3.1866 \(\int \frac{(d+e x)^4}{a d e+(c d^2+a e^2) x+c d e x^2} \, dx\)

Optimal. Leaf size=100 \[ \frac{e x \left (c d^2-a e^2\right )^2}{c^3 d^3}+\frac{(d+e x)^2 \left (c d^2-a e^2\right )}{2 c^2 d^2}+\frac{\left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^4 d^4}+\frac{(d+e x)^3}{3 c d} \]

[Out]

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

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Rubi [A]  time = 0.0452864, antiderivative size = 100, 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} \[ \frac{e x \left (c d^2-a e^2\right )^2}{c^3 d^3}+\frac{(d+e x)^2 \left (c d^2-a e^2\right )}{2 c^2 d^2}+\frac{\left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^4 d^4}+\frac{(d+e x)^3}{3 c d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0342907, size = 91, normalized size = 0.91 \[ \frac{c d e x \left (6 a^2 e^4-3 a c d e^2 (6 d+e x)+c^2 d^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+6 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{6 c^4 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 157, normalized size = 1.6 \begin{align*}{\frac{{e}^{3}{x}^{3}}{3\,cd}}-{\frac{{e}^{4}{x}^{2}a}{2\,{c}^{2}{d}^{2}}}+{\frac{3\,{e}^{2}{x}^{2}}{2\,c}}+{\frac{{a}^{2}{e}^{5}x}{{c}^{3}{d}^{3}}}-3\,{\frac{a{e}^{3}x}{{c}^{2}d}}+3\,{\frac{dex}{c}}-{\frac{\ln \left ( cdx+ae \right ){a}^{3}{e}^{6}}{{c}^{4}{d}^{4}}}+3\,{\frac{\ln \left ( cdx+ae \right ){a}^{2}{e}^{4}}{{c}^{3}{d}^{2}}}-3\,{\frac{\ln \left ( cdx+ae \right ) a{e}^{2}}{{c}^{2}}}+{\frac{{d}^{2}\ln \left ( cdx+ae \right ) }{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04188, size = 182, normalized size = 1.82 \begin{align*} \frac{2 \, c^{2} d^{2} e^{3} x^{3} + 3 \,{\left (3 \, c^{2} d^{3} e^{2} - a c d e^{4}\right )} x^{2} + 6 \,{\left (3 \, c^{2} d^{4} e - 3 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x}{6 \, c^{3} d^{3}} + \frac{{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\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)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.49823, size = 275, normalized size = 2.75 \begin{align*} \frac{2 \, c^{3} d^{3} e^{3} x^{3} + 3 \,{\left (3 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 6 \,{\left (3 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x + 6 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left (c d x + a e\right )}{6 \, c^{4} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.606648, size = 104, normalized size = 1.04 \begin{align*} \frac{e^{3} x^{3}}{3 c d} - \frac{x^{2} \left (a e^{4} - 3 c d^{2} e^{2}\right )}{2 c^{2} d^{2}} + \frac{x \left (a^{2} e^{5} - 3 a c d^{2} e^{3} + 3 c^{2} d^{4} e\right )}{c^{3} d^{3}} - \frac{\left (a e^{2} - c d^{2}\right )^{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)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

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

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Giac [B]  time = 1.21262, size = 374, normalized size = 3.74 \begin{align*} \frac{{\left (2 \, c^{2} d^{2} x^{3} e^{6} + 9 \, c^{2} d^{3} x^{2} e^{5} + 18 \, c^{2} d^{4} x e^{4} - 3 \, a c d x^{2} e^{7} - 18 \, a c d^{2} x e^{6} + 6 \, a^{2} x e^{8}\right )} e^{\left (-3\right )}}{6 \, c^{3} d^{3}} + \frac{{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\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{{\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 )} \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^{4} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*c^2*d^2*x^3*e^6 + 9*c^2*d^3*x^2*e^5 + 18*c^2*d^4*x*e^4 - 3*a*c*d*x^2*e^7 - 18*a*c*d^2*x*e^6 + 6*a^2*x*e
^8)*e^(-3)/(c^3*d^3) + 1/2*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*log(c*d*x^2*e + c*d^2*x + a
*x*e^2 + a*d*e)/(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)*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^4*d^4)