3.1.14 \(\int \frac {(e x)^m (A+B x^n) (c+d x^n)^2}{(a+b x^n)^3} \, dx\) [14]

Optimal. Leaf size=322 \[ \frac {d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n)) (e x)^{1+m}}{2 a^2 b^3 e (1+m) n^2}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}+\frac {(b c-a d) (e x)^{1+m} \left (c (a B (1+m)-A b (1+m-2 n))-d (A b (1+m)-a B (1+m+2 n)) x^n\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {(b c (a B (1+m)-A b (1+m-2 n)) (a d (1+m)-b c (1+m-n))-a d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{2 a^3 b^3 e (1+m) n^2} \]

[Out]

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

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Rubi [A]
time = 0.36, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {608, 470, 371} \begin {gather*} \frac {(e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right ) (b c (a B (m+1)-A b (m-2 n+1)) (a d (m+1)-b c (m-n+1))-a d (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1)))}{2 a^3 b^3 e (m+1) n^2}+\frac {d (e x)^{m+1} (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1))}{2 a^2 b^3 e (m+1) n^2}+\frac {(e x)^{m+1} (b c-a d) \left (c (a B (m+1)-A b (m-2 n+1))-d x^n (A b (m+1)-a B (m+2 n+1))\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(b*c*(1 + m) - a*d*(1 + m + n))*(A*b*(1 + m) - a*B*(1 + m + 2*n))*(e*x)^(1 + m))/(2*a^2*b^3*e*(1 + m)*n^2)
+ ((A*b - a*B)*(e*x)^(1 + m)*(c + d*x^n)^2)/(2*a*b*e*n*(a + b*x^n)^2) + ((b*c - a*d)*(e*x)^(1 + m)*(c*(a*B*(1
+ m) - A*b*(1 + m - 2*n)) - d*(A*b*(1 + m) - a*B*(1 + m + 2*n))*x^n))/(2*a^2*b^2*e*n^2*(a + b*x^n)) + ((b*c*(a
*B*(1 + m) - A*b*(1 + m - 2*n))*(a*d*(1 + m) - b*c*(1 + m - n)) - a*d*(b*c*(1 + m) - a*d*(1 + m + n))*(A*b*(1
+ m) - a*B*(1 + m + 2*n)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(2*a^3*
b^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*} \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2}{\left (a+b x^n\right )^3} \, dx &=\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}-\frac {\int \frac {(e x)^m \left (c+d x^n\right ) \left (-c (a B (1+m)-A b (1+m-2 n))+d (A b (1+m)-a B (1+m+2 n)) x^n\right )}{\left (a+b x^n\right )^2} \, dx}{2 a b n}\\ &=\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}+\frac {(b c-a d) (e x)^{1+m} \left (c (a B (1+m)-A b (1+m-2 n))-d (A b (1+m)-a B (1+m+2 n)) x^n\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {\int \frac {(e x)^m \left (c (a B (1+m)-A b (1+m-2 n)) (a d (1+m)-b c (1+m-n))+d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n)) x^n\right )}{a+b x^n} \, dx}{2 a^2 b^2 n^2}\\ &=\frac {d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n)) (e x)^{1+m}}{2 a^2 b^3 e (1+m) n^2}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}+\frac {(b c-a d) (e x)^{1+m} \left (c (a B (1+m)-A b (1+m-2 n))-d (A b (1+m)-a B (1+m+2 n)) x^n\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {\left (c (a B (1+m)-A b (1+m-2 n)) (a d (1+m)-b c (1+m-n))-\frac {a d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n))}{b}\right ) \int \frac {(e x)^m}{a+b x^n} \, dx}{2 a^2 b^2 n^2}\\ &=\frac {d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n)) (e x)^{1+m}}{2 a^2 b^3 e (1+m) n^2}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}+\frac {(b c-a d) (e x)^{1+m} \left (c (a B (1+m)-A b (1+m-2 n))-d (A b (1+m)-a B (1+m+2 n)) x^n\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {\left (c (a B (1+m)-A b (1+m-2 n)) (a d (1+m)-b c (1+m-n))-\frac {a d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n))}{b}\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^2 e (1+m) n^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1924\) vs. \(2(322)=644\).
