3.1847 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^6} \, dx\)

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

[Out]

1/3*(c*d*x+a*e)^3/(-a*e^2+c*d^2)/(e*x+d)^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} \[ \frac {(a e+c d x)^3}{3 (d+e x)^3 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.03, size = 59, normalized size = 1.69 \[ -\frac {a^2 e^4+a c d e^2 (d+3 e x)+c^2 d^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^6,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))/(e^3*(d + e*x)^3)

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fricas [B]  time = 1.00, size = 94, normalized size = 2.69 \[ -\frac {3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [B]  time = 0.18, size = 137, normalized size = 3.91 \[ -\frac {{\left (3 \, c^{2} d^{2} x^{4} e^{4} + 9 \, c^{2} d^{3} x^{3} e^{3} + 10 \, c^{2} d^{4} x^{2} e^{2} + 5 \, c^{2} d^{5} x e + c^{2} d^{6} + 3 \, a c d x^{3} e^{5} + 7 \, a c d^{2} x^{2} e^{4} + 5 \, a c d^{3} x e^{3} + a c d^{4} e^{2} + a^{2} x^{2} e^{6} + 2 \, a^{2} d x e^{5} + a^{2} d^{2} e^{4}\right )} e^{\left (-3\right )}}{3 \, {\left (x e + d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 83, normalized size = 2.37 \[ -\frac {c^{2} d^{2}}{\left (e x +d \right ) e^{3}}-\frac {\left (a \,e^{2}-c \,d^{2}\right ) c d}{\left (e x +d \right )^{2} e^{3}}-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{3 \left (e x +d \right )^{3} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.12, size = 94, normalized size = 2.69 \[ -\frac {3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [B]  time = 0.57, size = 65, normalized size = 1.86 \[ -\frac {\frac {a^2\,e}{3}-d\,\left (\frac {c^2\,x^3}{3}-a\,c\,x\right )+\frac {a\,c\,d^2}{3\,e}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.74, size = 99, normalized size = 2.83 \[ \frac {- a^{2} e^{4} - a c d^{2} e^{2} - c^{2} d^{4} - 3 c^{2} d^{2} e^{2} x^{2} + x \left (- 3 a c d e^{3} - 3 c^{2} d^{3} e\right )}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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