### 3.1887 $$\int \frac{(d+e x)^7}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx$$

Optimal. Leaf size=142 $\frac{e^3 x \left (4 c d^2-3 a e^2\right )}{c^4 d^4}-\frac{4 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^4}{2 c^5 d^5 (a e+c d x)^2}+\frac{6 e^2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^5 d^5}+\frac{e^4 x^2}{2 c^3 d^3}$

[Out]

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

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Rubi [A]  time = 0.135345, antiderivative size = 142, 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} $\frac{e^3 x \left (4 c d^2-3 a e^2\right )}{c^4 d^4}-\frac{4 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^4}{2 c^5 d^5 (a e+c d x)^2}+\frac{6 e^2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^5 d^5}+\frac{e^4 x^2}{2 c^3 d^3}$

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)^3,x]

[Out]

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

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

Mathematica [A]  time = 0.0702236, size = 191, normalized size = 1.35 $\frac{a^2 c^2 d^2 e^4 \left (18 d^2-16 d e x-11 e^2 x^2\right )+2 a^3 c d e^6 (e x-10 d)+7 a^4 e^8-4 a c^3 d^3 e^2 \left (-6 d^2 e x+d^3-4 d e^2 x^2+e^3 x^3\right )+12 e^2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2 \log (a e+c d x)+c^4 d^4 \left (-8 d^3 e x-d^4+8 d e^3 x^3+e^4 x^4\right )}{2 c^5 d^5 (a e+c d x)^2}$

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)^3,x]

[Out]

(7*a^4*e^8 + 2*a^3*c*d*e^6*(-10*d + e*x) + a^2*c^2*d^2*e^4*(18*d^2 - 16*d*e*x - 11*e^2*x^2) - 4*a*c^3*d^3*e^2*
(d^3 - 6*d^2*e*x - 4*d*e^2*x^2 + e^3*x^3) + c^4*d^4*(-d^4 - 8*d^3*e*x + 8*d*e^3*x^3 + e^4*x^4) + 12*e^2*(c*d^2
- a*e^2)^2*(a*e + c*d*x)^2*Log[a*e + c*d*x])/(2*c^5*d^5*(a*e + c*d*x)^2)

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Maple [B]  time = 0.05, size = 302, normalized size = 2.1 \begin{align*}{\frac{{e}^{4}{x}^{2}}{2\,{c}^{3}{d}^{3}}}-3\,{\frac{a{e}^{5}x}{{c}^{4}{d}^{4}}}+4\,{\frac{{e}^{3}x}{{c}^{3}{d}^{2}}}-{\frac{{a}^{4}{e}^{8}}{2\,{c}^{5}{d}^{5} \left ( cdx+ae \right ) ^{2}}}+2\,{\frac{{a}^{3}{e}^{6}}{{c}^{4}{d}^{3} \left ( cdx+ae \right ) ^{2}}}-3\,{\frac{{a}^{2}{e}^{4}}{{c}^{3}d \left ( cdx+ae \right ) ^{2}}}+2\,{\frac{ad{e}^{2}}{{c}^{2} \left ( cdx+ae \right ) ^{2}}}-{\frac{{d}^{3}}{2\,c \left ( cdx+ae \right ) ^{2}}}+4\,{\frac{{a}^{3}{e}^{7}}{{c}^{5}{d}^{5} \left ( cdx+ae \right ) }}-12\,{\frac{{a}^{2}{e}^{5}}{{c}^{4}{d}^{3} \left ( cdx+ae \right ) }}+12\,{\frac{a{e}^{3}}{{c}^{3}d \left ( cdx+ae \right ) }}-4\,{\frac{de}{{c}^{2} \left ( cdx+ae \right ) }}+6\,{\frac{{e}^{6}\ln \left ( cdx+ae \right ){a}^{2}}{{c}^{5}{d}^{5}}}-12\,{\frac{{e}^{4}\ln \left ( cdx+ae \right ) a}{{c}^{4}{d}^{3}}}+6\,{\frac{{e}^{2}\ln \left ( cdx+ae \right ) }{{c}^{3}d}} \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)^3,x)

[Out]

