3.1865 \(\int \frac{(d+e x)^5}{a d e+(c d^2+a e^2) x+c d e x^2} \, dx\)

Optimal. Leaf size=131 \[ \frac{e x \left (c d^2-a e^2\right )^3}{c^4 d^4}+\frac{(d+e x)^2 \left (c d^2-a e^2\right )^2}{2 c^3 d^3}+\frac{(d+e x)^3 \left (c d^2-a e^2\right )}{3 c^2 d^2}+\frac{\left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^5 d^5}+\frac{(d+e x)^4}{4 c d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0674149, antiderivative size = 131, 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} \[ \frac{e x \left (c d^2-a e^2\right )^3}{c^4 d^4}+\frac{(d+e x)^2 \left (c d^2-a e^2\right )^2}{2 c^3 d^3}+\frac{(d+e x)^3 \left (c d^2-a e^2\right )}{3 c^2 d^2}+\frac{\left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^5 d^5}+\frac{(d+e x)^4}{4 c d} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0529092, size = 134, normalized size = 1.02 \[ \frac{c d e x \left (6 a^2 c d e^4 (8 d+e x)-12 a^3 e^6-4 a c^2 d^2 e^2 \left (18 d^2+6 d e x+e^2 x^2\right )+c^3 d^3 \left (36 d^2 e x+48 d^3+16 d e^2 x^2+3 e^3 x^3\right )\right )+12 \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{12 c^5 d^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

(c*d*e*x*(-12*a^3*e^6 + 6*a^2*c*d*e^4*(8*d + e*x) - 4*a*c^2*d^2*e^2*(18*d^2 + 6*d*e*x + e^2*x^2) + c^3*d^3*(48
*d^3 + 36*d^2*e*x + 16*d*e^2*x^2 + 3*e^3*x^3)) + 12*(c*d^2 - a*e^2)^4*Log[a*e + c*d*x])/(12*c^5*d^5)

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 239, normalized size = 1.8 \begin{align*}{\frac{{e}^{4}{x}^{4}}{4\,cd}}-{\frac{{e}^{5}{x}^{3}a}{3\,{c}^{2}{d}^{2}}}+{\frac{4\,{e}^{3}{x}^{3}}{3\,c}}+{\frac{{e}^{6}{x}^{2}{a}^{2}}{2\,{c}^{3}{d}^{3}}}-2\,{\frac{{e}^{4}{x}^{2}a}{{c}^{2}d}}+3\,{\frac{d{e}^{2}{x}^{2}}{c}}-{\frac{{e}^{7}{a}^{3}x}{{c}^{4}{d}^{4}}}+4\,{\frac{{a}^{2}{e}^{5}x}{{c}^{3}{d}^{2}}}-6\,{\frac{a{e}^{3}x}{{c}^{2}}}+4\,{\frac{e{d}^{2}x}{c}}+{\frac{\ln \left ( cdx+ae \right ){a}^{4}{e}^{8}}{{c}^{5}{d}^{5}}}-4\,{\frac{\ln \left ( cdx+ae \right ){a}^{3}{e}^{6}}{{c}^{4}{d}^{3}}}+6\,{\frac{\ln \left ( cdx+ae \right ){a}^{2}{e}^{4}}{{c}^{3}d}}-4\,{\frac{d\ln \left ( cdx+ae \right ) a{e}^{2}}{{c}^{2}}}+{\frac{{d}^{3}\ln \left ( cdx+ae \right ) }{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.0299, size = 277, normalized size = 2.11 \begin{align*} \frac{3 \, c^{3} d^{3} e^{4} x^{4} + 4 \,{\left (4 \, c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{3} + 6 \,{\left (6 \, c^{3} d^{5} e^{2} - 4 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x^{2} + 12 \,{\left (4 \, c^{3} d^{6} e - 6 \, a c^{2} d^{4} e^{3} + 4 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} x}{12 \, c^{4} d^{4}} + \frac{{\left (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}\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)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.6241, size = 414, normalized size = 3.16 \begin{align*} \frac{3 \, c^{4} d^{4} e^{4} x^{4} + 4 \,{\left (4 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \,{\left (6 \, c^{4} d^{6} e^{2} - 4 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 12 \,{\left (4 \, c^{4} d^{7} e - 6 \, a c^{3} d^{5} e^{3} + 4 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x + 12 \,{\left (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}\right )} \log \left (c d x + a e\right )}{12 \, c^{5} d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.671662, size = 162, normalized size = 1.24 \begin{align*} \frac{e^{4} x^{4}}{4 c d} - \frac{x^{3} \left (a e^{5} - 4 c d^{2} e^{3}\right )}{3 c^{2} d^{2}} + \frac{x^{2} \left (a^{2} e^{6} - 4 a c d^{2} e^{4} + 6 c^{2} d^{4} e^{2}\right )}{2 c^{3} d^{3}} - \frac{x \left (a^{3} e^{7} - 4 a^{2} c d^{2} e^{5} + 6 a c^{2} d^{4} e^{3} - 4 c^{3} d^{6} e\right )}{c^{4} d^{4}} + \frac{\left (a e^{2} - c d^{2}\right )^{4} \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)**5/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.27892, size = 486, normalized size = 3.71 \begin{align*} \frac{{\left (3 \, c^{3} d^{3} x^{4} e^{8} + 16 \, c^{3} d^{4} x^{3} e^{7} + 36 \, c^{3} d^{5} x^{2} e^{6} + 48 \, c^{3} d^{6} x e^{5} - 4 \, a c^{2} d^{2} x^{3} e^{9} - 24 \, a c^{2} d^{3} x^{2} e^{8} - 72 \, a c^{2} d^{4} x e^{7} + 6 \, a^{2} c d x^{2} e^{10} + 48 \, a^{2} c d^{2} x e^{9} - 12 \, a^{3} x e^{11}\right )} e^{\left (-4\right )}}{12 \, c^{4} d^{4}} + \frac{{\left (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}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{5} d^{5}} + \frac{{\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\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 )}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}} c^{5} d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac")

[Out]

1/12*(3*c^3*d^3*x^4*e^8 + 16*c^3*d^4*x^3*e^7 + 36*c^3*d^5*x^2*e^6 + 48*c^3*d^6*x*e^5 - 4*a*c^2*d^2*x^3*e^9 - 2
4*a*c^2*d^3*x^2*e^8 - 72*a*c^2*d^4*x*e^7 + 6*a^2*c*d*x^2*e^10 + 48*a^2*c*d^2*x*e^9 - 12*a^3*x*e^11)*e^(-4)/(c^
4*d^4) + 1/2*(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)*log(c*d*x^2*e + c*d^2
*x + a*x*e^2 + a*d*e)/(c^5*d^5) + (c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^
4*c*d^2*e^8 - a^5*e^10)*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))/(sqrt(-c^
2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4)*c^5*d^5)