3.19.47 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^6} \, dx\) [1847]

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

[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 = {640, 37} \begin {gather*} \frac {(a e+c d x)^3}{3 (d+e x)^3 \left (c d^2-a e^2\right )} \end {gather*}

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 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^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 = 0.02, size = 59, normalized size = 1.69 \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 e^3 (d+e x)^3} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(82\) vs. \(2(33)=66\).
time = 0.67, size = 83, normalized size = 2.37

method result size
gosper \(-\frac {3 c^{2} d^{2} x^{2} e^{2}+3 a c d \,e^{3} x +3 c^{2} d^{3} e x +a^{2} e^{4}+a c \,d^{2} e^{2}+c^{2} d^{4}}{3 e^{3} \left (e x +d \right )^{3}}\) \(70\)
risch \(\frac {-\frac {c^{2} d^{2} x^{2}}{e}-\frac {c d \left (e^{2} a +c \,d^{2}\right ) x}{e^{2}}-\frac {a^{2} e^{4}+a c \,d^{2} e^{2}+c^{2} d^{4}}{3 e^{3}}}{\left (e x +d \right )^{3}}\) \(72\)
default \(-\frac {c d \left (e^{2} a -c \,d^{2}\right )}{e^{3} \left (e x +d \right )^{2}}-\frac {d^{2} c^{2}}{e^{3} \left (e x +d \right )}-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{3 e^{3} \left (e x +d \right )^{3}}\) \(83\)
norman \(\frac {-\frac {d^{2} \left (a^{2} e^{6}+a c \,d^{2} e^{4}+c^{2} d^{4} e^{2}\right )}{3 e^{5}}-\frac {\left (a^{2} e^{6}+7 a c \,d^{2} e^{4}+10 c^{2} d^{4} e^{2}\right ) x^{2}}{3 e^{3}}-e \,c^{2} d^{2} x^{4}-\frac {d \left (a c \,e^{4}+3 c^{2} d^{2} e^{2}\right ) x^{3}}{e^{2}}-\frac {d \left (2 a^{2} e^{6}+5 a c \,d^{2} e^{4}+5 c^{2} d^{4} e^{2}\right ) x}{3 e^{4}}}{\left (e x +d \right )^{5}}\) \(158\)

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,method=_RETURNVERBOSE)

[Out]

-c*d/e^3*(a*e^2-c*d^2)/(e*x+d)^2-d^2/e^3*c^2/(e*x+d)-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] Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (34) = 68\).
time = 0.27, size = 87, normalized size = 2.49 \begin {gather*} -\frac {3 \, c^{2} d^{2} x^{2} e^{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 (x^{3} e^{6} + 3 \, d x^{2} e^{5} + 3 \, d^{2} x e^{4} + d^{3} e^{3}\right )}} \end {gather*}

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*x^2*e^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)/(x^3*e^6 + 3*d*x^2*e^
5 + 3*d^2*x*e^4 + d^3*e^3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (34) = 68\).
time = 3.39, size = 87, normalized size = 2.49 \begin {gather*} -\frac {3 \, c^{2} d^{3} x e + c^{2} d^{4} + 3 \, a c d x e^{3} + a^{2} e^{4} + {\left (3 \, c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}}{3 \, {\left (x^{3} e^{6} + 3 \, d x^{2} e^{5} + 3 \, d^{2} x e^{4} + d^{3} e^{3}\right )}} \end {gather*}

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (27) = 54\).
time = 0.43, size = 99, normalized size = 2.83 \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 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \end {gather*}

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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Giac [A]
time = 0.89, size = 66, normalized size = 1.89 \begin {gather*} -\frac {{\left (3 \, c^{2} d^{2} x^{2} e^{2} + 3 \, c^{2} d^{3} x e + c^{2} d^{4} + 3 \, a c d x e^{3} + a c d^{2} e^{2} + a^{2} e^{4}\right )} e^{\left (-3\right )}}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}

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

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Mupad [B]
time = 0.57, size = 65, normalized size = 1.86 \begin {gather*} -\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} \end {gather*}

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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