3.19.100 \(\int \frac {(d+e x)^6}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\) [1900]

Optimal. Leaf size=35 \[ -\frac {(d+e x)^3}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 37} \begin {gather*} -\frac {(d+e x)^3}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[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*} \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx &=\int \frac {(d+e x)^2}{(a e+c d x)^4} \, dx\\ &=-\frac {(d+e x)^3}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 65, normalized size = 1.86 \begin {gather*} -\frac {a^2 e^4+a c d e^2 (d+3 e x)+c^2 d^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 c^3 d^3 (a e+c d x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(95\) vs. \(2(33)=66\).
time = 0.70, size = 96, normalized size = 2.74

method result size
gosper \(-\frac {3 e^{2} x^{2} c^{2} d^{2}+3 a c d \,e^{3} x +3 c^{2} d^{3} e x +a^{2} e^{4}+a c \,d^{2} e^{2}+c^{2} d^{4}}{3 c^{3} d^{3} \left (c d x +a e \right )^{3}}\) \(76\)
risch \(\frac {-\frac {e^{2} x^{2}}{d c}-\frac {e \left (e^{2} a +c \,d^{2}\right ) x}{c^{2} d^{2}}-\frac {a^{2} e^{4}+a c \,d^{2} e^{2}+c^{2} d^{4}}{3 c^{3} d^{3}}}{\left (c d x +a e \right )^{3}}\) \(80\)
default \(-\frac {e^{2}}{c^{3} d^{3} \left (c d x +a e \right )}-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{3 c^{3} d^{3} \left (c d x +a e \right )^{3}}+\frac {e \left (e^{2} a -c \,d^{2}\right )}{c^{3} d^{3} \left (c d x +a e \right )^{2}}\) \(96\)
norman \(\frac {-\frac {e^{5} x^{5}}{c d}+\frac {\left (-a^{2} e^{6}-2 a c \,d^{2} e^{4}-2 c^{2} d^{4} e^{2}\right ) x}{c^{3} d e}+\frac {\left (-a \,e^{8}-4 c \,e^{6} d^{2}\right ) x^{4}}{c^{2} d^{2} e^{2}}+\frac {\left (-a^{2} e^{8}-4 a c \,d^{2} e^{6}-5 c^{2} e^{4} d^{4}\right ) x^{2}}{c^{3} e^{2} d^{2}}+\frac {-a^{2} e^{4}-a c \,d^{2} e^{2}-c^{2} d^{4}}{3 c^{3}}+\frac {\left (-a^{2} e^{10}-10 a c \,d^{2} e^{8}-19 c^{2} d^{4} e^{6}\right ) x^{3}}{3 c^{3} d^{3} e^{3}}}{\left (c d x +a e \right )^{3} \left (e x +d \right )^{3}}\) \(223\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (34) = 68\).
time = 0.28, size = 109, normalized size = 3.11 \begin {gather*} -\frac {3 \, c^{2} d^{2} x^{2} e^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \, {\left (c^{6} d^{6} x^{3} + 3 \, a c^{5} d^{5} x^{2} e + 3 \, a^{2} c^{4} d^{4} x e^{2} + a^{3} c^{3} d^{3} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (34) = 68\).
time = 3.06, size = 109, normalized size = 3.11 \begin {gather*} -\frac {3 \, c^{2} d^{3} x e + c^{2} d^{4} + 3 \, a c d x e^{3} + a^{2} e^{4} + {\left (3 \, c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}}{3 \, {\left (c^{6} d^{6} x^{3} + 3 \, a c^{5} d^{5} x^{2} e + 3 \, a^{2} c^{4} d^{4} x e^{2} + a^{3} c^{3} d^{3} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (29) = 58\).
time = 0.69, size = 121, normalized size = 3.46 \begin {gather*} \frac {- a^{2} e^{4} - a c d^{2} e^{2} - c^{2} d^{4} - 3 c^{2} d^{2} e^{2} x^{2} + x \left (- 3 a c d e^{3} - 3 c^{2} d^{3} e\right )}{3 a^{3} c^{3} d^{3} e^{3} + 9 a^{2} c^{4} d^{4} e^{2} x + 9 a c^{5} d^{5} e x^{2} + 3 c^{6} d^{6} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

(-a**2*e**4 - a*c*d**2*e**2 - c**2*d**4 - 3*c**2*d**2*e**2*x**2 + x*(-3*a*c*d*e**3 - 3*c**2*d**3*e))/(3*a**3*c
**3*d**3*e**3 + 9*a**2*c**4*d**4*e**2*x + 9*a*c**5*d**5*e*x**2 + 3*c**6*d**6*x**3)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (34) = 68\).
time = 0.62, size = 73, normalized size = 2.09 \begin {gather*} -\frac {3 \, c^{2} d^{2} x^{2} e^{2} + 3 \, c^{2} d^{3} x e + c^{2} d^{4} + 3 \, a c d x e^{3} + a c d^{2} e^{2} + a^{2} e^{4}}{3 \, {\left (c d x + a e\right )}^{3} c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.06, size = 81, normalized size = 2.31 \begin {gather*} -\frac {\frac {d}{3\,c}+e\,\left (\frac {x}{c}-\frac {x^3}{3\,a}\right )+\frac {a\,e^2}{3\,c^2\,d}}{a^3\,e^3+3\,a^2\,c\,d\,e^2\,x+3\,a\,c^2\,d^2\,e\,x^2+c^3\,d^3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^6/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4,x)

[Out]

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

________________________________________________________________________________________