time = 1.05, size = 1924, normalized size = 5.98 \begin {gather*} \frac {x (e x)^m \left (a^2 A b^3 c^2 (1+m) n-a^3 b^2 B c^2 (1+m) n-2 a^3 A b^2 c d (1+m) n+2 a^4 b B c d (1+m) n+a^4 A b d^2 (1+m) n-a^5 B d^2 (1+m) n-a A b^3 c^2 (1+m) \left (a+b x^n\right )+a^2 b^2 B c^2 (1+m) \left (a+b x^n\right )+2 a^2 A b^2 c d (1+m) \left (a+b x^n\right )-2 a^3 b B c d (1+m) \left (a+b x^n\right )-a^3 A b d^2 (1+m) \left (a+b x^n\right )+a^4 B d^2 (1+m) \left (a+b x^n\right )-a A b^3 c^2 m (1+m) \left (a+b x^n\right )+a^2 b^2 B c^2 m (1+m) \left (a+b x^n\right )+2 a^2 A b^2 c d m (1+m) \left (a+b x^n\right )-2 a^3 b B c d m (1+m) \left (a+b x^n\right )-a^3 A b d^2 m (1+m) \left (a+b x^n\right )+a^4 B d^2 m (1+m) \left (a+b x^n\right )+2 a A b^3 c^2 (1+m) n \left (a+b x^n\right )-4 a^3 b B c d (1+m) n \left (a+b x^n\right )-2 a^3 A b d^2 (1+m) n \left (a+b x^n\right )+4 a^4 B d^2 (1+m) n \left (a+b x^n\right )+2 a^3 B d^2 n^2 \left (a+b x^n\right )^2+A b^3 c^2 \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )-a b^2 B c^2 \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )-2 a A b^2 c d \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )+2 a^2 b B c d \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )+a^2 A b d^2 \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )-a^3 B d^2 \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )+2 A b^3 c^2 m \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )-2 a b^2 B c^2 m \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )-4 a A b^2 c d m \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )+4 a^2 b B c d m \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )+2 a^2 A b d^2 m \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )-2 a^3 B d^2 m \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )+A b^3 c^2 m^2 \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )-a b^2 B c^2 m^2 \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )-2 a A b^2 c d m^2 \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )+2 a^2 b B c d m^2 \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )+a^2 A b d^2 m^2 \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )-a^3 B d^2 m^2 \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )-3 A b^3 c^2 n \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )+a b^2 B c^2 n \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )+2 a A b^2 c d n \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )+2 a^2 b B c d n \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )+a^2 A b d^2 n \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )-3 a^3 B d^2 n \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )-3 A b^3 c^2 m n \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )+a b^2 B c^2 m n \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )+2 a A b^2 c d m n \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )+2 a^2 b B c d m n \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )+a^2 A b d^2 m n \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )-3 a^3 B d^2 m n \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )+2 A b^3 c^2 n^2 \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )-2 a^3 B d^2 n^2 \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )\right )}{2 a^3 b^3 (1+m) n^2 \left (a+b x^n\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(e*x)^m*(a^2*A*b^3*c^2*(1 + m)*n - a^3*b^2*B*c^2*(1 + m)*n - 2*a^3*A*b^2*c*d*(1 + m)*n + 2*a^4*b*B*c*d*(1 +
m)*n + a^4*A*b*d^2*(1 + m)*n - a^5*B*d^2*(1 + m)*n - a*A*b^3*c^2*(1 + m)*(a + b*x^n) + a^2*b^2*B*c^2*(1 + m)*
(a + b*x^n) + 2*a^2*A*b^2*c*d*(1 + m)*(a + b*x^n) - 2*a^3*b*B*c*d*(1 + m)*(a + b*x^n) - a^3*A*b*d^2*(1 + m)*(a
+ b*x^n) + a^4*B*d^2*(1 + m)*(a + b*x^n) - a*A*b^3*c^2*m*(1 + m)*(a + b*x^n) + a^2*b^2*B*c^2*m*(1 + m)*(a + b
