\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^2} \, dx\) [1843]

Optimal result

Integrand size = 35, antiderivative size = 20 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^2} \, dx=\frac {(a e+c d x)^3}{3 c d} \]

[Out]

1/3*(c*d*x+a*e)^3/c/d

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 32} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^2} \, dx=\frac {(a e+c d x)^3}{3 c d} \]

[In]

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

[Out]

(a*e + c*d*x)^3/(3*c*d)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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)^2 \, dx \\ & = \frac {(a e+c d x)^3}{3 c d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^2} \, dx=\frac {(a e+c d x)^3}{3 c d} \]

[In]

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

[Out]

(a*e + c*d*x)^3/(3*c*d)

Maple [A] (verified)

Time = 2.69 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {\left (c d x +a e \right )^{3}}{3 c d}\)	\(19\)
parallelrisch	\(\frac {1}{3} c^{2} d^{2} x^{3}+a c d e \,x^{2}+a^{2} e^{2} x\)	\(29\)
gosper	\(\frac {x \left (c^{2} d^{2} x^{2}+3 a x c d e +3 a^{2} e^{2}\right )}{3}\)	\(30\)
risch	\(\frac {c^{2} d^{2} x^{3}}{3}+a c d e \,x^{2}+a^{2} e^{2} x +\frac {a^{3} e^{3}}{3 d c}\)	\(43\)
norman	\(\frac {\left (d \,e^{2} a c +\frac {1}{3} c^{2} d^{3}\right ) x^{3}+\left (a^{2} e^{3}+d^{2} e a c \right ) x^{2}+a^{2} d \,e^{2} x +\frac {e \,c^{2} d^{2} x^{4}}{3}}{e x +d}\)	\(70\)

[In]

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

[Out]

1/3*(c*d*x+a*e)^3/c/d

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^2} \, dx=\frac {1}{3} \, c^{2} d^{2} x^{3} + a c d e x^{2} + a^{2} e^{2} x \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (14) = 28\).

Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^2} \, dx=a^{2} e^{2} x + a c d e x^{2} + \frac {c^{2} d^{2} x^{3}}{3} \]

[In]

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

[Out]

a**2*e**2*x + a*c*d*e*x**2 + c**2*d**2*x**3/3

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^2} \, dx=\frac {1}{3} \, c^{2} d^{2} x^{3} + a c d e x^{2} + a^{2} e^{2} x \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (18) = 36\).

Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 4.85 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^2} \, dx=\frac {{\left (c^{2} d^{2} - \frac {3 \, c^{2} d^{3}}{e x + d} + \frac {3 \, c^{2} d^{4}}{{\left (e x + d\right )}^{2}} + \frac {3 \, a c d e^{2}}{e x + d} - \frac {6 \, a c d^{2} e^{2}}{{\left (e x + d\right )}^{2}} + \frac {3 \, a^{2} e^{4}}{{\left (e x + d\right )}^{2}}\right )} {\left (e x + d\right )}^{3}}{3 \, e^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^2} \, dx=a^2\,e^2\,x+a\,c\,d\,e\,x^2+\frac {c^2\,d^2\,x^3}{3} \]

[In]

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

[Out]

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