### 3.1899 $$\int \frac{(d+e x)^7}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx$$

Optimal. Leaf size=122 $-\frac{3 e^2 \left (c d^2-a e^2\right )}{c^4 d^4 (a e+c d x)}-\frac{3 e \left (c d^2-a e^2\right )^2}{2 c^4 d^4 (a e+c d x)^2}-\frac{\left (c d^2-a e^2\right )^3}{3 c^4 d^4 (a e+c d x)^3}+\frac{e^3 \log (a e+c d x)}{c^4 d^4}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0882848, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.057, Rules used = {626, 43} $-\frac{3 e^2 \left (c d^2-a e^2\right )}{c^4 d^4 (a e+c d x)}-\frac{3 e \left (c d^2-a e^2\right )^2}{2 c^4 d^4 (a e+c d x)^2}-\frac{\left (c d^2-a e^2\right )^3}{3 c^4 d^4 (a e+c d x)^3}+\frac{e^3 \log (a e+c d x)}{c^4 d^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

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

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^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]

Rule 43

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])

Rubi steps

\begin{align*} \int \frac{(d+e x)^7}{\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)^3}{(a e+c d x)^4} \, dx\\ &=\int \left (\frac{\left (c d^2-a e^2\right )^3}{c^3 d^3 (a e+c d x)^4}+\frac{3 e \left (c d^2-a e^2\right )^2}{c^3 d^3 (a e+c d x)^3}+\frac{3 \left (c d^2 e^2-a e^4\right )}{c^3 d^3 (a e+c d x)^2}+\frac{e^3}{c^3 d^3 (a e+c d x)}\right ) \, dx\\ &=-\frac{\left (c d^2-a e^2\right )^3}{3 c^4 d^4 (a e+c d x)^3}-\frac{3 e \left (c d^2-a e^2\right )^2}{2 c^4 d^4 (a e+c d x)^2}-\frac{3 e^2 \left (c d^2-a e^2\right )}{c^4 d^4 (a e+c d x)}+\frac{e^3 \log (a e+c d x)}{c^4 d^4}\\ \end{align*}

Mathematica [A]  time = 0.0516119, size = 99, normalized size = 0.81 $\frac{6 e^3 \log (a e+c d x)-\frac{\left (c d^2-a e^2\right ) \left (11 a^2 e^4+a c d e^2 (5 d+27 e x)+c^2 d^2 \left (2 d^2+9 d e x+18 e^2 x^2\right )\right )}{(a e+c d x)^3}}{6 c^4 d^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

(-(((c*d^2 - a*e^2)*(11*a^2*e^4 + a*c*d*e^2*(5*d + 27*e*x) + c^2*d^2*(2*d^2 + 9*d*e*x + 18*e^2*x^2)))/(a*e + c
*d*x)^3) + 6*e^3*Log[a*e + c*d*x])/(6*c^4*d^4)

________________________________________________________________________________________

Maple [A]  time = 0.045, size = 210, normalized size = 1.7 \begin{align*}{\frac{{a}^{3}{e}^{6}}{3\,{c}^{4}{d}^{4} \left ( cdx+ae \right ) ^{3}}}-{\frac{{a}^{2}{e}^{4}}{{c}^{3}{d}^{2} \left ( cdx+ae \right ) ^{3}}}+{\frac{a{e}^{2}}{{c}^{2} \left ( cdx+ae \right ) ^{3}}}-{\frac{{d}^{2}}{3\,c \left ( cdx+ae \right ) ^{3}}}-{\frac{3\,{a}^{2}{e}^{5}}{2\,{c}^{4}{d}^{4} \left ( cdx+ae \right ) ^{2}}}+3\,{\frac{a{e}^{3}}{{c}^{3}{d}^{2} \left ( cdx+ae \right ) ^{2}}}-{\frac{3\,e}{2\,{c}^{2} \left ( cdx+ae \right ) ^{2}}}+3\,{\frac{{e}^{4}a}{{c}^{4}{d}^{4} \left ( cdx+ae \right ) }}-3\,{\frac{{e}^{2}}{{c}^{3}{d}^{2} \left ( cdx+ae \right ) }}+{\frac{{e}^{3}\ln \left ( cdx+ae \right ) }{{c}^{4}{d}^{4}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.17454, size = 242, normalized size = 1.98 \begin{align*} -\frac{2 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6} + 18 \,{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 9 \,{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x}{6 \,{\left (c^{7} d^{7} x^{3} + 3 \, a c^{6} d^{6} e x^{2} + 3 \, a^{2} c^{5} d^{5} e^{2} x + a^{3} c^{4} d^{4} e^{3}\right )}} + \frac{e^{3} \log \left (c d x + a e\right )}{c^{4} d^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxima")

