^{2} + 18 \, c^{2} d^{4} e x - 18 \, a c d^{2} e^{3} x + 6 \, a^{2} e^{5} x}{6 \, c^{3} d^{3}} + \frac {{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{4} d^{4}} \]

[In]

integrate((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac")

[Out]

1/6*(2*c^2*d^2*e^3*x^3 + 9*c^2*d^3*e^2*x^2 - 3*a*c*d*e^4*x^2 + 18*c^2*d^4*e*x - 18*a*c*d^2*e^3*x + 6*a^2*e^5*x
)/(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)*log(abs(c*d*x + a*e))/(c^4*d^4)

Mupad [B] (verification not implemented)

Time = 9.72 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.38 \[ \int \frac {(d+e x)^4}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=x\,\left (\frac {3\,d\,e}{c}-\frac {a\,e\,\left (\frac {3\,e^2}{c}-\frac {a\,e^4}{c^2\,d^2}\right )}{c\,d}\right )+x^2\,\left (\frac {3\,e^2}{2\,c}-\frac {a\,e^4}{2\,c^2\,d^2}\right )+\frac {e^3\,x^3}{3\,c\,d}-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}{c^4\,d^4} \]

[In]

int((d + e*x)^4/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2),x)

[Out]

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