3.16.86 \(\int \frac {(d+e x)^6}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\)

Optimal. Leaf size=35 \[ -\frac {(d+e x)^3}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3} \]

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {626, 37} \begin {gather*} -\frac {(d+e x)^3}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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]

Rubi steps

\begin {align*} \int \frac {(d+e x)^6}{\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)^2}{(a e+c d x)^4} \, dx\\ &=-\frac {(d+e x)^3}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 65, normalized size = 1.86 \begin {gather*} -\frac {a^2 e^4+a c d e^2 (d+3 e x)+c^2 d^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 c^3 d^3 (a e+c d x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

IntegrateAlgebraic[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4, x]

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fricas [B]  time = 0.40, size = 113, normalized size = 3.23 \begin {gather*} -\frac {3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \, {\left (c^{6} d^{6} x^{3} + 3 \, a c^{5} d^{5} e x^{2} + 3 \, a^{2} c^{4} d^{4} e^{2} x + a^{3} c^{3} d^{3} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 112.40, size = 822, normalized size = 23.49 \begin {gather*} -\frac {3 \, c^{8} d^{14} x^{5} e^{5} + 12 \, c^{8} d^{15} x^{4} e^{4} + 19 \, c^{8} d^{16} x^{3} e^{3} + 15 \, c^{8} d^{17} x^{2} e^{2} + 6 \, c^{8} d^{18} x e + c^{8} d^{19} - 18 \, a c^{7} d^{12} x^{5} e^{7} - 69 \, a c^{7} d^{13} x^{4} e^{6} - 104 \, a c^{7} d^{14} x^{3} e^{5} - 78 \, a c^{7} d^{15} x^{2} e^{4} - 30 \, a c^{7} d^{16} x e^{3} - 5 \, a c^{7} d^{17} e^{2} + 45 \, a^{2} c^{6} d^{10} x^{5} e^{9} + 162 \, a^{2} c^{6} d^{11} x^{4} e^{8} + 226 \, a^{2} c^{6} d^{12} x^{3} e^{7} + 156 \, a^{2} c^{6} d^{13} x^{2} e^{6} + 57 \, a^{2} c^{6} d^{14} x e^{5} + 10 \, a^{2} c^{6} d^{15} e^{4} - 60 \, a^{3} c^{5} d^{8} x^{5} e^{11} - 195 \, a^{3} c^{5} d^{9} x^{4} e^{10} - 236 \, a^{3} c^{5} d^{10} x^{3} e^{9} - 138 \, a^{3} c^{5} d^{11} x^{2} e^{8} - 48 \, a^{3} c^{5} d^{12} x e^{7} - 11 \, a^{3} c^{5} d^{13} e^{6} + 45 \, a^{4} c^{4} d^{6} x^{5} e^{13} + 120 \, a^{4} c^{4} d^{7} x^{4} e^{12} + 100 \, a^{4} c^{4} d^{8} x^{3} e^{11} + 30 \, a^{4} c^{4} d^{9} x^{2} e^{10} + 15 \, a^{4} c^{4} d^{10} x e^{9} + 10 \, a^{4} c^{4} d^{11} e^{8} - 18 \, a^{5} c^{3} d^{4} x^{5} e^{15} - 27 \, a^{5} c^{3} d^{5} x^{4} e^{14} + 16 \, a^{5} c^{3} d^{6} x^{3} e^{13} + 30 \, a^{5} c^{3} d^{7} x^{2} e^{12} - 6 \, a^{5} c^{3} d^{8} x e^{11} - 11 \, a^{5} c^{3} d^{9} e^{10} + 3 \, a^{6} c^{2} d^{2} x^{5} e^{17} - 6 \, a^{6} c^{2} d^{3} x^{4} e^{16} - 26 \, a^{6} c^{2} d^{4} x^{3} e^{15} - 12 \, a^{6} c^{2} d^{5} x^{2} e^{14} + 15 \, a^{6} c^{2} d^{6} x e^{13} + 10 \, a^{6} c^{2} d^{7} e^{12} + 3 \, a^{7} c d x^{4} e^{18} + 4 \, a^{7} c d^{2} x^{3} e^{17} - 6 \, a^{7} c d^{3} x^{2} e^{16} - 12 \, a^{7} c d^{4} x e^{15} - 5 \, a^{7} c d^{5} e^{14} + a^{8} x^{3} e^{19} + 3 \, a^{8} d x^{2} e^{18} + 3 \, a^{8} d^{2} x e^{17} + a^{8} d^{3} e^{16}}{3 \, {\left (c^{9} d^{15} - 6 \, a c^{8} d^{13} e^{2} + 15 \, a^{2} c^{7} d^{11} e^{4} - 20 \, a^{3} c^{6} d^{9} e^{6} + 15 \, a^{4} c^{5} d^{7} e^{8} - 6 \, a^{5} c^{4} d^{5} e^{10} + a^{6} c^{3} d^{3} e^{12}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac")

[Out]

