3.1891 \(\int \frac {(d+e x)^3}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\)

Optimal. Leaf size=20 \[ -\frac {1}{2 c d (a e+c d x)^2} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 20, 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, 32} \[ -\frac {1}{2 c d (a e+c d x)^2} \]

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)^3,x]

[Out]

-1/(2*c*d*(a*e + c*d*x)^2)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[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)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac {1}{(a e+c d x)^3} \, dx\\ &=-\frac {1}{2 c d (a e+c d x)^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 20, normalized size = 1.00 \[ -\frac {1}{2 c d (a e+c d x)^2} \]

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)^3,x]

[Out]

-1/2*1/(c*d*(a*e + c*d*x)^2)

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fricas [A]  time = 1.26, size = 35, normalized size = 1.75 \[ -\frac {1}{2 \, {\left (c^{3} d^{3} x^{2} + 2 \, a c^{2} d^{2} e x + a^{2} c d e^{2}\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.23, size = 259, normalized size = 12.95 \[ -\frac {c^{4} d^{8} x^{2} e^{2} + 2 \, c^{4} d^{9} x e + c^{4} d^{10} - 4 \, a c^{3} d^{6} x^{2} e^{4} - 8 \, a c^{3} d^{7} x e^{3} - 4 \, a c^{3} d^{8} e^{2} + 6 \, a^{2} c^{2} d^{4} x^{2} e^{6} + 12 \, a^{2} c^{2} d^{5} x e^{5} + 6 \, a^{2} c^{2} d^{6} e^{4} - 4 \, a^{3} c d^{2} x^{2} e^{8} - 8 \, a^{3} c d^{3} x e^{7} - 4 \, a^{3} c d^{4} e^{6} + a^{4} x^{2} e^{10} + 2 \, a^{4} d x e^{9} + a^{4} d^{2} e^{8}}{2 \, {\left (c^{5} d^{9} - 4 \, a c^{4} d^{7} e^{2} + 6 \, a^{2} c^{3} d^{5} e^{4} - 4 \, a^{3} c^{2} d^{3} e^{6} + a^{4} c d e^{8}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{2}} \]

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)^3,x, algorithm="giac")

[Out]

-1/2*(c^4*d^8*x^2*e^2 + 2*c^4*d^9*x*e + c^4*d^10 - 4*a*c^3*d^6*x^2*e^4 - 8*a*c^3*d^7*x*e^3 - 4*a*c^3*d^8*e^2 +
6*a^2*c^2*d^4*x^2*e^6 + 12*a^2*c^2*d^5*x*e^5 + 6*a^2*c^2*d^6*e^4 - 4*a^3*c*d^2*x^2*e^8 - 8*a^3*c*d^3*x*e^7 -
4*a^3*c*d^4*e^6 + a^4*x^2*e^10 + 2*a^4*d*x*e^9 + a^4*d^2*e^8)/((c^5*d^9 - 4*a*c^4*d^7*e^2 + 6*a^2*c^3*d^5*e^4
- 4*a^3*c^2*d^3*e^6 + a^4*c*d*e^8)*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^2)

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maple [A]  time = 0.04, size = 19, normalized size = 0.95 \[ -\frac {1}{2 \left (c d x +a e \right )^{2} c d} \]

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)^3,x)

[Out]

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

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maxima [A]  time = 0.94, size = 35, normalized size = 1.75 \[ -\frac {1}{2 \, {\left (c^{3} d^{3} x^{2} + 2 \, a c^{2} d^{2} e x + a^{2} c d e^{2}\right )}} \]

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)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.03, size = 37, normalized size = 1.85 \[ -\frac {1}{2\,a^2\,c\,d\,e^2+4\,a\,c^2\,d^2\,e\,x+2\,c^3\,d^3\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.29, size = 39, normalized size = 1.95 \[ - \frac {1}{2 a^{2} c d e^{2} + 4 a c^{2} d^{2} e x + 2 c^{3} d^{3} x^{2}} \]

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)**3,x)

[Out]

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

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