\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{d+e x} \, dx\) [1854]

Optimal result

Integrand size = 35, antiderivative size = 91 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{d+e x} \, dx=\frac {\left (c d^2-a e^2\right )^2 (a e+c d x)^4}{4 c^3 d^3}+\frac {2 e \left (c d^2-a e^2\right ) (a e+c d x)^5}{5 c^3 d^3}+\frac {e^2 (a e+c d x)^6}{6 c^3 d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 91, 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} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{d+e x} \, dx=\frac {e^2 (a e+c d x)^6}{6 c^3 d^3}+\frac {2 e \left (c d^2-a e^2\right ) (a e+c d x)^5}{5 c^3 d^3}+\frac {\left (c d^2-a e^2\right )^2 (a e+c d x)^4}{4 c^3 d^3} \]

[In]

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

[Out]

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

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 (a e+c d x)^3 (d+e x)^2 \, dx \\ & = \int \left (\frac {\left (c d^2-a e^2\right )^2 (a e+c d x)^3}{c^2 d^2}+\frac {2 e \left (c d^2-a e^2\right ) (a e+c d x)^4}{c^2 d^2}+\frac {e^2 (a e+c d x)^5}{c^2 d^2}\right ) \, dx \\ & = \frac {\left (c d^2-a e^2\right )^2 (a e+c d x)^4}{4 c^3 d^3}+\frac {2 e \left (c d^2-a e^2\right ) (a e+c d x)^5}{5 c^3 d^3}+\frac {e^2 (a e+c d x)^6}{6 c^3 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{d+e x} \, dx=\frac {1}{60} x \left (20 a^3 e^3 \left (3 d^2+3 d e x+e^2 x^2\right )+15 a^2 c d e^2 x \left (6 d^2+8 d e x+3 e^2 x^2\right )+6 a c^2 d^2 e x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+c^3 d^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )\right ) \]

[In]

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

[Out]

(x*(20*a^3*e^3*(3*d^2 + 3*d*e*x + e^2*x^2) + 15*a^2*c*d*e^2*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + 6*a*c^2*d^2*e*x^
2*(10*d^2 + 15*d*e*x + 6*e^2*x^2) + c^3*d^3*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2)))/60

Maple [A] (verified)

