3.35 \(\int \frac{(e x)^m (A+B x^n)}{(a+b x^n)^3 (c+d x^n)^2} \, dx\)

Optimal. Leaf size=567 \[ \frac{b (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) \left (A b \left (a^2 d^2 \left (m^2+m (2-7 n)+12 n^2-7 n+1\right )-2 a b c d \left (m^2+m (2-5 n)+4 n^2-5 n+1\right )+b^2 c^2 \left (m^2+m (2-3 n)+2 n^2-3 n+1\right )\right )+a B \left (-a^2 d^2 \left (m^2+m (2-5 n)+6 n^2-5 n+1\right )+2 a b c d (m+1) (m-3 n+1)-b^2 c^2 (m+1) (m-n+1)\right )\right )}{2 a^3 e (m+1) n^2 (b c-a d)^4}+\frac{d (e x)^{m+1} \left (A \left (-2 a^2 d^2 n+a b c d (m-6 n+1)-b^2 c^2 (m-2 n+1)\right )+a B c (b c (m+1)-a d (m-6 n+1))\right )}{2 a^2 c e n^2 (b c-a d)^3 \left (c+d x^n\right )}+\frac{(e x)^{m+1} (A b (a d (m-5 n+1)-b c (m-2 n+1))+a B (b c (m+1)-a d (m-3 n+1)))}{2 a^2 e n^2 (b c-a d)^2 \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac{d^2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-4 n+1)-B c (m-3 n+1)))}{c^2 e (m+1) n (b c-a d)^4}+\frac{(e x)^{m+1} (A b-a B)}{2 a e n (b c-a d) \left (a+b x^n\right )^2 \left (c+d x^n\right )} \]

[Out]

(d*(a*B*c*(b*c*(1 + m) - a*d*(1 + m - 6*n)) + A*(a*b*c*d*(1 + m - 6*n) - b^2*c^2*(1 + m - 2*n) - 2*a^2*d^2*n))
*(e*x)^(1 + m))/(2*a^2*c*(b*c - a*d)^3*e*n^2*(c + d*x^n)) + ((A*b - a*B)*(e*x)^(1 + m))/(2*a*(b*c - a*d)*e*n*(
a + b*x^n)^2*(c + d*x^n)) + ((a*B*(b*c*(1 + m) - a*d*(1 + m - 3*n)) + A*b*(a*d*(1 + m - 5*n) - b*c*(1 + m - 2*
n)))*(e*x)^(1 + m))/(2*a^2*(b*c - a*d)^2*e*n^2*(a + b*x^n)*(c + d*x^n)) + (b*(a*B*(2*a*b*c*d*(1 + m)*(1 + m -
3*n) - b^2*c^2*(1 + m)*(1 + m - n) - a^2*d^2*(1 + m^2 + m*(2 - 5*n) - 5*n + 6*n^2)) + A*b*(b^2*c^2*(1 + m^2 +
m*(2 - 3*n) - 3*n + 2*n^2) - 2*a*b*c*d*(1 + m^2 + m*(2 - 5*n) - 5*n + 4*n^2) + a^2*d^2*(1 + m^2 + m*(2 - 7*n)
- 7*n + 12*n^2)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(2*a^3*(b*c - a*
d)^4*e*(1 + m)*n^2) + (d^2*(b*c*(A*d*(1 + m - 4*n) - B*c*(1 + m - 3*n)) + a*d*(B*c*(1 + m) - A*d*(1 + m - n)))
*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c^2*(b*c - a*d)^4*e*(1 + m)*n)

