3.1.82 \(\int \frac {d+e x^n}{(a+b x^n+c x^{2 n})^3} \, dx\) [82]

Optimal. Leaf size=713 \[ \frac {x \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^n\right )}{2 a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {x \left (a b^3 e-4 a^2 c^2 d (1-4 n)+5 a b^2 c d (1-3 n)-2 a^2 b c e (2-3 n)-b^4 d (1-2 n)+c \left (a b^2 e+2 a b c d (2-7 n)-4 a^2 c e (1-3 n)-b^3 d (1-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}+\frac {c \left (a b^2 \left (\sqrt {b^2-4 a c} e+6 c d (1-3 n)\right ) (1-n)+b^3 \left (a e-\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n)-b^4 d \left (1-3 n+2 n^2\right )-2 a b c \left (2 a e \left (1-n-3 n^2\right )-\sqrt {b^2-4 a c} d \left (2-9 n+7 n^2\right )\right )-4 a^2 c \left (\sqrt {b^2-4 a c} e \left (1-4 n+3 n^2\right )+2 c d \left (1-6 n+8 n^2\right )\right )\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) n^2}-\frac {c \left (a b^2 \left (\sqrt {b^2-4 a c} e-6 c d (1-3 n)\right ) (1-n)-b^3 \left (a e+\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n)+b^4 d \left (1-3 n+2 n^2\right )+2 a b c \left (2 a e \left (1-n-3 n^2\right )+\sqrt {b^2-4 a c} d \left (2-9 n+7 n^2\right )\right )-4 a^2 c \left (\sqrt {b^2-4 a c} e \left (1-4 n+3 n^2\right )-2 c d \left (1-6 n+8 n^2\right )\right )\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) n^2} \]

[Out]

1/2*x*(b^2*d-2*a*c*d-a*b*e+c*(-2*a*e+b*d)*x^n)/a/(-4*a*c+b^2)/n/(a+b*x^n+c*x^(2*n))^2+1/2*x*(a*b^3*e-4*a^2*c^2
*d*(1-4*n)+5*a*b^2*c*d*(1-3*n)-2*a^2*b*c*e*(2-3*n)-b^4*d*(1-2*n)+c*(a*b^2*e+2*a*b*c*d*(2-7*n)-4*a^2*c*e*(1-3*n
)-b^3*d*(1-2*n))*x^n)/a^2/(-4*a*c+b^2)^2/n^2/(a+b*x^n+c*x^(2*n))+1/2*c*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(
b-(-4*a*c+b^2)^(1/2)))*(-b^4*d*(2*n^2-3*n+1)+a*b^2*(1-n)*(6*c*d*(1-3*n)+e*(-4*a*c+b^2)^(1/2))+b^3*(1-n)*(a*e-d
*(1-2*n)*(-4*a*c+b^2)^(1/2))-4*a^2*c*(2*c*d*(8*n^2-6*n+1)+e*(3*n^2-4*n+1)*(-4*a*c+b^2)^(1/2))-2*a*b*c*(2*a*e*(
-3*n^2-n+1)-d*(7*n^2-9*n+2)*(-4*a*c+b^2)^(1/2)))/a^2/(-4*a*c+b^2)^2/n^2/(b^2-4*a*c-b*(-4*a*c+b^2)^(1/2))-1/2*c
*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)))*(b^4*d*(2*n^2-3*n+1)+a*b^2*(1-n)*(-6*c*d*(1-3*n
)+e*(-4*a*c+b^2)^(1/2))-b^3*(1-n)*(a*e+d*(1-2*n)*(-4*a*c+b^2)^(1/2))-4*a^2*c*(-2*c*d*(8*n^2-6*n+1)+e*(3*n^2-4*
n+1)*(-4*a*c+b^2)^(1/2))+2*a*b*c*(2*a*e*(-3*n^2-n+1)+d*(7*n^2-9*n+2)*(-4*a*c+b^2)^(1/2)))/a^2/(-4*a*c+b^2)^2/n
^2/(b^2-4*a*c+b*(-4*a*c+b^2)^(1/2))

