3.1888 \(\int \frac{(d+e x)^6}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\)

Optimal. Leaf size=111 \[ -\frac{3 e \left (c d^2-a e^2\right )^2}{c^4 d^4 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^3}{2 c^4 d^4 (a e+c d x)^2}+\frac{3 e^2 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^4 d^4}+\frac{e^3 x}{c^3 d^3} \]

[Out]

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

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Rubi [A]  time = 0.0920715, antiderivative size = 111, 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} \[ -\frac{3 e \left (c d^2-a e^2\right )^2}{c^4 d^4 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^3}{2 c^4 d^4 (a e+c d x)^2}+\frac{3 e^2 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^4 d^4}+\frac{e^3 x}{c^3 d^3} \]

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)^3,x]

[Out]

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

Mathematica [A]  time = 0.0531893, size = 139, normalized size = 1.25 \[ \frac{a^2 c d e^4 (9 d-4 e x)-5 a^3 e^6+a c^2 d^2 e^2 \left (-3 d^2+12 d e x+4 e^2 x^2\right )-6 e^2 \left (a e^2-c d^2\right ) (a e+c d x)^2 \log (a e+c d x)-c^3 \left (-2 d^3 e^3 x^3+6 d^5 e x+d^6\right )}{2 c^4 d^4 (a e+c d x)^2} \]

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)^3,x]

[Out]

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

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Maple [A]  time = 0.048, size = 201, normalized size = 1.8 \begin{align*}{\frac{{e}^{3}x}{{c}^{3}{d}^{3}}}+{\frac{{a}^{3}{e}^{6}}{2\,{c}^{4}{d}^{4} \left ( cdx+ae \right ) ^{2}}}-{\frac{3\,{a}^{2}{e}^{4}}{2\,{c}^{3}{d}^{2} \left ( cdx+ae \right ) ^{2}}}+{\frac{3\,a{e}^{2}}{2\,{c}^{2} \left ( cdx+ae \right ) ^{2}}}-{\frac{{d}^{2}}{2\,c \left ( cdx+ae \right ) ^{2}}}-3\,{\frac{{a}^{2}{e}^{5}}{{c}^{4}{d}^{4} \left ( cdx+ae \right ) }}+6\,{\frac{a{e}^{3}}{{c}^{3}{d}^{2} \left ( cdx+ae \right ) }}-3\,{\frac{e}{{c}^{2} \left ( cdx+ae \right ) }}-3\,{\frac{{e}^{4}\ln \left ( cdx+ae \right ) a}{{c}^{4}{d}^{4}}}+3\,{\frac{{e}^{2}\ln \left ( cdx+ae \right ) }{{c}^{3}{d}^{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)^3,x)

[Out]

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

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Maxima [A]  time = 1.0967, size = 211, normalized size = 1.9 \begin{align*} -\frac{c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} - 9 \, a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6} + 6 \,{\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \,{\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}} + \frac{e^{3} x}{c^{3} d^{3}} + \frac{3 \,{\left (c d^{2} e^{2} - a e^{4}\right )} \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)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.85677, size = 443, normalized size = 3.99 \begin{align*} \frac{2 \, c^{3} d^{3} e^{3} x^{3} + 4 \, a c^{2} d^{2} e^{4} x^{2} - c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6} - 2 \,{\left (3 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + 2 \, a^{2} c d e^{5}\right )} x + 6 \,{\left (a^{2} c d^{2} e^{4} - a^{3} e^{6} +{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x\right )} \log \left (c d x + a e\right )}{2 \,{\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.49333, size = 163, normalized size = 1.47 \begin{align*} - \frac{5 a^{3} e^{6} - 9 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6} + x \left (6 a^{2} c d e^{5} - 12 a c^{2} d^{3} e^{3} + 6 c^{3} d^{5} e\right )}{2 a^{2} c^{4} d^{4} e^{2} + 4 a c^{5} d^{5} e x + 2 c^{6} d^{6} x^{2}} + \frac{e^{3} x}{c^{3} d^{3}} - \frac{3 e^{2} \left (a e^{2} - c d^{2}\right ) \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)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

-(5*a**3*e**6 - 9*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6 + x*(6*a**2*c*d*e**5 - 12*a*c**2*d**3*e**3
+ 6*c**3*d**5*e))/(2*a**2*c**4*d**4*e**2 + 4*a*c**5*d**5*e*x + 2*c**6*d**6*x**2) + e**3*x/(c**3*d**3) - 3*e**
2*(a*e**2 - c*d**2)*log(a*e + c*d*x)/(c**4*d**4)

