3.1876 \(\int \frac{(d+e x)^6}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\)

Optimal. Leaf size=145 \[ \frac{e^2 x \left (3 a^2 e^4-8 a c d^2 e^2+6 c^2 d^4\right )}{c^4 d^4}+\frac{e^3 x^2 \left (2 c d^2-a e^2\right )}{c^3 d^3}-\frac{\left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)}+\frac{4 e \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^5 d^5}+\frac{e^4 x^3}{3 c^2 d^2} \]

[Out]

(e^2*(6*c^2*d^4 - 8*a*c*d^2*e^2 + 3*a^2*e^4)*x)/(c^4*d^4) + (e^3*(2*c*d^2 - a*e^2)*x^2)/(c^3*d^3) + (e^4*x^3)/
(3*c^2*d^2) - (c*d^2 - a*e^2)^4/(c^5*d^5*(a*e + c*d*x)) + (4*e*(c*d^2 - a*e^2)^3*Log[a*e + c*d*x])/(c^5*d^5)

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Rubi [A]  time = 0.151851, antiderivative size = 145, 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} \[ \frac{e^2 x \left (3 a^2 e^4-8 a c d^2 e^2+6 c^2 d^4\right )}{c^4 d^4}+\frac{e^3 x^2 \left (2 c d^2-a e^2\right )}{c^3 d^3}-\frac{\left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)}+\frac{4 e \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^5 d^5}+\frac{e^4 x^3}{3 c^2 d^2} \]

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)^2,x]

[Out]

(e^2*(6*c^2*d^4 - 8*a*c*d^2*e^2 + 3*a^2*e^4)*x)/(c^4*d^4) + (e^3*(2*c*d^2 - a*e^2)*x^2)/(c^3*d^3) + (e^4*x^3)/
(3*c^2*d^2) - (c*d^2 - a*e^2)^4/(c^5*d^5*(a*e + c*d*x)) + (4*e*(c*d^2 - a*e^2)^3*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)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac{(d+e x)^4}{(a e+c d x)^2} \, dx\\ &=\int \left (\frac{6 c^2 d^4 e^2-8 a c d^2 e^4+3 a^2 e^6}{c^4 d^4}+\frac{2 e^3 \left (2 c d^2-a e^2\right ) x}{c^3 d^3}+\frac{e^4 x^2}{c^2 d^2}+\frac{\left (c d^2-a e^2\right )^4}{c^4 d^4 (a e+c d x)^2}+\frac{4 e \left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)}\right ) \, dx\\ &=\frac{e^2 \left (6 c^2 d^4-8 a c d^2 e^2+3 a^2 e^4\right ) x}{c^4 d^4}+\frac{e^3 \left (2 c d^2-a e^2\right ) x^2}{c^3 d^3}+\frac{e^4 x^3}{3 c^2 d^2}-\frac{\left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)}+\frac{4 e \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^5 d^5}\\ \end{align*}

Mathematica [A]  time = 0.0756049, size = 196, normalized size = 1.35 \[ \frac{-6 a^2 c^2 d^2 e^4 \left (3 d^2+4 d e x-e^2 x^2\right )+3 a^3 c d e^6 (4 d+3 e x)-3 a^4 e^8+2 a c^3 d^3 e^2 \left (9 d^2 e x+6 d^3-9 d e^2 x^2-e^3 x^3\right )-12 e \left (a e^2-c d^2\right )^3 (a e+c d x) \log (a e+c d x)+c^4 d^4 \left (18 d^2 e^2 x^2-3 d^4+6 d e^3 x^3+e^4 x^4\right )}{3 c^5 d^5 (a e+c d x)} \]

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)^2,x]

[Out]

(-3*a^4*e^8 + 3*a^3*c*d*e^6*(4*d + 3*e*x) - 6*a^2*c^2*d^2*e^4*(3*d^2 + 4*d*e*x - e^2*x^2) + 2*a*c^3*d^3*e^2*(6
*d^3 + 9*d^2*e*x - 9*d*e^2*x^2 - e^3*x^3) + c^4*d^4*(-3*d^4 + 18*d^2*e^2*x^2 + 6*d*e^3*x^3 + e^4*x^4) - 12*e*(
-(c*d^2) + a*e^2)^3*(a*e + c*d*x)*Log[a*e + c*d*x])/(3*c^5*d^5*(a*e + c*d*x))

