### 3.1867 $$\int \frac{(d+e x)^3}{a d e+(c d^2+a e^2) x+c d e x^2} \, dx$$

Optimal. Leaf size=69 $\frac{e x \left (c d^2-a e^2\right )}{c^2 d^2}+\frac{\left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^3 d^3}+\frac{(d+e x)^2}{2 c d}$

[Out]

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

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Rubi [A]  time = 0.0331473, antiderivative size = 69, 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} $\frac{e x \left (c d^2-a e^2\right )}{c^2 d^2}+\frac{\left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^3 d^3}+\frac{(d+e x)^2}{2 c d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.020929, size = 58, normalized size = 0.84 $\frac{2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)+c d e x \left (c d (4 d+e x)-2 a e^2\right )}{2 c^3 d^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.041, size = 93, normalized size = 1.4 \begin{align*}{\frac{{e}^{2}{x}^{2}}{2\,cd}}-{\frac{a{e}^{3}x}{{c}^{2}{d}^{2}}}+2\,{\frac{ex}{c}}+{\frac{\ln \left ( cdx+ae \right ){a}^{2}{e}^{4}}{{c}^{3}{d}^{3}}}-2\,{\frac{\ln \left ( cdx+ae \right ) a{e}^{2}}{{c}^{2}d}}+{\frac{d\ln \left ( cdx+ae \right ) }{c}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05332, size = 104, normalized size = 1.51 \begin{align*} \frac{c d e^{2} x^{2} + 2 \,{\left (2 \, c d^{2} e - a e^{3}\right )} x}{2 \, c^{2} d^{2}} + \frac{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (c d x + a e\right )}{c^{3} d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.57952, size = 165, normalized size = 2.39 \begin{align*} \frac{c^{2} d^{2} e^{2} x^{2} + 2 \,{\left (2 \, c^{2} d^{3} e - a c d e^{3}\right )} x + 2 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (c d x + a e\right )}{2 \, c^{3} d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.439244, size = 61, normalized size = 0.88 \begin{align*} \frac{e^{2} x^{2}}{2 c d} - \frac{x \left (a e^{3} - 2 c d^{2} e\right )}{c^{2} d^{2}} + \frac{\left (a e^{2} - c d^{2}\right )^{2} \log{\left (a e + c d x \right )}}{c^{3} d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.27223, size = 282, normalized size = 4.09 \begin{align*} \frac{{\left (c d x^{2} e^{4} + 4 \, c d^{2} x e^{3} - 2 \, a x e^{5}\right )} e^{\left (-2\right )}}{2 \, c^{2} d^{2}} + \frac{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, 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 )} \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^{3} d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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