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Rubi [A]  time = 2.34061, antiderivative size = 567, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {595, 597, 364} \[ \frac{b (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) \left (A b \left (a^2 d^2 \left (m^2+m (2-7 n)+12 n^2-7 n+1\right )-2 a b c d \left (m^2+m (2-5 n)+4 n^2-5 n+1\right )+b^2 c^2 \left (m^2+m (2-3 n)+2 n^2-3 n+1\right )\right )+a B \left (-a^2 d^2 \left (m^2+m (2-5 n)+6 n^2-5 n+1\right )+2 a b c d (m+1) (m-3 n+1)-b^2 c^2 (m+1) (m-n+1)\right )\right )}{2 a^3 e (m+1) n^2 (b c-a d)^4}+\frac{d (e x)^{m+1} \left (A \left (-2 a^2 d^2 n+a b c d (m-6 n+1)-b^2 c^2 (m-2 n+1)\right )+a B c (b c (m+1)-a d (m-6 n+1))\right )}{2 a^2 c e n^2 (b c-a d)^3 \left (c+d x^n\right )}+\frac{(e x)^{m+1} (A b (a d (m-5 n+1)-b c (m-2 n+1))+a B (b c (m+1)-a d (m-3 n+1)))}{2 a^2 e n^2 (b c-a d)^2 \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac{d^2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-4 n+1)-B c (m-3 n+1)))}{c^2 e (m+1) n (b c-a d)^4}+\frac{(e x)^{m+1} (A b-a B)}{2 a e n (b c-a d) \left (a+b x^n\right )^2 \left (c+d x^n\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(a*B*c*(b*c*(1 + m) - a*d*(1 + m - 6*n)) + A*(a*b*c*d*(1 + m - 6*n) - b^2*c^2*(1 + m - 2*n) - 2*a^2*d^2*n))
*(e*x)^(1 + m))/(2*a^2*c*(b*c - a*d)^3*e*n^2*(c + d*x^n)) + ((A*b - a*B)*(e*x)^(1 + m))/(2*a*(b*c - a*d)*e*n*(
a + b*x^n)^2*(c + d*x^n)) + ((a*B*(b*c*(1 + m) - a*d*(1 + m - 3*n)) + A*b*(a*d*(1 + m - 5*n) - b*c*(1 + m - 2*
n)))*(e*x)^(1 + m))/(2*a^2*(b*c - a*d)^2*e*n^2*(a + b*x^n)*(c + d*x^n)) + (b*(a*B*(2*a*b*c*d*(1 + m)*(1 + m -
3*n) - b^2*c^2*(1 + m)*(1 + m - n) - a^2*d^2*(1 + m^2 + m*(2 - 5*n) - 5*n + 6*n^2)) + A*b*(b^2*c^2*(1 + m^2 +
m*(2 - 3*n) - 3*n + 2*n^2) - 2*a*b*c*d*(1 + m^2 + m*(2 - 5*n) - 5*n + 4*n^2) + a^2*d^2*(1 + m^2 + m*(2 - 7*n)
- 7*n + 12*n^2)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(2*a^3*(b*c - a*
d)^4*e*(1 + m)*n^2) + (d^2*(b*c*(A*d*(1 + m - 4*n) - B*c*(1 + m - 3*n)) + a*d*(B*c*(1 + m) - A*d*(1 + m - n)))
*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c^2*(b*c - a*d)^4*e*(1 + m)*n)

Rule 595

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p +
1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(
m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
e, f, g, m, n, q}, x] && LtQ[p, -1]

Rule 597

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, n, p}, x]