[Out]

-1/6*(2*c^3*d^6 + 3*a*c^2*d^4*e^2 + 6*a^2*c*d^2*e^4 - 11*a^3*e^6 + 18*(c^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2 + 9*(c
^3*d^5*e + 2*a*c^2*d^3*e^3 - 3*a^2*c*d*e^5)*x)/(c^7*d^7*x^3 + 3*a*c^6*d^6*e*x^2 + 3*a^2*c^5*d^5*e^2*x + a^3*c^
4*d^4*e^3) + e^3*log(c*d*x + a*e)/(c^4*d^4)

________________________________________________________________________________________

Fricas [A]  time = 1.80898, size = 433, normalized size = 3.55 \begin{align*} -\frac{2 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6} + 18 \,{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 9 \,{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x - 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{4} x^{2} + 3 \, a^{2} c d e^{5} x + a^{3} e^{6}\right )} \log \left (c d x + a e\right )}{6 \,{\left (c^{7} d^{7} x^{3} + 3 \, a c^{6} d^{6} e x^{2} + 3 \, a^{2} c^{5} d^{5} e^{2} x + a^{3} c^{4} d^{4} e^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fricas")

[Out]

-1/6*(2*c^3*d^6 + 3*a*c^2*d^4*e^2 + 6*a^2*c*d^2*e^4 - 11*a^3*e^6 + 18*(c^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2 + 9*(c
^3*d^5*e + 2*a*c^2*d^3*e^3 - 3*a^2*c*d*e^5)*x - 6*(c^3*d^3*e^3*x^3 + 3*a*c^2*d^2*e^4*x^2 + 3*a^2*c*d*e^5*x + a
^3*e^6)*log(c*d*x + a*e))/(c^7*d^7*x^3 + 3*a*c^6*d^6*e*x^2 + 3*a^2*c^5*d^5*e^2*x + a^3*c^4*d^4*e^3)

________________________________________________________________________________________

Sympy [A]  time = 3.17158, size = 189, normalized size = 1.55 \begin{align*} \frac{11 a^{3} e^{6} - 6 a^{2} c d^{2} e^{4} - 3 a c^{2} d^{4} e^{2} - 2 c^{3} d^{6} + x^{2} \left (18 a c^{2} d^{2} e^{4} - 18 c^{3} d^{4} e^{2}\right ) + x \left (27 a^{2} c d e^{5} - 18 a c^{2} d^{3} e^{3} - 9 c^{3} d^{5} e\right )}{6 a^{3} c^{4} d^{4} e^{3} + 18 a^{2} c^{5} d^{5} e^{2} x + 18 a c^{6} d^{6} e x^{2} + 6 c^{7} d^{7} x^{3}} + \frac{e^{3} \log{\left (a e + c d x \right )}}{c^{4} d^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**7/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

(11*a**3*e**6 - 6*a**2*c*d**2*e**4 - 3*a*c**2*d**4*e**2 - 2*c**3*d**6 + x**2*(18*a*c**2*d**2*e**4 - 18*c**3*d*
*4*e**2) + x*(27*a**2*c*d*e**5 - 18*a*c**2*d**3*e**3 - 9*c**3*d**5*e))/(6*a**3*c**4*d**4*e**3 + 18*a**2*c**5*d
**5*e**2*x + 18*a*c**6*d**6*e*x**2 + 6*c**7*d**7*x**3) + e**3*log(a*e + c*d*x)/(c**4*d**4)

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac")

[Out]

Timed out