### 3.1857 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^4} \, dx$$

Optimal. Leaf size=89 $\frac{c d x \left (c d^2-a e^2\right )^2}{e^3}+\frac{1}{2} \left (a-\frac{c d^2}{e^2}\right ) (a e+c d x)^2-\frac{\left (c d^2-a e^2\right )^3 \log (d+e x)}{e^4}+\frac{(a e+c d x)^3}{3 e}$

[Out]

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

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Rubi [A]  time = 0.0486573, antiderivative size = 89, 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} $\frac{c d x \left (c d^2-a e^2\right )^2}{e^3}+\frac{1}{2} \left (a-\frac{c d^2}{e^2}\right ) (a e+c d x)^2-\frac{\left (c d^2-a e^2\right )^3 \log (d+e x)}{e^4}+\frac{(a e+c d x)^3}{3 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(c*d*(c*d^2 - a*e^2)^2*x)/e^3 + ((a - (c*d^2)/e^2)*(a*e + c*d*x)^2)/2 + (a*e + c*d*x)^3/(3*e) - ((c*d^2 - a*e^
2)^3*Log[d + e*x])/e^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{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^4} \, dx &=\int \frac{(a e+c d x)^3}{d+e x} \, dx\\ &=\int \left (\frac{c d \left (c d^2-a e^2\right )^2}{e^3}-\frac{c d \left (c d^2-a e^2\right ) (a e+c d x)}{e^2}+\frac{c d (a e+c d x)^2}{e}+\frac{\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{c d \left (c d^2-a e^2\right )^2 x}{e^3}+\frac{1}{2} \left (a-\frac{c d^2}{e^2}\right ) (a e+c d x)^2+\frac{(a e+c d x)^3}{3 e}-\frac{\left (c d^2-a e^2\right )^3 \log (d+e x)}{e^4}\\ \end{align*}

Mathematica [A]  time = 0.0358542, size = 85, normalized size = 0.96 $\frac{c d e x \left (18 a^2 e^4+9 a c d e^2 (e x-2 d)+c^2 d^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 \left (c d^2-a e^2\right )^3 \log (d+e x)}{6 e^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 138, normalized size = 1.6 \begin{align*}{\frac{{x}^{3}{c}^{3}{d}^{3}}{3\,e}}+{\frac{3\,{c}^{2}{d}^{2}{x}^{2}a}{2}}-{\frac{{c}^{3}{d}^{4}{x}^{2}}{2\,{e}^{2}}}+3\,cde{a}^{2}x-3\,{\frac{{c}^{2}{d}^{3}ax}{e}}+{\frac{{c}^{3}{d}^{5}x}{{e}^{3}}}+{e}^{2}\ln \left ( ex+d \right ){a}^{3}-3\,\ln \left ( ex+d \right ){a}^{2}c{d}^{2}+3\,{\frac{\ln \left ( ex+d \right ) a{c}^{2}{d}^{4}}{{e}^{2}}}-{\frac{\ln \left ( ex+d \right ){c}^{3}{d}^{6}}{{e}^{4}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^4,x)

[Out]

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

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Maxima [A]  time = 1.12567, size = 177, normalized size = 1.99 \begin{align*} \frac{2 \, c^{3} d^{3} e^{2} x^{3} - 3 \,{\left (c^{3} d^{4} e - 3 \, a c^{2} d^{2} e^{3}\right )} x^{2} + 6 \,{\left (c^{3} d^{5} - 3 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x}{6 \, e^{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 (e x + d\right )}{e^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.67437, size = 262, normalized size = 2.94 \begin{align*} \frac{2 \, c^{3} d^{3} e^{3} x^{3} - 3 \,{\left (c^{3} d^{4} e^{2} - 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 6 \,{\left (c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3} + 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 (e x + d\right )}{6 \, e^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.651007, size = 100, normalized size = 1.12 \begin{align*} \frac{c^{3} d^{3} x^{3}}{3 e} + \frac{x^{2} \left (3 a c^{2} d^{2} e^{2} - c^{3} d^{4}\right )}{2 e^{2}} + \frac{x \left (3 a^{2} c d e^{4} - 3 a c^{2} d^{3} e^{2} + c^{3} d^{5}\right )}{e^{3}} + \frac{\left (a e^{2} - c d^{2}\right )^{3} \log{\left (d + e x \right )}}{e^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**4,x)

[Out]

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

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Giac [A]  time = 1.34463, size = 173, normalized size = 1.94 \begin{align*} -{\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 )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, c^{3} d^{3} x^{3} e^{11} - 3 \, c^{3} d^{4} x^{2} e^{10} + 6 \, c^{3} d^{5} x e^{9} + 9 \, a c^{2} d^{2} x^{2} e^{12} - 18 \, a c^{2} d^{3} x e^{11} + 18 \, a^{2} c d x e^{13}\right )} e^{\left (-12\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^4,x, algorithm="giac")

[Out]

-(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*e^(-4)*log(abs(x*e + d)) + 1/6*(2*c^3*d^3*x^3*e^11 -
3*c^3*d^4*x^2*e^10 + 6*c^3*d^5*x*e^9 + 9*a*c^2*d^2*x^2*e^12 - 18*a*c^2*d^3*x*e^11 + 18*a^2*c*d*x*e^13)*e^(-12)