\(\int \frac {(e x)^m (a+b x^n)^2 (A+B x^n)}{(c+d x^n)^3} \, dx\) [36]

Optimal result

Integrand size = 31, antiderivative size = 322 \[ \int \frac {(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx=\frac {b (a d (1+m)-b c (1+m+n)) (A d (1+m)-B c (1+m+2 n)) (e x)^{1+m}}{2 c^2 d^3 e (1+m) n^2}-\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^2}{2 c d e n \left (c+d x^n\right )^2}-\frac {(b c-a d) (e x)^{1+m} \left (a (B c (1+m)-A d (1+m-2 n))-b (A d (1+m)-B c (1+m+2 n)) x^n\right )}{2 c^2 d^2 e n^2 \left (c+d x^n\right )}+\frac {(a d (B c (1+m)-A d (1+m-2 n)) (b c (1+m)-a d (1+m-n))-b c (a d (1+m)-b c (1+m+n)) (A d (1+m)-B c (1+m+2 n))) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{2 c^3 d^3 e (1+m) n^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {608, 470, 371} \[ \int \frac {(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx=\frac {(e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {d x^n}{c}\right ) (a d (b c (m+1)-a d (m-n+1)) (B c (m+1)-A d (m-2 n+1))-b c (a d (m+1)-b c (m+n+1)) (A d (m+1)-B c (m+2 n+1)))}{2 c^3 d^3 e (m+1) n^2}+\frac {b (e x)^{m+1} (a d (m+1)-b c (m+n+1)) (A d (m+1)-B c (m+2 n+1))}{2 c^2 d^3 e (m+1) n^2}-\frac {(e x)^{m+1} (b c-a d) \left (a (B c (m+1)-A d (m-2 n+1))-b x^n (A d (m+1)-B c (m+2 n+1))\right )}{2 c^2 d^2 e n^2 \left (c+d x^n\right )}-\frac {(e x)^{m+1} \left (a+b x^n\right )^2 (B c-A d)}{2 c d e n \left (c+d x^n\right )^2} \]

[In]

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

[Out]

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 608

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/(a*b*g*n*(p + 1))), x] + Dis
t[1/(a*b*n*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + (b*e - a*f)*(
m + 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n},
x] && LtQ[p, -1] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])

Rubi steps \begin{align*} \text {integral}& = -\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^2}{2 c d e n \left (c+d x^n\right )^2}-\frac {\int \frac {(e x)^m \left (a+b x^n\right ) \left (-a (B c (1+m)-A d (1+m-2 n))+b (A d (1+m)-B c (1+m+2 n)) x^n\right )}{\left (c+d x^n\right )^2} \, dx}{2 c d n} \\ & = -\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^2}{2 c d e n \left (c+d x^n\right )^2}-\frac {(b c-a d) (e x)^{1+m} \left (a (B c (1+m)-A d (1+m-2 n))-b (A d (1+m)-B c (1+m+2 n)) x^n\right )}{2 c^2 d^2 e n^2 \left (c+d x^n\right )}+\frac {\int \frac {(e x)^m \left (a (B c (1+m)-A d (1+m-2 n)) (b c (1+m)-a d (1+m-n))+b (a d (1+m)-b c (1+m+n)) (A d (1+m)-B c (1+m+2 n)) x^n\right )}{c+d x^n} \, dx}{2 c^2 d^2 n^2} \\ & = \frac {b (a d (1+m)-b c (1+m+n)) (A d (1+m)-B c (1+m+2 n)) (e x)^{1+m}}{2 c^2 d^3 e (1+m) n^2}-\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^2}{2 c d e n \left (c+d x^n\right )^2}-\frac {(b c-a d) (e x)^{1+m} \left (a (B c (1+m)-A d (1+m-2 n))-b (A d (1+m)-B c (1+m+2 n)) x^n\right )}{2 c^2 d^2 e n^2 \left (c+d x^n\right )}+\frac {\left (a (B c (1+m)-A d (1+m-2 n)) (b c (1+m)-a d (1+m-n))-\frac {b c (a d (1+m)-b c (1+m+n)) (A d (1+m)-B c (1+m+2 n))}{d}\right ) \int \frac {(e x)^m}{c+d x^n} \, dx}{2 c^2 d^2 n^2} \\ & = \frac {b (a d (1+m)-b c (1+m+n)) (A d (1+m)-B c (1+m+2 n)) (e x)^{1+m}}{2 c^2 d^3 e (1+m) n^2}-\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^2}{2 c d e n \left (c+d x^n\right )^2}-\frac {(b c-a d) (e x)^{1+m} \left (a (B c (1+m)-A d (1+m-2 n))-b (A d (1+m)-B c (1+m+2 n)) x^n\right )}{2 c^2 d^2 e n^2 \left (c+d x^n\right )}+\frac {\left (a (B c (1+m)-A d (1+m-2 n)) (b c (1+m)-a d (1+m-n))-\frac {b c (a d (1+m)-b c (1+m+n)) (A d (1+m)-B c (1+m+2 n))}{d}\right ) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{2 c^3 d^2 e (1+m) n^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.53 \[ \int \frac {(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx=\frac {x (e x)^m \left (b^2 B-\frac {b (3 b B c-A b d-2 a B d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c}+\frac {(b c-a d) (3 b B c-2 A b d-a B d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c^2}-\frac {(b c-a d)^2 (B c-A d) \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c^3}\right )}{d^3 (1+m)} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (e x \right )^{m} \left (a +b \,x^{n}\right )^{2} \left (A +B \,x^{n}\right )}{\left (c +d \,x^{n}\right )^{3}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{2} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{2} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{2} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,{\left (a+b\,x^n\right )}^2}{{\left (c+d\,x^n\right )}^3} \,d x \]

[In]

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

[Out]

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