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Maple [A]  time = 0.048, size = 275, normalized size = 1.9 \begin{align*}{\frac{{e}^{4}{x}^{3}}{3\,{c}^{2}{d}^{2}}}-{\frac{{e}^{5}{x}^{2}a}{{c}^{3}{d}^{3}}}+2\,{\frac{{e}^{3}{x}^{2}}{{c}^{2}d}}+3\,{\frac{{a}^{2}{e}^{6}x}{{c}^{4}{d}^{4}}}-8\,{\frac{a{e}^{4}x}{{c}^{3}{d}^{2}}}+6\,{\frac{{e}^{2}x}{{c}^{2}}}-{\frac{{a}^{4}{e}^{8}}{{c}^{5}{d}^{5} \left ( cdx+ae \right ) }}+4\,{\frac{{a}^{3}{e}^{6}}{{c}^{4}{d}^{3} \left ( cdx+ae \right ) }}-6\,{\frac{{a}^{2}{e}^{4}}{{c}^{3}d \left ( cdx+ae \right ) }}+4\,{\frac{ad{e}^{2}}{{c}^{2} \left ( cdx+ae \right ) }}-{\frac{{d}^{3}}{c \left ( cdx+ae \right ) }}-4\,{\frac{{e}^{7}\ln \left ( cdx+ae \right ){a}^{3}}{{c}^{5}{d}^{5}}}+12\,{\frac{{e}^{5}\ln \left ( cdx+ae \right ){a}^{2}}{{c}^{4}{d}^{3}}}-12\,{\frac{{e}^{3}\ln \left ( cdx+ae \right ) a}{{c}^{3}d}}+4\,{\frac{de\ln \left ( cdx+ae \right ) }{{c}^{2}}} \end{align*}

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)^2,x)

[Out]

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

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Maxima [A]  time = 1.032, size = 289, normalized size = 1.99 \begin{align*} -\frac{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{c^{6} d^{6} x + a c^{5} d^{5} e} + \frac{c^{2} d^{2} e^{4} x^{3} + 3 \,{\left (2 \, c^{2} d^{3} e^{3} - a c d e^{5}\right )} x^{2} + 3 \,{\left (6 \, c^{2} d^{4} e^{2} - 8 \, a c d^{2} e^{4} + 3 \, a^{2} e^{6}\right )} x}{3 \, c^{4} d^{4}} + \frac{4 \,{\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\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)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.94051, size = 602, normalized size = 4.15 \begin{align*} \frac{c^{4} d^{4} e^{4} x^{4} - 3 \, c^{4} d^{8} + 12 \, a c^{3} d^{6} e^{2} - 18 \, a^{2} c^{2} d^{4} e^{4} + 12 \, a^{3} c d^{2} e^{6} - 3 \, a^{4} e^{8} + 2 \,{\left (3 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \,{\left (3 \, c^{4} d^{6} e^{2} - 3 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 3 \,{\left (6 \, a c^{3} d^{5} e^{3} - 8 \, a^{2} c^{2} d^{3} e^{5} + 3 \, a^{3} c d e^{7}\right )} x + 12 \,{\left (a c^{3} d^{6} e^{2} - 3 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{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\right )} \log \left (c d x + a e\right )}{3 \,{\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}} \end{align*}

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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.31122, size = 187, normalized size = 1.29 \begin{align*} - \frac{a^{4} e^{8} - 4 a^{3} c d^{2} e^{6} + 6 a^{2} c^{2} d^{4} e^{4} - 4 a c^{3} d^{6} e^{2} + c^{4} d^{8}}{a c^{5} d^{5} e + c^{6} d^{6} x} + \frac{e^{4} x^{3}}{3 c^{2} d^{2}} - \frac{x^{2} \left (a e^{5} - 2 c d^{2} e^{3}\right )}{c^{3} d^{3}} + \frac{x \left (3 a^{2} e^{6} - 8 a c d^{2} e^{4} + 6 c^{2} d^{4} e^{2}\right )}{c^{4} d^{4}} - \frac{4 e \left (a e^{2} - c d^{2}\right )^{3} \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)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

