### 3.1901 $$\int \frac{(d+e x)^5}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx$$

Optimal. Leaf size=54 $-\frac{c d^2-a e^2}{3 c^2 d^2 (a e+c d x)^3}-\frac{e}{2 c^2 d^2 (a e+c d x)^2}$

[Out]

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

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Rubi [A]  time = 0.0390207, antiderivative size = 54, 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} $-\frac{c d^2-a e^2}{3 c^2 d^2 (a e+c d x)^3}-\frac{e}{2 c^2 d^2 (a e+c d x)^2}$

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)^4,x]

[Out]

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

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

Mathematica [A]  time = 0.0169026, size = 37, normalized size = 0.69 $-\frac{a e^2+c d (2 d+3 e x)}{6 c^2 d^2 (a e+c d x)^3}$

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)^4,x]

[Out]

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

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Maple [A]  time = 0.043, size = 51, normalized size = 0.9 \begin{align*} -{\frac{-a{e}^{2}+c{d}^{2}}{3\,{c}^{2}{d}^{2} \left ( cdx+ae \right ) ^{3}}}-{\frac{e}{2\,{c}^{2}{d}^{2} \left ( cdx+ae \right ) ^{2}}} \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)^4,x)

[Out]

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

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Maxima [A]  time = 1.0737, size = 100, normalized size = 1.85 \begin{align*} -\frac{3 \, c d e x + 2 \, c d^{2} + a e^{2}}{6 \,{\left (c^{5} d^{5} x^{3} + 3 \, a c^{4} d^{4} e x^{2} + 3 \, a^{2} c^{3} d^{3} e^{2} x + a^{3} c^{2} d^{2} e^{3}\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)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.98335, size = 149, normalized size = 2.76 \begin{align*} -\frac{3 \, c d e x + 2 \, c d^{2} + a e^{2}}{6 \,{\left (c^{5} d^{5} x^{3} + 3 \, a c^{4} d^{4} e x^{2} + 3 \, a^{2} c^{3} d^{3} e^{2} x + a^{3} c^{2} d^{2} e^{3}\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)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.851355, size = 80, normalized size = 1.48 \begin{align*} - \frac{a e^{2} + 2 c d^{2} + 3 c d e x}{6 a^{3} c^{2} d^{2} e^{3} + 18 a^{2} c^{3} d^{3} e^{2} x + 18 a c^{4} d^{4} e x^{2} + 6 c^{5} d^{5} x^{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)**4,x)

[Out]

-(a*e**2 + 2*c*d**2 + 3*c*d*e*x)/(6*a**3*c**2*d**2*e**3 + 18*a**2*c**3*d**3*e**2*x + 18*a*c**4*d**4*e*x**2 + 6
*c**5*d**5*x**3)

