3.31 \(\int \frac{(e x)^m (a+b x^n) (A+B x^n)}{(c+d x^n)^2} \, dx\)
Optimal. Leaf size=178 \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (A d (b c (m+1)-a d (m-n+1))+B c (a d (m+1)-b c (m+n+1)))}{c^2 d^2 e (m+1) n}-\frac{(e x)^{m+1} (b c-a d) \left (A+B x^n\right )}{c d e n \left (c+d x^n\right )}-\frac{B (e x)^{m+1} (a d (m+1)-b c (m+n+1))}{c d^2 e (m+1) n} \]
[Out]
-((B*(a*d*(1 + m) - b*c*(1 + m + n))*(e*x)^(1 + m))/(c*d^2*e*(1 + m)*n)) - ((b*c - a*d)*(e*x)^(1 + m)*(A + B*x
^n))/(c*d*e*n*(c + d*x^n)) + ((A*d*(b*c*(1 + m) - a*d*(1 + m - n)) + B*c*(a*d*(1 + m) - b*c*(1 + m + n)))*(e*x
)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c^2*d^2*e*(1 + m)*n)
________________________________________________________________________________________
Rubi [A] time = 0.25265, antiderivative size = 178, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used =
{594, 459, 364} \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (A d (b c (m+1)-a d (m-n+1))+B c (a d (m+1)-b c (m+n+1)))}{c^2 d^2 e (m+1) n}-\frac{(e x)^{m+1} (b c-a d) \left (A+B x^n\right )}{c d e n \left (c+d x^n\right )}-\frac{B (e x)^{m+1} (a d (m+1)-b c (m+n+1))}{c d^2 e (m+1) n} \]
Antiderivative was successfully verified.
[In]
Int[((e*x)^m*(a + b*x^n)*(A + B*x^n))/(c + d*x^n)^2,x]
[Out]
-((B*(a*d*(1 + m) - b*c*(1 + m + n))*(e*x)^(1 + m))/(c*d^2*e*(1 + m)*n)) - ((b*c - a*d)*(e*x)^(1 + m)*(A + B*x
^n))/(c*d*e*n*(c + d*x^n)) + ((A*d*(b*c*(1 + m) - a*d*(1 + m - n)) + B*c*(a*d*(1 + m) - b*c*(1 + m + n)))*(e*x
)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c^2*d^2*e*(1 + m)*n)
Rule 594
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] + Dist[
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])
Rule 459
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 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 )}{\left (c+d x^n\right )^2} \, dx &=-\frac{(b c-a d) (e x)^{1+m} \left (A+B x^n\right )}{c d e n \left (c+d x^n\right )}-\frac{\int \frac{(e x)^m \left (-A (b c (1+m)-a d (1+m-n))+B (a d (1+m)-b c (1+m+n)) x^n\right )}{c+d x^n} \, dx}{c d n}\\ &=-\frac{B (a d (1+m)-b c (1+m+n)) (e x)^{1+m}}{c d^2 e (1+m) n}-\frac{(b c-a d) (e x)^{1+m} \left (A+B x^n\right )}{c d e n \left (c+d x^n\right )}+\frac{(A d (b c (1+m)-a d (1+m-n))+B c (a d (1+m)-b c (1+m+n))) \int \frac{(e x)^m}{c+d x^n} \, dx}{c d^2 n}\\ &=-\frac{B (a d (1+m)-b c (1+m+n)) (e x)^{1+m}}{c d^2 e (1+m) n}-\frac{(b c-a d) (e x)^{1+m} \left (A+B x^n\right )}{c d e n \left (c+d x^n\right )}+\frac{(A d (b c (1+m)-a d (1+m-n))+B c (a d (1+m)-b c (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 d^2 e (1+m) n}\\ \end{align*}
Mathematica [A] time = 0.15861, size = 110, normalized size = 0.62 \[ \frac{x (e x)^m \left (c (a B d+A b d-2 b B c) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )+(b c-a d) (B c-A d) \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )+b B c^2\right )}{c^2 d^2 (m+1)} \]
Antiderivative was successfully verified.