1/2*e^4*x^2/c^3/d^3-3*e^5/c^4/d^4*a*x+4*e^3/c^3/d^2*x-1/2/c^5/d^5/(c*d*x+a*e)^2*a^4*e^8+2/c^4/d^3/(c*d*x+a*e)^
2*a^3*e^6-3/c^3/d/(c*d*x+a*e)^2*a^2*e^4+2/c^2*d/(c*d*x+a*e)^2*a*e^2-1/2/c*d^3/(c*d*x+a*e)^2+4/d^5*e^7/c^5/(c*d
*x+a*e)*a^3-12/d^3*e^5/c^4/(c*d*x+a*e)*a^2+12/d*e^3/c^3/(c*d*x+a*e)*a-4*d*e/c^2/(c*d*x+a*e)+6/c^5/d^5*e^6*ln(c
*d*x+a*e)*a^2-12/c^4/d^3*e^4*ln(c*d*x+a*e)*a+6/c^3/d*e^2*ln(c*d*x+a*e)

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Maxima [A]  time = 1.07075, size = 302, normalized size = 2.13 \begin{align*} -\frac{c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} - 18 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 7 \, a^{4} e^{8} + 8 \,{\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x}{2 \,{\left (c^{7} d^{7} x^{2} + 2 \, a c^{6} d^{6} e x + a^{2} c^{5} d^{5} e^{2}\right )}} + \frac{c d e^{4} x^{2} + 2 \,{\left (4 \, c d^{2} e^{3} - 3 \, a e^{5}\right )} x}{2 \, c^{4} d^{4}} + \frac{6 \,{\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} \log \left (c d x + a e\right )}{c^{5} d^{5}} \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)^3,x, algorithm="maxima")

[Out]

-1/2*(c^4*d^8 + 4*a*c^3*d^6*e^2 - 18*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 7*a^4*e^8 + 8*(c^4*d^7*e - 3*a*c^3*d
^5*e^3 + 3*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)/(c^7*d^7*x^2 + 2*a*c^6*d^6*e*x + a^2*c^5*d^5*e^2) + 1/2*(c*d*e^4*
x^2 + 2*(4*c*d^2*e^3 - 3*a*e^5)*x)/(c^4*d^4) + 6*(c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*log(c*d*x + a*e)/(c^5
*d^5)

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Fricas [B]  time = 1.89035, size = 664, normalized size = 4.68 \begin{align*} \frac{c^{4} d^{4} e^{4} x^{4} - c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 18 \, a^{2} c^{2} d^{4} e^{4} - 20 \, a^{3} c d^{2} e^{6} + 7 \, a^{4} e^{8} + 4 \,{\left (2 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} +{\left (16 \, a c^{3} d^{4} e^{4} - 11 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} - 2 \,{\left (4 \, c^{4} d^{7} e - 12 \, a c^{3} d^{5} e^{3} + 8 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x + 12 \,{\left (a^{2} c^{2} d^{4} e^{4} - 2 \, a^{3} c d^{2} e^{6} + a^{4} e^{8} +{\left (c^{4} d^{6} e^{2} - 2 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \,{\left (a c^{3} d^{5} e^{3} - 2 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x\right )} \log \left (c d x + a e\right )}{2 \,{\left (c^{7} d^{7} x^{2} + 2 \, a c^{6} d^{6} e x + a^{2} c^{5} d^{5} e^{2}\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)^3,x, algorithm="fricas")

[Out]

1/2*(c^4*d^4*e^4*x^4 - c^4*d^8 - 4*a*c^3*d^6*e^2 + 18*a^2*c^2*d^4*e^4 - 20*a^3*c*d^2*e^6 + 7*a^4*e^8 + 4*(2*c^
4*d^5*e^3 - a*c^3*d^3*e^5)*x^3 + (16*a*c^3*d^4*e^4 - 11*a^2*c^2*d^2*e^6)*x^2 - 2*(4*c^4*d^7*e - 12*a*c^3*d^5*e
^3 + 8*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x + 12*(a^2*c^2*d^4*e^4 - 2*a^3*c*d^2*e^6 + a^4*e^8 + (c^4*d^6*e^2 - 2*a
*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 2*(a*c^3*d^5*e^3 - 2*a^2*c^2*d^3*e^5 + a^3*c*d*e^7)*x)*log(c*d*x + a*e))
/(c^7*d^7*x^2 + 2*a*c^6*d^6*e*x + a^2*c^5*d^5*e^2)

