### 3.1889 $$\int \frac{(d+e x)^5}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx$$

Optimal. Leaf size=85 $-\frac{2 e \left (c d^2-a e^2\right )}{c^3 d^3 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^2}{2 c^3 d^3 (a e+c d x)^2}+\frac{e^2 \log (a e+c d x)}{c^3 d^3}$

[Out]

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

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Rubi [A]  time = 0.0614312, antiderivative size = 85, 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} $-\frac{2 e \left (c d^2-a e^2\right )}{c^3 d^3 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^2}{2 c^3 d^3 (a e+c d x)^2}+\frac{e^2 \log (a e+c d x)}{c^3 d^3}$

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)^3,x]

[Out]

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

Mathematica [A]  time = 0.0336488, size = 65, normalized size = 0.76 $\frac{2 e^2 \log (a e+c d x)-\frac{\left (c d^2-a e^2\right ) \left (3 a e^2+c d (d+4 e x)\right )}{(a e+c d x)^2}}{2 c^3 d^3}$

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)^3,x]

[Out]

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

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Maple [A]  time = 0.046, size = 123, normalized size = 1.5 \begin{align*} -{\frac{{a}^{2}{e}^{4}}{2\,{c}^{3}{d}^{3} \left ( cdx+ae \right ) ^{2}}}+{\frac{a{e}^{2}}{{c}^{2}d \left ( cdx+ae \right ) ^{2}}}-{\frac{d}{2\,c \left ( cdx+ae \right ) ^{2}}}+2\,{\frac{a{e}^{3}}{{c}^{3}{d}^{3} \left ( cdx+ae \right ) }}-2\,{\frac{e}{{c}^{2}d \left ( cdx+ae \right ) }}+{\frac{{e}^{2}\ln \left ( cdx+ae \right ) }{{c}^{3}{d}^{3}}} \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)^3,x)

[Out]

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

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Maxima [A]  time = 1.07021, size = 142, normalized size = 1.67 \begin{align*} -\frac{c^{2} d^{4} + 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 4 \,{\left (c^{2} d^{3} e - a c d e^{3}\right )} x}{2 \,{\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} e^{2}\right )}} + \frac{e^{2} \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)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.98946, size = 255, normalized size = 3. \begin{align*} -\frac{c^{2} d^{4} + 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 4 \,{\left (c^{2} d^{3} e - a c d e^{3}\right )} x - 2 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \log \left (c d x + a e\right )}{2 \,{\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} e^{2}\right )}} \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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.852945, size = 109, normalized size = 1.28 \begin{align*} \frac{3 a^{2} e^{4} - 2 a c d^{2} e^{2} - c^{2} d^{4} + x \left (4 a c d e^{3} - 4 c^{2} d^{3} e\right )}{2 a^{2} c^{3} d^{3} e^{2} + 4 a c^{4} d^{4} e x + 2 c^{5} d^{5} x^{2}} + \frac{e^{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)**5/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

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

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Giac [B]  time = 1.33168, size = 811, normalized size = 9.54 \begin{align*} \frac{{\left (c^{5} d^{10} e^{2} - 5 \, a c^{4} d^{8} e^{4} + 10 \, a^{2} c^{3} d^{6} e^{6} - 10 \, a^{3} c^{2} d^{4} e^{8} + 5 \, a^{4} c d^{2} e^{10} - a^{5} e^{12}\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^{7} d^{11} - 4 \, a c^{6} d^{9} e^{2} + 6 \, a^{2} c^{5} d^{7} e^{4} - 4 \, a^{3} c^{4} d^{5} e^{6} + a^{4} c^{3} d^{3} e^{8}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac{e^{2} \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{c^{6} d^{14} - 2 \, a c^{5} d^{12} e^{2} - 5 \, a^{2} c^{4} d^{10} e^{4} + 20 \, a^{3} c^{3} d^{8} e^{6} - 25 \, a^{4} c^{2} d^{6} e^{8} + 14 \, a^{5} c d^{4} e^{10} - 3 \, a^{6} d^{2} e^{12} + 4 \,{\left (c^{6} d^{11} e^{3} - 5 \, a c^{5} d^{9} e^{5} + 10 \, a^{2} c^{4} d^{7} e^{7} - 10 \, a^{3} c^{3} d^{5} e^{9} + 5 \, a^{4} c^{2} d^{3} e^{11} - a^{5} c d e^{13}\right )} x^{3} + 3 \,{\left (3 \, c^{6} d^{12} e^{2} - 14 \, a c^{5} d^{10} e^{4} + 25 \, a^{2} c^{4} d^{8} e^{6} - 20 \, a^{3} c^{3} d^{6} e^{8} + 5 \, a^{4} c^{2} d^{4} e^{10} + 2 \, a^{5} c d^{2} e^{12} - a^{6} e^{14}\right )} x^{2} + 6 \,{\left (c^{6} d^{13} e - 4 \, a c^{5} d^{11} e^{3} + 5 \, a^{2} c^{4} d^{9} e^{5} - 5 \, a^{4} c^{2} d^{5} e^{9} + 4 \, a^{5} c d^{3} e^{11} - a^{6} d e^{13}\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^{3} d^{3}} \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)^3,x, algorithm="giac")

[Out]

(c^5*d^10*e^2 - 5*a*c^4*d^8*e^4 + 10*a^2*c^3*d^6*e^6 - 10*a^3*c^2*d^4*e^8 + 5*a^4*c*d^2*e^10 - a^5*e^12)*arcta
n((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^7*d^11 - 4*a*c^6*d^9*e^2 + 6*a^2*c
^5*d^7*e^4 - 4*a^3*c^4*d^5*e^6 + a^4*c^3*d^3*e^8)*sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4)) + 1/2*e^2*log(c*d*
x^2*e + c*d^2*x + a*x*e^2 + a*d*e)/(c^3*d^3) - 1/2*(c^6*d^14 - 2*a*c^5*d^12*e^2 - 5*a^2*c^4*d^10*e^4 + 20*a^3*
c^3*d^8*e^6 - 25*a^4*c^2*d^6*e^8 + 14*a^5*c*d^4*e^10 - 3*a^6*d^2*e^12 + 4*(c^6*d^11*e^3 - 5*a*c^5*d^9*e^5 + 10
*a^2*c^4*d^7*e^7 - 10*a^3*c^3*d^5*e^9 + 5*a^4*c^2*d^3*e^11 - a^5*c*d*e^13)*x^3 + 3*(3*c^6*d^12*e^2 - 14*a*c^5*
d^10*e^4 + 25*a^2*c^4*d^8*e^6 - 20*a^3*c^3*d^6*e^8 + 5*a^4*c^2*d^4*e^10 + 2*a^5*c*d^2*e^12 - a^6*e^14)*x^2 + 6
*(c^6*d^13*e - 4*a*c^5*d^11*e^3 + 5*a^2*c^4*d^9*e^5 - 5*a^4*c^2*d^5*e^9 + 4*a^5*c*d^3*e^11 - a^6*d*e^13)*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^3*d^3)