[In]
Integrate[((e*x)^m*(a + b*x^n)*(A + B*x^n))/(c + d*x^n)^2,x]
[Out]
(x*(e*x)^m*(b*B*c^2 + c*(-2*b*B*c + A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)
] + (b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(c^2*d^2*(1 + m))
________________________________________________________________________________________
Maple [F] time = 0.349, 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 ) }{ \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)/(c+d*x^n)^2,x)
[Out]
int((e*x)^m*(a+b*x^n)*(A+B*x^n)/(c+d*x^n)^2,x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -{\left ({\left (a d^{2} e^{m}{\left (m - n + 1\right )} - b c d e^{m}{\left (m + 1\right )}\right )} A +{\left (b c^{2} e^{m}{\left (m + n + 1\right )} - a c d e^{m}{\left (m + 1\right )}\right )} B\right )} \int \frac{x^{m}}{c d^{3} n x^{n} + c^{2} d^{2} n}\,{d x} + \frac{B b c d e^{m} n x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )} -{\left ({\left (b c d e^{m}{\left (m + 1\right )} - a d^{2} e^{m}{\left (m + 1\right )}\right )} A -{\left (b c^{2} e^{m}{\left (m + n + 1\right )} - a c d e^{m}{\left (m + 1\right )}\right )} B\right )} x x^{m}}{{\left (m n + n\right )} c d^{3} x^{n} +{\left (m n + n\right )} c^{2} d^{2}} \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)/(c+d*x^n)^2,x, algorithm="maxima")
[Out]
-((a*d^2*e^m*(m - n + 1) - b*c*d*e^m*(m + 1))*A + (b*c^2*e^m*(m + n + 1) - a*c*d*e^m*(m + 1))*B)*integrate(x^m
/(c*d^3*n*x^n + c^2*d^2*n), x) + (B*b*c*d*e^m*n*x*e^(m*log(x) + n*log(x)) - ((b*c*d*e^m*(m + 1) - a*d^2*e^m*(m
+ 1))*A - (b*c^2*e^m*(m + n + 1) - a*c*d*e^m*(m + 1))*B)*x*x^m)/((m*n + n)*c*d^3*x^n + (m*n + n)*c^2*d^2)
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B b x^{2 \, n} + A a +{\left (B a + A b\right )} x^{n}\right )} \left (e x\right )^{m}}{d^{2} x^{2 \, n} + 2 \, c d x^{n} + c^{2}}, 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)/(c+d*x^n)^2,x, algorithm="fricas")
[Out]
integral((B*b*x^(2*n) + A*a + (B*a + A*b)*x^n)*(e*x)^m/(d^2*x^(2*n) + 2*c*d*x^n + c^2), x)
________________________________________________________________________________________
Sympy [C] time = 47.9481, size = 4129, normalized size = 23.2 \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)/(c+d*x**n)**2,x)
[Out]
A*a*(-e**m*m**2*x*x**m*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c*(c*n**3*gamma(m/n
+ 1 + 1/n) + d*n**3*x**n*gamma(m/n + 1 + 1/n))) + e**m*m*n*x*x**m*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n +
1/n)*gamma(m/n + 1/n)/(c*(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma(m/n + 1 + 1/n))) + e**m*m*n*x*x**m*g
amma(m/n + 1/n)/(c*(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma(m/n + 1 + 1/n))) - 2*e**m*m*x*x**m*lerchph
i(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c*(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma
(m/n + 1 + 1/n))) + e**m*n*x*x**m*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c*(c*n**3
*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma(m/n + 1 + 1/n))) + e**m*n*x*x**m*gamma(m/n + 1/n)/(c*(c*n**3*gamma(m
/n + 1 + 1/n) + d*n**3*x**n*gamma(m/n + 1 + 1/n))) - e**m*x*x**m*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1
/n)*gamma(m/n + 1/n)/(c*(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma(m/n + 1 + 1/n))) - d*e**m*m**2*x*x**m
*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c**2*(c*n**3*gamma(m/n + 1 + 1/n) + d
*n**3*x**n*gamma(m/n + 1 + 1/n))) + d*e**m*m*n*x*x**m*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*ga