-1/3*(3*c^8*d^14*x^5*e^5 + 12*c^8*d^15*x^4*e^4 + 19*c^8*d^16*x^3*e^3 + 15*c^8*d^17*x^2*e^2 + 6*c^8*d^18*x*e +
c^8*d^19 - 18*a*c^7*d^12*x^5*e^7 - 69*a*c^7*d^13*x^4*e^6 - 104*a*c^7*d^14*x^3*e^5 - 78*a*c^7*d^15*x^2*e^4 - 30
*a*c^7*d^16*x*e^3 - 5*a*c^7*d^17*e^2 + 45*a^2*c^6*d^10*x^5*e^9 + 162*a^2*c^6*d^11*x^4*e^8 + 226*a^2*c^6*d^12*x
^3*e^7 + 156*a^2*c^6*d^13*x^2*e^6 + 57*a^2*c^6*d^14*x*e^5 + 10*a^2*c^6*d^15*e^4 - 60*a^3*c^5*d^8*x^5*e^11 - 19
5*a^3*c^5*d^9*x^4*e^10 - 236*a^3*c^5*d^10*x^3*e^9 - 138*a^3*c^5*d^11*x^2*e^8 - 48*a^3*c^5*d^12*x*e^7 - 11*a^3*
c^5*d^13*e^6 + 45*a^4*c^4*d^6*x^5*e^13 + 120*a^4*c^4*d^7*x^4*e^12 + 100*a^4*c^4*d^8*x^3*e^11 + 30*a^4*c^4*d^9*
x^2*e^10 + 15*a^4*c^4*d^10*x*e^9 + 10*a^4*c^4*d^11*e^8 - 18*a^5*c^3*d^4*x^5*e^15 - 27*a^5*c^3*d^5*x^4*e^14 + 1
6*a^5*c^3*d^6*x^3*e^13 + 30*a^5*c^3*d^7*x^2*e^12 - 6*a^5*c^3*d^8*x*e^11 - 11*a^5*c^3*d^9*e^10 + 3*a^6*c^2*d^2*
x^5*e^17 - 6*a^6*c^2*d^3*x^4*e^16 - 26*a^6*c^2*d^4*x^3*e^15 - 12*a^6*c^2*d^5*x^2*e^14 + 15*a^6*c^2*d^6*x*e^13
+ 10*a^6*c^2*d^7*e^12 + 3*a^7*c*d*x^4*e^18 + 4*a^7*c*d^2*x^3*e^17 - 6*a^7*c*d^3*x^2*e^16 - 12*a^7*c*d^4*x*e^15
- 5*a^7*c*d^5*e^14 + a^8*x^3*e^19 + 3*a^8*d*x^2*e^18 + 3*a^8*d^2*x*e^17 + a^8*d^3*e^16)/((c^9*d^15 - 6*a*c^8*
d^13*e^2 + 15*a^2*c^7*d^11*e^4 - 20*a^3*c^6*d^9*e^6 + 15*a^4*c^5*d^7*e^8 - 6*a^5*c^4*d^5*e^10 + a^6*c^3*d^3*e^
12)*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^3)

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maple [B]  time = 0.05, size = 96, normalized size = 2.74 \begin {gather*} -\frac {e^{2}}{\left (c d x +a e \right ) c^{3} d^{3}}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) e}{\left (c d x +a e \right )^{2} c^{3} d^{3}}-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{3 \left (c d x +a e \right )^{3} c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x)

[Out]

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

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maxima [B]  time = 1.13, size = 113, normalized size = 3.23 \begin {gather*} -\frac {3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \, {\left (c^{6} d^{6} x^{3} + 3 \, a c^{5} d^{5} e x^{2} + 3 \, a^{2} c^{4} d^{4} e^{2} x + a^{3} c^{3} d^{3} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.06, size = 81, normalized size = 2.31 \begin {gather*} -\frac {\frac {d}{3\,c}+e\,\left (\frac {x}{c}-\frac {x^3}{3\,a}\right )+\frac {a\,e^2}{3\,c^2\,d}}{a^3\,e^3+3\,a^2\,c\,d\,e^2\,x+3\,a\,c^2\,d^2\,e\,x^2+c^3\,d^3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^6/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4,x)

[Out]

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

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sympy [B]  time = 0.86, size = 121, normalized size = 3.46 \begin {gather*} \frac {- a^{2} e^{4} - a c d^{2} e^{2} - c^{2} d^{4} - 3 c^{2} d^{2} e^{2} x^{2} + x \left (- 3 a c d e^{3} - 3 c^{2} d^{3} e\right )}{3 a^{3} c^{3} d^{3} e^{3} + 9 a^{2} c^{4} d^{4} e^{2} x + 9 a c^{5} d^{5} e x^{2} + 3 c^{6} d^{6} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

(-a**2*e**4 - a*c*d**2*e**2 - c**2*d**4 - 3*c**2*d**2*e**2*x**2 + x*(-3*a*c*d*e**3 - 3*c**2*d**3*e))/(3*a**3*c
**3*d**3*e**3 + 9*a**2*c**4*d**4*e**2*x + 9*a*c**5*d**5*e*x**2 + 3*c**6*d**6*x**3)

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