Rule 364

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

Rubi steps

\begin{align*} \int \frac{(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^3 \left (c+d x^n\right )^2} \, dx &=\frac{(A b-a B) (e x)^{1+m}}{2 a (b c-a d) e n \left (a+b x^n\right )^2 \left (c+d x^n\right )}-\frac{\int \frac{(e x)^m \left (-a B c (1+m)+A b c (1+m-2 n)+2 a A d n+(A b-a B) d (1+m-3 n) x^n\right )}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx}{2 a (b c-a d) n}\\ &=\frac{(A b-a B) (e x)^{1+m}}{2 a (b c-a d) e n \left (a+b x^n\right )^2 \left (c+d x^n\right )}+\frac{(a B (b c (1+m)-a d (1+m-3 n))+A b (a d (1+m-5 n)-b c (1+m-2 n))) (e x)^{1+m}}{2 a^2 (b c-a d)^2 e n^2 \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac{\int \frac{(e x)^m \left (-c (1+m) (a B (b c (1+m)-a d (1+m-3 n))+A b (a d (1+m-5 n)-b c (1+m-2 n)))+(b c-a d) n (a B c (1+m)-A b c (1+m-2 n)-2 a A d n)-d (a B (b c (1+m)-a d (1+m-3 n))+A b (a d (1+m-5 n)-b c (1+m-2 n))) (1+m-2 n) x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )^2} \, dx}{2 a^2 (b c-a d)^2 n^2}\\ &=\frac{d \left (a B c (b c (1+m)-a d (1+m-6 n))+A \left (a b c d (1+m-6 n)-b^2 c^2 (1+m-2 n)-2 a^2 d^2 n\right )\right ) (e x)^{1+m}}{2 a^2 c (b c-a d)^3 e n^2 \left (c+d x^n\right )}+\frac{(A b-a B) (e x)^{1+m}}{2 a (b c-a d) e n \left (a+b x^n\right )^2 \left (c+d x^n\right )}+\frac{(a B (b c (1+m)-a d (1+m-3 n))+A b (a d (1+m-5 n)-b c (1+m-2 n))) (e x)^{1+m}}{2 a^2 (b c-a d)^2 e n^2 \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac{\int \frac{(e x)^m \left (-n \left (a d (1+m) \left (a B c (b c (1+m)-a d (1+m-6 n))+A \left (a b c d (1+m-6 n)-b^2 c^2 (1+m-2 n)-2 a^2 d^2 n\right )\right )+(b c-a d) (c (1+m) (a B (b c (1+m)-a d (1+m-3 n))+A b (a d (1+m-5 n)-b c (1+m-2 n)))-(b c-a d) n (a B c (1+m)-A (b c (1+m-2 n)+2 a d n)))\right )-b d (1+m-n) n \left (a B c (b c (1+m)-a d (1+m-6 n))+A \left (a b c d (1+m-6 n)-b^2 c^2 (1+m-2 n)-2 a^2 d^2 n\right )\right ) x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx}{2 a^2 c (b c-a d)^3 n^3}\\ &=\frac{d \left (a B c (b c (1+m)-a d (1+m-6 n))+A \left (a b c d (1+m-6 n)-b^2 c^2 (1+m-2 n)-2 a^2 d^2 n\right )\right ) (e x)^{1+m}}{2 a^2 c (b c-a d)^3 e n^2 \left (c+d x^n\right )}+\frac{(A b-a B) (e x)^{1+m}}{2 a (b c-a d) e n \left (a+b x^n\right )^2 \left (c+d x^n\right )}+\frac{(a B (b c (1+m)-a d (1+m-3 n))+A b (a d (1+m-5 n)-b c (1+m-2 n))) (e x)^{1+m}}{2 a^2 (b c-a d)^2 e n^2 \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac{\int \left (\frac{b c n \left (a B \left (2 a b c d (1+m) (1+m-3 