mma(m/n + 1/n)/(c**2*(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma(m/n + 1 + 1/n))) - 2*d*e**m*m*x*x**m*x**
n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c**2*(c*n**3*gamma(m/n + 1 + 1/n) + d*n**
3*x**n*gamma(m/n + 1 + 1/n))) + d*e**m*n*x*x**m*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/
n + 1/n)/(c**2*(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma(m/n + 1 + 1/n))) - d*e**m*x*x**m*x**n*lerchphi
(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c**2*(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gam
ma(m/n + 1 + 1/n)))) + A*b*(-e**m*m**2*x*x**m*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(
m/n + 1 + 1/n)/(c*(c*n**3*gamma(m/n + 2 + 1/n) + d*n**3*x**n*gamma(m/n + 2 + 1/n))) - e**m*m*n*x*x**m*x**n*ler
chphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(c*(c*n**3*gamma(m/n + 2 + 1/n) + d*n**
3*x**n*gamma(m/n + 2 + 1/n))) + e**m*m*n*x*x**m*x**n*gamma(m/n + 1 + 1/n)/(c*(c*n**3*gamma(m/n + 2 + 1/n) + d*
n**3*x**n*gamma(m/n + 2 + 1/n))) - 2*e**m*m*x*x**m*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*g
amma(m/n + 1 + 1/n)/(c*(c*n**3*gamma(m/n + 2 + 1/n) + d*n**3*x**n*gamma(m/n + 2 + 1/n))) + e**m*n**2*x*x**m*x*
*n*gamma(m/n + 1 + 1/n)/(c*(c*n**3*gamma(m/n + 2 + 1/n) + d*n**3*x**n*gamma(m/n + 2 + 1/n))) - e**m*n*x*x**m*x
**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(c*(c*n**3*gamma(m/n + 2 + 1/n)
+ d*n**3*x**n*gamma(m/n + 2 + 1/n))) + e**m*n*x*x**m*x**n*gamma(m/n + 1 + 1/n)/(c*(c*n**3*gamma(m/n + 2 + 1/n)
+ d*n**3*x**n*gamma(m/n + 2 + 1/n))) - e**m*x*x**m*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*
gamma(m/n + 1 + 1/n)/(c*(c*n**3*gamma(m/n + 2 + 1/n) + d*n**3*x**n*gamma(m/n + 2 + 1/n))) - d*e**m*m**2*x*x**m
*x**(2*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(c**2*(c*n**3*gamma(m/n +
2 + 1/n) + d*n**3*x**n*gamma(m/n + 2 + 1/n))) - d*e**m*m*n*x*x**m*x**(2*n)*lerchphi(d*x**n*exp_polar(I*pi)/c,
1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(c**2*(c*n**3*gamma(m/n + 2 + 1/n) + d*n**3*x**n*gamma(m/n + 2 + 1/n)))
- 2*d*e**m*m*x*x**m*x**(2*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(c**2*
(c*n**3*gamma(m/n + 2 + 1/n) + d*n**3*x**n*gamma(m/n + 2 + 1/n))) - d*e**m*n*x*x**m*x**(2*n)*lerchphi(d*x**n*e
xp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(c**2*(c*n**3*gamma(m/n + 2 + 1/n) + d*n**3*x**n*gamm
a(m/n + 2 + 1/n))) - d*e**m*x*x**m*x**(2*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1
+ 1/n)/(c**2*(c*n**3*gamma(m/n + 2 + 1/n) + d*n**3*x**n*gamma(m/n + 2 + 1/n)))) + B*a*(-e**m*m**2*x*x**m*x**n
*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(c*(c*n**3*gamma(m/n + 2 + 1/n) + d
*n**3*x**n*gamma(m/n + 2 + 1/n))) - e**m*m*n*x*x**m*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*
gamma(m/n + 1 + 1/n)/(c*(c*n**3*gamma(m/n + 2 + 1/n) + d*n**3*x**n*gamma(m/n + 2 + 1/n))) + e**m*m*n*x*x**m*x*
*n*gamma(m/n + 1 + 1/n)/(c*(c*n**3*gamma(m/n + 2 + 1/n) + d*n**3*x**n*gamma(m/n + 2 + 1/n))) - 2*e**m*m*x*x**m
*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(c*(c*n**3*gamma(m/n + 2 + 1/n
) + d*n**3*x**n*gamma(m/n + 2 + 1/n))) + e**m*n**2*x*x**m*x**n*gamma(m/n + 1 + 1/n)/(c*(c*n**3*gamma(m/n + 2 +
1/n) + d*n**3*x**n*gamma(m/n + 2 + 1/n))) - e**m*n*x*x**m*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1
+ 1/n)*gamma(m/n + 1 + 1/n)/(c*(c*n**3*gamma(m/n + 2 + 1/n) + d*n**3*x**n*gamma(m/n + 2 + 1/n))) + e**m*n*x*x*
*m*x**n*gamma(m/n + 1 + 1/n)/(c*(c*n**3*gamma(m/n + 2 + 1/n) + d*n**3*x**n*gamma(m/n + 2 + 1/n))) - e**m*x*x**
m*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(c*(c*n**3*gamma(m/n + 2 + 1/