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Rubi [A]
time = 1.14, antiderivative size = 713, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1444, 1436, 251} \begin {gather*} \frac {x \left (c x^n \left (-4 a^2 c e (1-3 n)+a b^2 e+2 a b c d (2-7 n)+b^3 (-d) (1-2 n)\right )-2 a^2 b c e (2-3 n)-4 a^2 c^2 d (1-4 n)+a b^3 e+5 a b^2 c d (1-3 n)-b^4 d (1-2 n)\right )}{2 a^2 n^2 \left (b^2-4 a c\right )^2 \left (a+b x^n+c x^{2 n}\right )}+\frac {c x \left (-4 a^2 c \left (e \left (3 n^2-4 n+1\right ) \sqrt {b^2-4 a c}+2 c d \left (8 n^2-6 n+1\right )\right )-2 a b c \left (2 a e \left (-3 n^2-n+1\right )-d \left (7 n^2-9 n+2\right ) \sqrt {b^2-4 a c}\right )+a b^2 (1-n) \left (e \sqrt {b^2-4 a c}+6 c d (1-3 n)\right )+b^3 (1-n) \left (a e-d (1-2 n) \sqrt {b^2-4 a c}\right )+b^4 (-d) \left (2 n^2-3 n+1\right )\right ) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2 n^2 \left (b^2-4 a c\right )^2 \left (-b \sqrt {b^2-4 a c}-4 a c+b^2\right )}-\frac {c x \left (-4 a^2 c \left (e \left (3 n^2-4 n+1\right ) \sqrt {b^2-4 a c}-2 c d \left (8 n^2-6 n+1\right )\right )+2 a b c \left (d \left (7 n^2-9 n+2\right ) \sqrt {b^2-4 a c}+2 a e \left (-3 n^2-n+1\right )\right )+a b^2 (1-n) \left (e \sqrt {b^2-4 a c}-6 c d (1-3 n)\right )-b^3 (1-n) \left (d (1-2 n) \sqrt {b^2-4 a c}+a e\right )+b^4 d \left (2 n^2-3 n+1\right )\right ) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 a^2 n^2 \left (b^2-4 a c\right )^2 \left (b \sqrt {b^2-4 a c}-4 a c+b^2\right )}+\frac {x \left (c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{2 a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^n)/(a + b*x^n + c*x^(2*n))^3,x]

[Out]

(x*(b^2*d - 2*a*c*d - a*b*e + c*(b*d - 2*a*e)*x^n))/(2*a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))^2) + (x*(a*b^
3*e - 4*a^2*c^2*d*(1 - 4*n) + 5*a*b^2*c*d*(1 - 3*n) - 2*a^2*b*c*e*(2 - 3*n) - b^4*d*(1 - 2*n) + c*(a*b^2*e + 2
*a*b*c*d*(2 - 7*n) - 4*a^2*c*e*(1 - 3*n) - b^3*d*(1 - 2*n))*x^n))/(2*a^2*(b^2 - 4*a*c)^2*n^2*(a + b*x^n + c*x^
(2*n))) + (c*(a*b^2*(Sqrt[b^2 - 4*a*c]*e + 6*c*d*(1 - 3*n))*(1 - n) + b^3*(a*e - Sqrt[b^2 - 4*a*c]*d*(1 - 2*n)
)*(1 - n) - b^4*d*(1 - 3*n + 2*n^2) - 2*a*b*c*(2*a*e*(1 - n - 3*n^2) - Sqrt[b^2 - 4*a*c]*d*(2 - 9*n + 7*n^2))
- 4*a^2*c*(Sqrt[b^2 - 4*a*c]*e*(1 - 4*n + 3*n^2) + 2*c*d*(1 - 6*n + 8*n^2)))*x*Hypergeometric2F1[1, n^(-1), 1
+ n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^2*(b^2 - 4*a*c)^2*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n^2)
- (c*(a*b^2*(Sqrt[b^2 - 4*a*c]*e - 6*c*d*(1 - 3*n))*(1 - n) - b^3*(a*e + Sqrt[b^2 - 4*a*c]*d*(1 - 2*n))*(1 -
n) + b^4*d*(1 - 3*n + 2*n^2) + 2*a*b*c*(2*a*e*(1 - n - 3*n^2) + Sqrt[b^2 - 4*a*c]*d*(2 - 9*n + 7*n^2)) - 4*a^2
*c*(Sqrt[b^2 - 4*a*c]*e*(1 - 4*n + 3*n^2) - 2*c*d*(1 - 6*n + 8*n^2)))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1
), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(2*a^2*(b^2 - 4*a*c)^2*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n^2)

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 1436

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*
c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In
t[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] ||  !IGtQ[n/2, 0])