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Sympy [A]  time = 2.65236, size = 224, normalized size = 1.58 \begin{align*} \frac{7 a^{4} e^{8} - 20 a^{3} c d^{2} e^{6} + 18 a^{2} c^{2} d^{4} e^{4} - 4 a c^{3} d^{6} e^{2} - c^{4} d^{8} + x \left (8 a^{3} c d e^{7} - 24 a^{2} c^{2} d^{3} e^{5} + 24 a c^{3} d^{5} e^{3} - 8 c^{4} d^{7} e\right )}{2 a^{2} c^{5} d^{5} e^{2} + 4 a c^{6} d^{6} e x + 2 c^{7} d^{7} x^{2}} + \frac{e^{4} x^{2}}{2 c^{3} d^{3}} - \frac{x \left (3 a e^{5} - 4 c d^{2} e^{3}\right )}{c^{4} d^{4}} + \frac{6 e^{2} \left (a e^{2} - c d^{2}\right )^{2} \log{\left (a e + c d x \right )}}{c^{5} d^{5}} \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)**3,x)

[Out]

(7*a**4*e**8 - 20*a**3*c*d**2*e**6 + 18*a**2*c**2*d**4*e**4 - 4*a*c**3*d**6*e**2 - c**4*d**8 + x*(8*a**3*c*d*e
**7 - 24*a**2*c**2*d**3*e**5 + 24*a*c**3*d**5*e**3 - 8*c**4*d**7*e))/(2*a**2*c**5*d**5*e**2 + 4*a*c**6*d**6*e*
x + 2*c**7*d**7*x**2) + e**4*x**2/(2*c**3*d**3) - x*(3*a*e**5 - 4*c*d**2*e**3)/(c**4*d**4) + 6*e**2*(a*e**2 -
c*d**2)**2*log(a*e + c*d*x)/(c**5*d**5)

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Giac [B]  time = 14.7952, size = 1084, normalized size = 7.63 \begin{align*} \text{result too large to display} \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)^3,x, algorithm="giac")

[Out]

6*(c^7*d^14*e^2 - 7*a*c^6*d^12*e^4 + 21*a^2*c^5*d^10*e^6 - 35*a^3*c^4*d^8*e^8 + 35*a^4*c^3*d^6*e^10 - 21*a^5*c
^2*d^4*e^12 + 7*a^6*c*d^2*e^14 - a^7*e^16)*arctan((2*c*d*x*e + c*d^2 + a*e^2)/sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 -
a^2*e^4))/((c^9*d^13 - 4*a*c^8*d^11*e^2 + 6*a^2*c^7*d^9*e^4 - 4*a^3*c^6*d^7*e^6 + a^4*c^5*d^5*e^8)*sqrt(-c^2*d
^4 + 2*a*c*d^2*e^2 - a^2*e^4)) + 3*(c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*log(c*d*x^2*e + c*d^2*x + a*x*e^2 +
a*d*e)/(c^5*d^5) + 1/2*(c^3*d^3*x^2*e^10 + 8*c^3*d^4*x*e^9 - 6*a*c^2*d^2*x*e^11)*e^(-6)/(c^6*d^6) - 1/2*(c^8*
d^18 - 28*a^2*c^6*d^14*e^4 + 112*a^3*c^5*d^12*e^6 - 210*a^4*c^4*d^10*e^8 + 224*a^5*c^3*d^8*e^10 - 140*a^6*c^2*
d^6*e^12 + 48*a^7*c*d^4*e^14 - 7*a^8*d^2*e^16 + 8*(c^8*d^15*e^3 - 7*a*c^7*d^13*e^5 + 21*a^2*c^6*d^11*e^7 - 35*
a^3*c^5*d^9*e^9 + 35*a^4*c^4*d^7*e^11 - 21*a^5*c^3*d^5*e^13 + 7*a^6*c^2*d^3*e^15 - a^7*c*d*e^17)*x^3 + (17*c^8
*d^16*e^2 - 112*a*c^7*d^14*e^4 + 308*a^2*c^6*d^12*e^6 - 448*a^3*c^5*d^10*e^8 + 350*a^4*c^4*d^8*e^10 - 112*a^5*
c^3*d^6*e^12 - 28*a^6*c^2*d^4*e^14 + 32*a^7*c*d^2*e^16 - 7*a^8*e^18)*x^2 + 2*(5*c^8*d^17*e - 28*a*c^7*d^15*e^3
+ 56*a^2*c^6*d^13*e^5 - 28*a^3*c^5*d^11*e^7 - 70*a^4*c^4*d^9*e^9 + 140*a^5*c^3*d^7*e^11 - 112*a^6*c^2*d^5*e^1
3 + 44*a^7*c*d^3*e^15 - 7*a^8*d*e^17)*x)/((c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)^2*(c*d*x^2*e + c*d^2*x + a*x*e^2
+ a*d*e)^2*c^5*d^5)