n) + d*n**3*x**n*gamma(m/n + 2 + 1/n))) - d*e**m*m**2*x*x**m*x**(2*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/
n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(c**2*(c*n**3*gamma(m/n + 2 + 1/n) + d*n**3*x**n*gamma(m/n + 2 + 1/n))) - d*
e**m*m*n*x*x**m*x**(2*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(c**2*(c*n*
*3*gamma(m/n + 2 + 1/n) + d*n**3*x**n*gamma(m/n + 2 + 1/n))) - 2*d*e**m*m*x*x**m*x**(2*n)*lerchphi(d*x**n*exp_
polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(c**2*(c*n**3*gamma(m/n + 2 + 1/n) + d*n**3*x**n*gamma(m
/n + 2 + 1/n))) - d*e**m*n*x*x**m*x**(2*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1
+ 1/n)/(c**2*(c*n**3*gamma(m/n + 2 + 1/n) + d*n**3*x**n*gamma(m/n + 2 + 1/n))) - d*e**m*x*x**m*x**(2*n)*lerchp
hi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(c**2*(c*n**3*gamma(m/n + 2 + 1/n) + d*n**
3*x**n*gamma(m/n + 2 + 1/n)))) + B*b*(-e**m*m**2*x*x**m*x**(2*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 2
+ 1/n)*gamma(m/n + 2 + 1/n)/(c*(c*n**3*gamma(m/n + 3 + 1/n) + d*n**3*x**n*gamma(m/n + 3 + 1/n))) - 3*e**m*m*n
*x*x**m*x**(2*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(c*(c*n**3*gamma(m/
n + 3 + 1/n) + d*n**3*x**n*gamma(m/n + 3 + 1/n))) + e**m*m*n*x*x**m*x**(2*n)*gamma(m/n + 2 + 1/n)/(c*(c*n**3*g
amma(m/n + 3 + 1/n) + d*n**3*x**n*gamma(m/n + 3 + 1/n))) - 2*e**m*m*x*x**m*x**(2*n)*lerchphi(d*x**n*exp_polar(
I*pi)/c, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(c*(c*n**3*gamma(m/n + 3 + 1/n) + d*n**3*x**n*gamma(m/n + 3 +
1/n))) - 2*e**m*n**2*x*x**m*x**(2*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)
/(c*(c*n**3*gamma(m/n + 3 + 1/n) + d*n**3*x**n*gamma(m/n + 3 + 1/n))) + 2*e**m*n**2*x*x**m*x**(2*n)*gamma(m/n
+ 2 + 1/n)/(c*(c*n**3*gamma(m/n + 3 + 1/n) + d*n**3*x**n*gamma(m/n + 3 + 1/n))) - 3*e**m*n*x*x**m*x**(2*n)*ler
chphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(c*(c*n**3*gamma(m/n + 3 + 1/n) + d*n**
3*x**n*gamma(m/n + 3 + 1/n))) + e**m*n*x*x**m*x**(2*n)*gamma(m/n + 2 + 1/n)/(c*(c*n**3*gamma(m/n + 3 + 1/n) +
d*n**3*x**n*gamma(m/n + 3 + 1/n))) - e**m*x*x**m*x**(2*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 2 + 1/n)
*gamma(m/n + 2 + 1/n)/(c*(c*n**3*gamma(m/n + 3 + 1/n) + d*n**3*x**n*gamma(m/n + 3 + 1/n))) - d*e**m*m**2*x*x**
m*x**(3*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(c**2*(c*n**3*gamma(m/n +
3 + 1/n) + d*n**3*x**n*gamma(m/n + 3 + 1/n))) - 3*d*e**m*m*n*x*x**m*x**(3*n)*lerchphi(d*x**n*exp_polar(I*pi)/
c, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(c**2*(c*n**3*gamma(m/n + 3 + 1/n) + d*n**3*x**n*gamma(m/n + 3 + 1/n
))) - 2*d*e**m*m*x*x**m*x**(3*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(c*
*2*(c*n**3*gamma(m/n + 3 + 1/n) + d*n**3*x**n*gamma(m/n + 3 + 1/n))) - 2*d*e**m*n**2*x*x**m*x**(3*n)*lerchphi(
d*x**n*exp_polar(I*pi)/c, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(c**2*(c*n**3*gamma(m/n + 3 + 1/n) + d*n**3*x
**n*gamma(m/n + 3 + 1/n))) - 3*d*e**m*n*x*x**m*x**(3*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 2 + 1/n)*g
amma(m/n + 2 + 1/n)/(c**2*(c*n**3*gamma(m/n + 3 + 1/n) + d*n**3*x**n*gamma(m/n + 3 + 1/n))) - d*e**m*x*x**m*x*
*(3*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(c**2*(c*n**3*gamma(m/n + 3 +
1/n) + d*n**3*x**n*gamma(m/n + 3 + 1/n))))
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{n} + A\right )}{\left (b x^{n} + a\right )} \left (e x\right )^{m}}{{\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)/(c+d*x^n)^2,x, algorithm="giac")
[Out]
integrate((B*x^n + A)*(b*x^n + a)*(e*x)^m/(d*x^n + c)^2, x)