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

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Giac [B]  time = 1.47024, size = 757, normalized size = 5.22 \begin{align*} \frac{4 \,{\left (c^{6} d^{12} e - 6 \, a c^{5} d^{10} e^{3} + 15 \, a^{2} c^{4} d^{8} e^{5} - 20 \, a^{3} c^{3} d^{6} e^{7} + 15 \, a^{4} c^{2} d^{4} e^{9} - 6 \, a^{5} c d^{2} e^{11} + a^{6} e^{13}\right )} \arctan \left (\frac{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}}}\right )}{{\left (c^{7} d^{9} - 2 \, a c^{6} d^{7} e^{2} + a^{2} c^{5} d^{5} e^{4}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac{2 \,{\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{c^{5} d^{5}} + \frac{{\left (c^{4} d^{4} x^{3} e^{10} + 6 \, c^{4} d^{5} x^{2} e^{9} + 18 \, c^{4} d^{6} x e^{8} - 3 \, a c^{3} d^{3} x^{2} e^{11} - 24 \, a c^{3} d^{4} x e^{10} + 9 \, a^{2} c^{2} d^{2} x e^{12}\right )} e^{\left (-6\right )}}{3 \, c^{6} d^{6}} - \frac{c^{6} d^{13} - 6 \, a c^{5} d^{11} e^{2} + 15 \, a^{2} c^{4} d^{9} e^{4} - 20 \, a^{3} c^{3} d^{7} e^{6} + 15 \, a^{4} c^{2} d^{5} e^{8} - 6 \, a^{5} c d^{3} e^{10} + a^{6} d e^{12} +{\left (c^{6} d^{12} e - 6 \, a c^{5} d^{10} e^{3} + 15 \, a^{2} c^{4} d^{8} e^{5} - 20 \, a^{3} c^{3} d^{6} e^{7} + 15 \, a^{4} c^{2} d^{4} e^{9} - 6 \, a^{5} c d^{2} e^{11} + a^{6} e^{13}\right )} x}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )} c^{5} d^{5}} \end{align*}

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)^2,x, algorithm="giac")

[Out]

4*(c^6*d^12*e - 6*a*c^5*d^10*e^3 + 15*a^2*c^4*d^8*e^5 - 20*a^3*c^3*d^6*e^7 + 15*a^4*c^2*d^4*e^9 - 6*a^5*c*d^2*
e^11 + a^6*e^13)*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^7*d^9 - 2*a*
c^6*d^7*e^2 + a^2*c^5*d^5*e^4)*sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4)) + 2*(c^3*d^6*e - 3*a*c^2*d^4*e^3 + 3*
a^2*c*d^2*e^5 - a^3*e^7)*log(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)/(c^5*d^5) + 1/3*(c^4*d^4*x^3*e^10 + 6*c^4*
d^5*x^2*e^9 + 18*c^4*d^6*x*e^8 - 3*a*c^3*d^3*x^2*e^11 - 24*a*c^3*d^4*x*e^10 + 9*a^2*c^2*d^2*x*e^12)*e^(-6)/(c^
6*d^6) - (c^6*d^13 - 6*a*c^5*d^11*e^2 + 15*a^2*c^4*d^9*e^4 - 20*a^3*c^3*d^7*e^6 + 15*a^4*c^2*d^5*e^8 - 6*a^5*c
*d^3*e^10 + a^6*d*e^12 + (c^6*d^12*e - 6*a*c^5*d^10*e^3 + 15*a^2*c^4*d^8*e^5 - 20*a^3*c^3*d^6*e^7 + 15*a^4*c^2
*d^4*e^9 - 6*a^5*c*d^2*e^11 + a^6*e^13)*x)/((c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(c*d*x^2*e + c*d^2*x + a*x*e^2
+ a*d*e)*c^5*d^5)