*x^n) + 2*a^2*A*b^2*c*d*m*(1 + m)*(a + b*x^n) - 2*a^3*b*B*c*d*m*(1 + m)*(a + b*x^n) - a^3*A*b*d^2*m*(1 + m)*(a
+ b*x^n) + a^4*B*d^2*m*(1 + m)*(a + b*x^n) + 2*a*A*b^3*c^2*(1 + m)*n*(a + b*x^n) - 4*a^3*b*B*c*d*(1 + m)*n*(a
+ b*x^n) - 2*a^3*A*b*d^2*(1 + m)*n*(a + b*x^n) + 4*a^4*B*d^2*(1 + m)*n*(a + b*x^n) + 2*a^3*B*d^2*n^2*(a + b*x
^n)^2 + A*b^3*c^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - a*b^2*B*c^2*(a
+ b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 2*a*A*b^2*c*d*(a + b*x^n)^2*Hypergeo
metric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 2*a^2*b*B*c*d*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m
)/n, (1 + m + n)/n, -((b*x^n)/a)] + a^2*A*b*d^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -
((b*x^n)/a)] - a^3*B*d^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 2*A*b^3*
c^2*m*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 2*a*b^2*B*c^2*m*(a + b*x^n)
^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 4*a*A*b^2*c*d*m*(a + b*x^n)^2*Hypergeometric
2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 4*a^2*b*B*c*d*m*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n,
(1 + m + n)/n, -((b*x^n)/a)] + 2*a^2*A*b*d^2*m*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -
((b*x^n)/a)] - 2*a^3*B*d^2*m*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + A*b^
3*c^2*m^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - a*b^2*B*c^2*m^2*(a + b*
x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 2*a*A*b^2*c*d*m^2*(a + b*x^n)^2*Hypergeo
metric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 2*a^2*b*B*c*d*m^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1
+ m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a^2*A*b*d^2*m^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m +
n)/n, -((b*x^n)/a)] - a^3*B*d^2*m^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)
] - 3*A*b^3*c^2*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a*b^2*B*c^2*n*(
a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 2*a*A*b^2*c*d*n*(a + b*x^n)^2*Hype
rgeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 2*a^2*b*B*c*d*n*(a + b*x^n)^2*Hypergeometric2F1[1,
(1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a^2*A*b*d^2*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m +
n)/n, -((b*x^n)/a)] - 3*a^3*B*d^2*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)
] - 3*A*b^3*c^2*m*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a*b^2*B*c^2*m
*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 2*a*A*b^2*c*d*m*n*(a + b*x^n)^
2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 2*a^2*b*B*c*d*m*n*(a + b*x^n)^2*Hypergeometri
c2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a^2*A*b*d^2*m*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n
, (1 + m + n)/n, -((b*x^n)/a)] - 3*a^3*B*d^2*m*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n,
-((b*x^n)/a)] + 2*A*b^3*c^2*n^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 2
*a^3*B*d^2*n^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]))/(2*a^3*b^3*(1 + m)
*n^2*(a + b*x^n)^2)

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (A +B \,x^{n}\right ) \left (c +d \,x^{n}\right )^{2}}{\left (a +b \,x^{n}\right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

[Out]

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

[Out]

integrate((B*x^n + A)*(d*x^n + c)^2*(x*e)^m/(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 {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,{\left (c+d\,x^n\right )}^2}{{\left (a+b\,x^n\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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