n)-b^2 c^2 (1+m) (1+m-n)-a^2 d^2 \left (1+m^2+m (2-5 n)-5 n+6 n^2\right )\right )+A b \left (b^2 c^2 \left (1+m^2+m (2-3 n)-3 n+2 n^2\right )-2 a b c d \left (1+m^2+m (2-5 n)-5 n+4 n^2\right )+a^2 d^2 \left (1+m^2+m (2-7 n)-7 n+12 n^2\right )\right )\right ) (e x)^m}{(b c-a d) \left (a+b x^n\right )}+\frac{2 a^2 d^2 (b c (A d (1+m-4 n)-B c (1+m-3 n))+a d (B c (1+m)-A d (1+m-n))) n^2 (e x)^m}{(b c-a d) \left (c+d x^n\right )}\right ) \, dx}{2 a^2 c (b c-a d)^3 n^3}\\ &=\frac{d \left (a B c (b c (1+m)-a d (1+m-6 n))+A \left (a b c d (1+m-6 n)-b^2 c^2 (1+m-2 n)-2 a^2 d^2 n\right )\right ) (e x)^{1+m}}{2 a^2 c (b c-a d)^3 e n^2 \left (c+d x^n\right )}+\frac{(A b-a B) (e x)^{1+m}}{2 a (b c-a d) e n \left (a+b x^n\right )^2 \left (c+d x^n\right )}+\frac{(a B (b c (1+m)-a d (1+m-3 n))+A b (a d (1+m-5 n)-b c (1+m-2 n))) (e x)^{1+m}}{2 a^2 (b c-a d)^2 e n^2 \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac{\left (d^2 (b c (A d (1+m-4 n)-B c (1+m-3 n))+a d (B c (1+m)-A d (1+m-n)))\right ) \int \frac{(e x)^m}{c+d x^n} \, dx}{c (b c-a d)^4 n}+\frac{\left (b \left (a B \left (2 a b c d (1+m) (1+m-3 n)-b^2 c^2 (1+m) (1+m-n)-a^2 d^2 \left (1+m^2+m (2-5 n)-5 n+6 n^2\right )\right )+A b \left (b^2 c^2 \left (1+m^2+m (2-3 n)-3 n+2 n^2\right )-2 a b c d \left (1+m^2+m (2-5 n)-5 n+4 n^2\right )+a^2 d^2 \left (1+m^2+m (2-7 n)-7 n+12 n^2\right )\right )\right )\right ) \int \frac{(e x)^m}{a+b x^n} \, dx}{2 a^2 (b c-a d)^4 n^2}\\ &=\frac{d \left (a B c (b c (1+m)-a d (1+m-6 n))+A \left (a b c d (1+m-6 n)-b^2 c^2 (1+m-2 n)-2 a^2 d^2 n\right )\right ) (e x)^{1+m}}{2 a^2 c (b c-a d)^3 e n^2 \left (c+d x^n\right )}+\frac{(A b-a B) (e x)^{1+m}}{2 a (b c-a d) e n \left (a+b x^n\right )^2 \left (c+d x^n\right )}+\frac{(a B (b c (1+m)-a d (1+m-3 n))+A b (a d (1+m-5 n)-b c (1+m-2 n))) (e x)^{1+m}}{2 a^2 (b c-a d)^2 e n^2 \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac{b \left (a B \left (2 a b c d (1+m) (1+m-3 n)-b^2 c^2 (1+m) (1+m-n)-a^2 d^2 \left (1+m^2+m (2-5 n)-5 n+6 n^2\right )\right )+A b \left (b^2 c^2 \left (1+m^2+m (2-3 n)-3 n+2 n^2\right )-2 a b c d \left (1+m^2+m (2-5 n)-5 n+4 n^2\right )+a^2 d^2 \left (1+m^2+m (2-7 n)-7 n+12 n^2\right )\right )\right ) (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{n};\frac{1+m+n}{n};-\frac{b x^n}{a}\right )}{2 a^3 (b c-a d)^4 e (1+m) n^2}+\frac{d^2 (b c (A d (1+m-4 n)-B c (1+m-3 n))+a d (B c (1+m)-A d (1+m-n))) (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{n};\frac{1+m+n}{n};-\frac{d x^n}{c}\right )}{c^2 (b c-a d)^4 e (1+m) n}\\ \end{align*}