Rule 1444

Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[(-x)*(d*b^2 -
a*b*e - 2*a*c*d + (b*d - 2*a*e)*c*x^n)*((a + b*x^n + c*x^(2*n))^(p + 1)/(a*n*(p + 1)*(b^2 - 4*a*c))), x] + Di
st[1/(a*n*(p + 1)*(b^2 - 4*a*c)), Int[Simp[(n*p + n + 1)*d*b^2 - a*b*e - 2*a*c*d*(2*n*p + 2*n + 1) + (2*n*p +
3*n + 1)*(d*b - 2*a*e)*c*x^n, x]*(a + b*x^n + c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && Eq
Q[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {d+e x^n}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx &=\frac {x \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^n\right )}{2 a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}-\frac {\int \frac {-a b e-2 a c d (1-4 n)+b^2 (d-2 d n)+c (b d-2 a e) (1-3 n) x^n}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx}{2 a \left (b^2-4 a c\right ) n}\\ &=\frac {x \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^n\right )}{2 a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {x \left (a b^3 e-4 a^2 c^2 d (1-4 n)+5 a b^2 c d (1-3 n)-2 a^2 b c e (2-3 n)-b^4 d (1-2 n)+c \left (a b^2 e+2 a b c d (2-7 n)-4 a^2 c e (1-3 n)-b^3 d (1-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}+\frac {\int \frac {2 a^2 b c e (2-5 n)-a b^3 e (1-n)+b^4 d \left (1-3 n+2 n^2\right )+4 a^2 c^2 d \left (1-6 n+8 n^2\right )-a b^2 c d \left (5-21 n+16 n^2\right )-c \left (a b^2 e+2 a b c d (2-7 n)-4 a^2 c e (1-3 n)-b^3 d (1-2 n)\right ) (1-n) x^n}{a+b x^n+c x^{2 n}} \, dx}{2 a^2 \left (b^2-4 a c\right )^2 n^2}\\ &=\frac {x \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^n\right )}{2 a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {x \left (a b^3 e-4 a^2 c^2 d (1-4 n)+5 a b^2 c d (1-3 n)-2 a^2 b c e (2-3 n)-b^4 d (1-2 n)+c \left (a b^2 e+2 a b c d (2-7 n)-4 a^2 c e (1-3 n)-b^3 d (1-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac {\left (c \left (a b^2 \left (\sqrt {b^2-4 a c} e-6 c d (1-3 n)\right ) (1-n)-b^3 \left (a e+\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n)+b^4 d \left (1-3 n+2 n^2\right )+2 a b c \left (2 a e \left (1-n-3 n^2\right )+\sqrt {b^2-4 a c} d \left (2-9 n+7 n^2\right )\right )-4 a^2 c \left (\sqrt {b^2-4 a c} e \left (1-4 n+3 n^2\right )-2 c d \left (1-6 n+8 n^2\right )\right )\right )\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{4 a^2 \left (b^2-4 a c\right )^{5/2} n^2}-\frac {\left (c \left (a b^2 \left (\sqrt {b^2-4 a c} e+6 c d (1-3 n)\right ) (1-n)+b^3 \left (a e-\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n)-b^4 d \left (1-3 n+2 n^2\right )-2 a b c \left (2 a e \left (1-n-3 n^2\right )-\sqrt {b^2-4 a c} d \left (2-9 n+7 n^2\right )\right )-4 a^2 c \left (\sqrt {b^2-4 a c} e \left (1-4 n+3 n^2\right )+2 c d \left (1-6 n+8 n^2\right )\right )\right )\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{4 a^2 \left (b^2-4 a c\right )^{5/2} n^2}\\ &=\frac {x \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^n\right )}{2 a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {x \left (a b^3 e-4 a^2 c^2 d (1-4 n)+5 a b^2 c d (1-3 n)-2 a^2 b c e (2-3 n)-b^4 d (1-2 n)+c \left (a b^2 e+2 a b c d (2-7 n)-4 a^2 c e (1-3 n)-b^3 d (1-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac {c \left (a b^2 \left (\sqrt {b^2-4 a c} e+6 c d (1-3 n)\right ) (1-n)+b^3 \left (a e-\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n)-b^4 d \left (1-3 n+2 n^2\right )-2 a b c \left (2 a e \left (1-n-3 n^2\right )-\sqrt {b^2-4 a c} d \left (2-9 n+7 n^2\right )\right )-4 a^2 c \left (\sqrt {b^2-4 a c} e \left (1-4 n+3 n^2\right )+2 c d \left (1-6 n+8 n^2\right )\right )\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{5/2} \left (b-\sqrt {b^2-4 a c}\right ) n^2}-\frac {c \left (a b^2 \left (\sqrt {b^2-4 a c} e-6 c d (1-3 n)\right ) (1-n)-b^3 \left (a e+\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n)+b^4 d \left (1-3 n+2 n^2\right )+2 a b c \left (2 a e \left (1-n-3 n^2\right )+\sqrt {b^2-4 a c} d \left (2-9 n+7 n^2\right )\right )-4 a^2 c \left (\sqrt {b^2-4 a c} e \left (1-4 n+3 n^2\right )-2 c d \left (1-6 n+8 n^2\right )\right )\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{5/2} \left (b+\sqrt {b^2-4 a c}\right ) n^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(8593\) vs. \(2(713)=1426\).
time = 6.77, size = 8593, normalized size = 12.05 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {d +e \,x^{n}}{\left (a +b \,x^{n}+c \,x^{2 n}\right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^n)/(a+b*x^n+c*x^(2*n))^3,x, algorithm="maxima")