Time = 2.43 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.62

method	result	size
norman	\(\frac {e^{2} c^{3} d^{3} x^{6}}{6}+\left (\frac {3}{5} e^{3} a \,c^{2} d^{2}+\frac {2}{5} d^{4} e \,c^{3}\right ) x^{5}+\left (\frac {3}{4} d \,e^{4} a^{2} c +\frac {3}{2} d^{3} e^{2} c^{2} a +\frac {1}{4} d^{5} c^{3}\right ) x^{4}+\left (\frac {1}{3} a^{3} e^{5}+2 d^{2} e^{3} a^{2} c +d^{4} a \,c^{2} e \right ) x^{3}+\left (d \,e^{4} a^{3}+\frac {3}{2} d^{3} e^{2} a^{2} c \right ) x^{2}+d^{2} a^{3} e^{3} x\)	\(147\)
risch	\(\frac {1}{6} e^{2} c^{3} d^{3} x^{6}+\frac {3}{5} x^{5} e^{3} a \,c^{2} d^{2}+\frac {2}{5} x^{5} d^{4} e \,c^{3}+\frac {3}{4} x^{4} d \,e^{4} a^{2} c +\frac {3}{2} x^{4} d^{3} e^{2} c^{2} a +\frac {1}{4} x^{4} d^{5} c^{3}+\frac {1}{3} x^{3} a^{3} e^{5}+2 x^{3} d^{2} e^{3} a^{2} c +x^{3} d^{4} a \,c^{2} e +x^{2} d \,e^{4} a^{3}+\frac {3}{2} x^{2} d^{3} e^{2} a^{2} c +d^{2} a^{3} e^{3} x\)	\(157\)
parallelrisch	\(\frac {1}{6} e^{2} c^{3} d^{3} x^{6}+\frac {3}{5} x^{5} e^{3} a \,c^{2} d^{2}+\frac {2}{5} x^{5} d^{4} e \,c^{3}+\frac {3}{4} x^{4} d \,e^{4} a^{2} c +\frac {3}{2} x^{4} d^{3} e^{2} c^{2} a +\frac {1}{4} x^{4} d^{5} c^{3}+\frac {1}{3} x^{3} a^{3} e^{5}+2 x^{3} d^{2} e^{3} a^{2} c +x^{3} d^{4} a \,c^{2} e +x^{2} d \,e^{4} a^{3}+\frac {3}{2} x^{2} d^{3} e^{2} a^{2} c +d^{2} a^{3} e^{3} x\)	\(157\)
gosper	\(\frac {x \left (10 e^{2} c^{3} d^{3} x^{5}+36 x^{4} e^{3} a \,c^{2} d^{2}+24 x^{4} d^{4} e \,c^{3}+45 x^{3} d \,e^{4} a^{2} c +90 x^{3} d^{3} e^{2} c^{2} a +15 x^{3} d^{5} c^{3}+20 x^{2} a^{3} e^{5}+120 x^{2} d^{2} e^{3} a^{2} c +60 x^{2} d^{4} a \,c^{2} e +60 x d \,e^{4} a^{3}+90 x \,d^{3} e^{2} a^{2} c +60 d^{2} a^{3} e^{3}\right )}{60}\)	\(158\)
default	\(\frac {e^{2} c^{3} d^{3} x^{6}}{6}+\frac {\left (e^{3} a \,c^{2} d^{2}+2 c^{2} d^{2} e \left (e^{2} a +c \,d^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 a \,e^{2} c d \left (e^{2} a +c \,d^{2}\right )+c d \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right ) x^{4}}{4}+\frac {\left (a e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )+2 c \,d^{2} a e \left (e^{2} a +c \,d^{2}\right )\right ) x^{3}}{3}+\frac {\left (2 a^{2} e^{2} d \left (e^{2} a +c \,d^{2}\right )+d^{3} e^{2} a^{2} c \right ) x^{2}}{2}+d^{2} a^{3} e^{3} x\)	\(205\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.65 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{d+e x} \, dx=\frac {1}{6} \, c^{3} d^{3} e^{2} x^{6} + a^{3} d^{2} e^{3} x + \frac {1}{5} \, {\left (2 \, c^{3} d^{4} e + 3 \, a c^{2} d^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (c^{3} d^{5} + 6 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a c^{2} d^{4} e + 6 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} c d^{3} e^{2} + 2 \, a^{3} d e^{4}\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{d+e x} \, dx=a^{3} d^{2} e^{3} x + \frac {c^{3} d^{3} e^{2} x^{6}}{6} + x^{5} \cdot \left (\frac {3 a c^{2} d^{2} e^{3}}{5} + \frac {2 c^{3} d^{4} e}{5}\right ) + x^{4} \cdot \left (\frac {3 a^{2} c d e^{4}}{4} + \frac {3 a c^{2} d^{3} e^{2}}{2} + \frac {c^{3} d^{5}}{4}\right ) + x^{3} \left (\frac {a^{3} e^{5}}{3} + 2 a^{2} c d^{2} e^{3} + a c^{2} d^{4} e\right ) + x^{2} \left (a^{3} d e^{4} + \frac {3 a^{2} c d^{3} e^{2}}{2}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.65 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{d+e x} \, dx=\frac {1}{6} \, c^{3} d^{3} e^{2} x^{6} + a^{3} d^{2} e^{3} x + \frac {1}{5} \, {\left (2 \, c^{3} d^{4} e + 3 \, a c^{2} d^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (c^{3} d^{5} + 6 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a c^{2} d^{4} e + 6 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} c d^{3} e^{2} + 2 \, a^{3} d e^{4}\right )} x^{2} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.71 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{d+e x} \, dx=\frac {1}{6} \, c^{3} d^{3} e^{2} x^{6} + \frac {2}{5} \, c^{3} d^{4} e x^{5} + \frac {3}{5} \, a c^{2} d^{2} e^{3} x^{5} + \frac {1}{4} \, c^{3} d^{5} x^{4} + \frac {3}{2} \, a c^{2} d^{3} e^{2} x^{4} + \frac {3}{4} \, a^{2} c d e^{4} x^{4} + a c^{2} d^{4} e x^{3} + 2 \, a^{2} c d^{2} e^{3} x^{3} + \frac {1}{3} \, a^{3} e^{5} x^{3} + \frac {3}{2} \, a^{2} c d^{3} e^{2} x^{2} + a^{3} d e^{4} x^{2} + a^{3} d^{2} e^{3} x \]

[In]

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

[Out]

1/6*c^3*d^3*e^2*x^6 + 2/5*c^3*d^4*e*x^5 + 3/5*a*c^2*d^2*e^3*x^5 + 1/4*c^3*d^5*x^4 + 3/2*a*c^2*d^3*e^2*x^4 + 3/
4*a^2*c*d*e^4*x^4 + a*c^2*d^4*e*x^3 + 2*a^2*c*d^2*e^3*x^3 + 1/3*a^3*e^5*x^3 + 3/2*a^2*c*d^3*e^2*x^2 + a^3*d*e^
4*x^2 + a^3*d^2*e^3*x

Mupad [B] (verification not implemented)

Time = 9.89 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.59 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{d+e x} \, dx=x^3\,\left (\frac {a^3\,e^5}{3}+2\,a^2\,c\,d^2\,e^3+a\,c^2\,d^4\,e\right )+x^4\,\left (\frac {3\,a^2\,c\,d\,e^4}{4}+\frac {3\,a\,c^2\,d^3\,e^2}{2}+\frac {c^3\,d^5}{4}\right )+a^3\,d^2\,e^3\,x+\frac {c^3\,d^3\,e^2\,x^6}{6}+\frac {a^2\,d\,e^2\,x^2\,\left (3\,c\,d^2+2\,a\,e^2\right )}{2}+\frac {c^2\,d^2\,e\,x^5\,\left (2\,c\,d^2+3\,a\,e^2\right )}{5} \]

[In]

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

[Out]

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