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Giac [B]  time = 1.34645, size = 944, normalized size = 8.5 \begin{align*} \frac{3 \,{\left (c^{6} d^{12} e^{2} - 6 \, a c^{5} d^{10} e^{4} + 15 \, a^{2} c^{4} d^{8} e^{6} - 20 \, a^{3} c^{3} d^{6} e^{8} + 15 \, a^{4} c^{2} d^{4} e^{10} - 6 \, a^{5} c d^{2} e^{12} + a^{6} e^{14}\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^{8} d^{12} - 4 \, a c^{7} d^{10} e^{2} + 6 \, a^{2} c^{6} d^{8} e^{4} - 4 \, a^{3} c^{5} d^{6} e^{6} + a^{4} c^{4} d^{4} e^{8}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac{x e^{3}}{c^{3} d^{3}} + \frac{3 \,{\left (c d^{2} e^{2} - a e^{4}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{4} d^{4}} - \frac{c^{7} d^{16} - a c^{6} d^{14} e^{2} - 15 \, a^{2} c^{5} d^{12} e^{4} + 55 \, a^{3} c^{4} d^{10} e^{6} - 85 \, a^{4} c^{3} d^{8} e^{8} + 69 \, a^{5} c^{2} d^{6} e^{10} - 29 \, a^{6} c d^{4} e^{12} + 5 \, a^{7} d^{2} e^{14} + 6 \,{\left (c^{7} d^{13} e^{3} - 6 \, a c^{6} d^{11} e^{5} + 15 \, a^{2} c^{5} d^{9} e^{7} - 20 \, a^{3} c^{4} d^{7} e^{9} + 15 \, a^{4} c^{3} d^{5} e^{11} - 6 \, a^{5} c^{2} d^{3} e^{13} + a^{6} c d e^{15}\right )} x^{3} +{\left (13 \, c^{7} d^{14} e^{2} - 73 \, a c^{6} d^{12} e^{4} + 165 \, a^{2} c^{5} d^{10} e^{6} - 185 \, a^{3} c^{4} d^{8} e^{8} + 95 \, a^{4} c^{3} d^{6} e^{10} - 3 \, a^{5} c^{2} d^{4} e^{12} - 17 \, a^{6} c d^{2} e^{14} + 5 \, a^{7} e^{16}\right )} x^{2} + 2 \,{\left (4 \, c^{7} d^{15} e - 19 \, a c^{6} d^{13} e^{3} + 30 \, a^{2} c^{5} d^{11} e^{5} - 5 \, a^{3} c^{4} d^{9} e^{7} - 40 \, a^{4} c^{3} d^{7} e^{9} + 51 \, a^{5} c^{2} d^{5} e^{11} - 26 \, a^{6} c d^{3} e^{13} + 5 \, a^{7} d e^{15}\right )} x}{2 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}^{2}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{2} c^{4} d^{4}} \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)^3,x, algorithm="giac")

[Out]

3*(c^6*d^12*e^2 - 6*a*c^5*d^10*e^4 + 15*a^2*c^4*d^8*e^6 - 20*a^3*c^3*d^6*e^8 + 15*a^4*c^2*d^4*e^10 - 6*a^5*c*d
^2*e^12 + a^6*e^14)*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^8*d^12 -
4*a*c^7*d^10*e^2 + 6*a^2*c^6*d^8*e^4 - 4*a^3*c^5*d^6*e^6 + a^4*c^4*d^4*e^8)*sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^
2*e^4)) + x*e^3/(c^3*d^3) + 3/2*(c*d^2*e^2 - a*e^4)*log(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)/(c^4*d^4) - 1/2
*(c^7*d^16 - a*c^6*d^14*e^2 - 15*a^2*c^5*d^12*e^4 + 55*a^3*c^4*d^10*e^6 - 85*a^4*c^3*d^8*e^8 + 69*a^5*c^2*d^6*
e^10 - 29*a^6*c*d^4*e^12 + 5*a^7*d^2*e^14 + 6*(c^7*d^13*e^3 - 6*a*c^6*d^11*e^5 + 15*a^2*c^5*d^9*e^7 - 20*a^3*c
^4*d^7*e^9 + 15*a^4*c^3*d^5*e^11 - 6*a^5*c^2*d^3*e^13 + a^6*c*d*e^15)*x^3 + (13*c^7*d^14*e^2 - 73*a*c^6*d^12*e
^4 + 165*a^2*c^5*d^10*e^6 - 185*a^3*c^4*d^8*e^8 + 95*a^4*c^3*d^6*e^10 - 3*a^5*c^2*d^4*e^12 - 17*a^6*c*d^2*e^14
+ 5*a^7*e^16)*x^2 + 2*(4*c^7*d^15*e - 19*a*c^6*d^13*e^3 + 30*a^2*c^5*d^11*e^5 - 5*a^3*c^4*d^9*e^7 - 40*a^4*c^
3*d^7*e^9 + 51*a^5*c^2*d^5*e^11 - 26*a^6*c*d^3*e^13 + 5*a^7*d*e^15)*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^4*d^4)