[Out]

-1/2*((2*a*b*c^3*d*(7*n - 2) - b^3*c^2*d*(2*n - 1) - (4*a^2*c^3*(3*n - 1) + a*b^2*c^2)*e)*x*x^(3*n) + (a*b^2*c
^2*d*(29*n - 9) - 4*a^2*c^3*d*(4*n - 1) - 2*b^4*c*d*(2*n - 1) - 2*(a^2*b*c^2*(9*n - 4) + a*b^3*c)*e)*x*x^(2*n)
+ (4*a*b^3*c*d*(3*n - 1) - b^5*d*(2*n - 1) + 2*a^2*b*c^2*d*n - (4*a^3*c^2*(5*n - 1) + a^2*b^2*c*(4*n - 3) + a
*b^4)*e)*x*x^n + (a^2*b^2*c*d*(21*n - 5) - 4*a^3*c^2*d*(6*n - 1) - a*b^4*d*(3*n - 1) - (2*a^3*b*c*(5*n - 2) -
a^2*b^3*(n - 1))*e)*x)/(a^4*b^4*n^2 - 8*a^5*b^2*c*n^2 + 16*a^6*c^2*n^2 + (a^2*b^4*c^2*n^2 - 8*a^3*b^2*c^3*n^2
+ 16*a^4*c^4*n^2)*x^(4*n) + 2*(a^2*b^5*c*n^2 - 8*a^3*b^3*c^2*n^2 + 16*a^4*b*c^3*n^2)*x^(3*n) + (a^2*b^6*n^2 -
6*a^3*b^4*c*n^2 + 32*a^5*c^3*n^2)*x^(2*n) + 2*(a^3*b^5*n^2 - 8*a^4*b^3*c*n^2 + 16*a^5*b*c^2*n^2)*x^n) + integr
ate(1/2*((2*n^2 - 3*n + 1)*b^4*d - (16*n^2 - 21*n + 5)*a*b^2*c*d + 4*(8*n^2 - 6*n + 1)*a^2*c^2*d + ((2*n^2 - 3
*n + 1)*b^3*c*d - 2*(7*n^2 - 9*n + 2)*a*b*c^2*d + (4*(3*n^2 - 4*n + 1)*a^2*c^2 + a*b^2*c*(n - 1))*e)*x^n - (2*
a^2*b*c*(5*n - 2) - a*b^3*(n - 1))*e)/(a^3*b^4*n^2 - 8*a^4*b^2*c*n^2 + 16*a^5*c^2*n^2 + (a^2*b^4*c*n^2 - 8*a^3
*b^2*c^2*n^2 + 16*a^4*c^3*n^2)*x^(2*n) + (a^2*b^5*n^2 - 8*a^3*b^3*c*n^2 + 16*a^4*b*c^2*n^2)*x^n), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^n)/(a+b*x^n+c*x^(2*n))^3,x, algorithm="fricas")

[Out]

integral((x^n*e + d)/(c^3*x^(6*n) + b^3*x^(3*n) + 3*a*b^2*x^(2*n) + 3*a^2*b*x^n + a^3 + 3*(b*c^2*x^n + a*c^2)*
x^(4*n) + 3*(b^2*c*x^(2*n) + 2*a*b*c*x^n + a^2*c)*x^(2*n)), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x**n)/(a+b*x**n+c*x**(2*n))**3,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^n)/(a+b*x^n+c*x^(2*n))^3,x, algorithm="giac")

[Out]

integrate((x^n*e + d)/(c*x^(2*n) + b*x^n + a)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {d+e\,x^n}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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