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Giac [B]  time = 1.5247, size = 865, normalized size = 16.02 \begin{align*} -\frac{3 \, c^{7} d^{13} x^{4} e^{4} + 11 \, c^{7} d^{14} x^{3} e^{3} + 15 \, c^{7} d^{15} x^{2} e^{2} + 9 \, c^{7} d^{16} x e + 2 \, c^{7} d^{17} - 18 \, a c^{6} d^{11} x^{4} e^{6} - 65 \, a c^{6} d^{12} x^{3} e^{5} - 87 \, a c^{6} d^{13} x^{2} e^{4} - 51 \, a c^{6} d^{14} x e^{3} - 11 \, a c^{6} d^{15} e^{2} + 45 \, a^{2} c^{5} d^{9} x^{4} e^{8} + 159 \, a^{2} c^{5} d^{10} x^{3} e^{7} + 207 \, a^{2} c^{5} d^{11} x^{2} e^{6} + 117 \, a^{2} c^{5} d^{12} x e^{5} + 24 \, a^{2} c^{5} d^{13} e^{4} - 60 \, a^{3} c^{4} d^{7} x^{4} e^{10} - 205 \, a^{3} c^{4} d^{8} x^{3} e^{9} - 255 \, a^{3} c^{4} d^{9} x^{2} e^{8} - 135 \, a^{3} c^{4} d^{10} x e^{7} - 25 \, a^{3} c^{4} d^{11} e^{6} + 45 \, a^{4} c^{3} d^{5} x^{4} e^{12} + 145 \, a^{4} c^{3} d^{6} x^{3} e^{11} + 165 \, a^{4} c^{3} d^{7} x^{2} e^{10} + 75 \, a^{4} c^{3} d^{8} x e^{9} + 10 \, a^{4} c^{3} d^{9} e^{8} - 18 \, a^{5} c^{2} d^{3} x^{4} e^{14} - 51 \, a^{5} c^{2} d^{4} x^{3} e^{13} - 45 \, a^{5} c^{2} d^{5} x^{2} e^{12} - 9 \, a^{5} c^{2} d^{6} x e^{11} + 3 \, a^{5} c^{2} d^{7} e^{10} + 3 \, a^{6} c d x^{4} e^{16} + 5 \, a^{6} c d^{2} x^{3} e^{15} - 3 \, a^{6} c d^{3} x^{2} e^{14} - 9 \, a^{6} c d^{4} x e^{13} - 4 \, a^{6} c d^{5} e^{12} + a^{7} x^{3} e^{17} + 3 \, a^{7} d x^{2} e^{16} + 3 \, a^{7} d^{2} x e^{15} + a^{7} d^{3} e^{14}}{6 \,{\left (c^{8} d^{14} - 6 \, a c^{7} d^{12} e^{2} + 15 \, a^{2} c^{6} d^{10} e^{4} - 20 \, a^{3} c^{5} d^{8} e^{6} + 15 \, a^{4} c^{4} d^{6} e^{8} - 6 \, a^{5} c^{3} d^{4} e^{10} + a^{6} c^{2} d^{2} e^{12}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{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)^4,x, algorithm="giac")

[Out]

-1/6*(3*c^7*d^13*x^4*e^4 + 11*c^7*d^14*x^3*e^3 + 15*c^7*d^15*x^2*e^2 + 9*c^7*d^16*x*e + 2*c^7*d^17 - 18*a*c^6*
d^11*x^4*e^6 - 65*a*c^6*d^12*x^3*e^5 - 87*a*c^6*d^13*x^2*e^4 - 51*a*c^6*d^14*x*e^3 - 11*a*c^6*d^15*e^2 + 45*a^
2*c^5*d^9*x^4*e^8 + 159*a^2*c^5*d^10*x^3*e^7 + 207*a^2*c^5*d^11*x^2*e^6 + 117*a^2*c^5*d^12*x*e^5 + 24*a^2*c^5*
d^13*e^4 - 60*a^3*c^4*d^7*x^4*e^10 - 205*a^3*c^4*d^8*x^3*e^9 - 255*a^3*c^4*d^9*x^2*e^8 - 135*a^3*c^4*d^10*x*e^
7 - 25*a^3*c^4*d^11*e^6 + 45*a^4*c^3*d^5*x^4*e^12 + 145*a^4*c^3*d^6*x^3*e^11 + 165*a^4*c^3*d^7*x^2*e^10 + 75*a
^4*c^3*d^8*x*e^9 + 10*a^4*c^3*d^9*e^8 - 18*a^5*c^2*d^3*x^4*e^14 - 51*a^5*c^2*d^4*x^3*e^13 - 45*a^5*c^2*d^5*x^2
*e^12 - 9*a^5*c^2*d^6*x*e^11 + 3*a^5*c^2*d^7*e^10 + 3*a^6*c*d*x^4*e^16 + 5*a^6*c*d^2*x^3*e^15 - 3*a^6*c*d^3*x^
2*e^14 - 9*a^6*c*d^4*x*e^13 - 4*a^6*c*d^5*e^12 + a^7*x^3*e^17 + 3*a^7*d*x^2*e^16 + 3*a^7*d^2*x*e^15 + a^7*d^3*
e^14)/((c^8*d^14 - 6*a*c^7*d^12*e^2 + 15*a^2*c^6*d^10*e^4 - 20*a^3*c^5*d^8*e^6 + 15*a^4*c^4*d^6*e^8 - 6*a^5*c^
3*d^4*e^10 + a^6*c^2*d^2*e^12)*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^3)