Mathematica [A]  time = 0.45545, size = 270, normalized size = 0.48 \[ \frac{x (e x)^m \left (\frac{b (b c-a d) (a B d-2 A b d+b B c) \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a^2}+\frac{b (A b-a B) (b c-a d)^2 \, _2F_1\left (3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a^3}+\frac{d^2 (b c-a d) (B c-A d) \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c^2}+\frac{d^2 (a B d-3 A b d+2 b B c) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c}-\frac{b d (a B d-3 A b d+2 b B c) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a}\right )}{(m+1) (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(e*x)^m*(-((b*d*(2*b*B*c - 3*A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/a
) + (d^2*(2*b*B*c - 3*A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c + (b*(b*c
- a*d)*(b*B*c - 2*A*b*d + a*B*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/a^2 + (d^2*(b*
c - a*d)*(B*c - A*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c^2 + (b*(A*b - a*B)*(b*c -
a*d)^2*Hypergeometric2F1[3, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/a^3))/((b*c - a*d)^4*(1 + m))

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Maple [F]  time = 0.731, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) }{ \left ( a+b{x}^{n} \right ) ^{3} \left ( c+d{x}^{n} \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((m^2 - m*(3*n - 2) + 2*n^2 - 3*n + 1)*b^4*c^2*e^m - 2*(m^2 - m*(5*n - 2) + 4*n^2 - 5*n + 1)*a*b^3*c*d*e^m +
(m^2 - m*(7*n - 2) + 12*n^2 - 7*n + 1)*a^2*b^2*d^2*e^m)*A - ((m^2 - m*(n - 2) - n + 1)*a*b^3*c^2*e^m - 2*(m^2
- m*(3*n - 2) - 3*n + 1)*a^2*b^2*c*d*e^m + (m^2 - m*(5*n - 2) + 6*n^2 - 5*n + 1)*a^3*b*d^2*e^m)*B)*integrate(1
/2*x^m/(a^3*b^4*c^4*n^2 - 4*a^4*b^3*c^3*d*n^2 + 6*a^5*b^2*c^2*d^2*n^2 - 4*a^6*b*c*d^3*n^2 + a^7*d^4*n^2 + (a^2
*b^5*c^4*n^2 - 4*a^3*b^4*c^3*d*n^2 + 6*a^4*b^3*c^2*d^2*n^2 - 4*a^5*b^2*c*d^3*n^2 + a^6*b*d^4*n^2)*x^n), x) - (
(a*d^4*e^m*(m - n + 1) - b*c*d^3*e^m*(m - 4*n + 1))*A + (b*c^2*d^2*e^m*(m - 3*n + 1) - a*c*d^3*e^m*(m + 1))*B)
*integrate(x^m/(b^4*c^6*n - 4*a*b^3*c^5*d*n + 6*a^2*b^2*c^4*d^2*n - 4*a^3*b*c^3*d^3*n + a^4*c^2*d^4*n + (b^4*c
^5*d*n - 4*a*b^3*c^4*d^2*n + 6*a^2*b^2*c^3*d^3*n - 4*a^3*b*c^2*d^4*n + a^4*c*d^5*n)*x^n), x) - 1/2*(((a*b^3*c^
3*e^m*(m - 3*n + 1) - a^2*b^2*c^2*d*e^m*(m - 7*n + 1) + 2*a^4*d^3*e^m*n)*A - (a^2*b^2*c^3*e^m*(m - n + 1) - a^
3*b*c^2*d*e^m*(m - 5*n + 1) + 2*a^4*c*d^2*e^m*n)*B)*x*x^m + ((b^4*c^2*d*e^m*(m - 2*n + 1) - a*b^3*c*d^2*e^m*(m
- 6*n + 1) + 2*a^2*b^2*d^3*e^m*n)*A + (a^2*b^2*c*d^2*e^m*(m - 6*n + 1) - a*b^3*c^2*d*e^m*(m + 1))*B)*x*e^(m*l
og(x) + 2*n*log(x)) + ((b^4*c^3*e^m*(m - 2*n + 1) - a^2*b^2*c*d^2*e^m*(m - 7*n + 1) + 3*a*b^3*c^2*d*e^m*n + 4*
a^3*b*d^3*e^m*n)*A + (a^3*b*c*d^2*e^m*(m - 9*n + 1) - a*b^3*c^3*e^m*(m + 1) - 3*a^2*b^2*c^2*d*e^m*n)*B)*x*e^(m
*log(x) + n*log(x)))/(a^4*b^3*c^5*n^2 - 3*a^5*b^2*c^4*d*n^2 + 3*a^6*b*c^3*d^2*n^2 - a^7*c^2*d^3*n^2 + (a^2*b^5
*c^4*d*n^2 - 3*a^3*b^4*c^3*d^2*n^2 + 3*a^4*b^3*c^2*d^3*n^2 - a^5*b^2*c*d^4*n^2)*x^(3*n) + (a^2*b^5*c^5*n^2 - a
^3*b^4*c^4*d*n^2 - 3*a^4*b^3*c^3*d^2*n^2 + 5*a^5*b^2*c^2*d^3*n^2 - 2*a^6*b*c*d^4*n^2)*x^(2*n) + (2*a^3*b^4*c^5
*n^2 - 5*a^4*b^3*c^4*d*n^2 + 3*a^5*b^2*c^3*d^2*n^2 + a^6*b*c^2*d^3*n^2 - a^7*c*d^4*n^2)*x^n)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{b^{3} d^{2} x^{5 \, n} + a^{3} c^{2} +{\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{4 \, n} +{\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{3 \, n} +{\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2 \, n} +{\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x^{n}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*x^n + A)*(e*x)^m/(b^3*d^2*x^(5*n) + a^3*c^2 + (2*b^3*c*d + 3*a*b^2*d^2)*x^(4*n) + (b^3*c^2 + 6*a*b
^2*c*d + 3*a^2*b*d^2)*x^(3*n) + (3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*x^(2*n) + (3*a^2*b*c^2 + 2*a^3*c*d)*x^n)
, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{3}{\left (d x^{n} + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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