\(\int \frac {(e x)^m (a+b x^n) (A+B x^n)}{c+d x^n} \, dx\) [24]

Optimal result

Integrand size = 29, antiderivative size = 122 \[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{c+d x^n} \, dx=\frac {b B x^{1+n} (e x)^m}{d (1+m+n)}-\frac {(b B c-A b d-a B d) (e x)^{1+m}}{d^2 e (1+m)}+\frac {(b c-a d) (B c-A d) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c d^2 e (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {584, 20, 30, 371} \[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{c+d x^n} \, dx=\frac {(e x)^{m+1} (b c-a d) (B c-A d) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {d x^n}{c}\right )}{c d^2 e (m+1)}-\frac {(e x)^{m+1} (-a B d-A b d+b B c)}{d^2 e (m+1)}+\frac {b B x^{n+1} (e x)^m}{d (m+n+1)} \]

[In]

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

[Out]

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

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[b^IntPart[n]*((b*v)^FracPart[n]/(a^IntPart[n]
*(a*v)^FracPart[n])), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 584

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.78 \[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{c+d x^n} \, dx=\frac {x (e x)^m \left (\frac {-b B c+A b d+a B d}{1+m}+\frac {b B d x^n}{1+m+n}+\frac {(b c-a d) (B c-A d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c (1+m)}\right )}{d^2} \]

[In]

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

[Out]

(x*(e*x)^m*((-(b*B*c) + A*b*d + a*B*d)/(1 + m) + (b*B*d*x^n)/(1 + m + n) + ((b*c - a*d)*(B*c - A*d)*Hypergeome
tric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c*(1 + m))))/d^2

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.57 (sec) , antiderivative size = 872, normalized size of antiderivative = 7.15 \[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{c+d x^n} \, dx=\text {Too large to display} \]

[In]

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

[Out]

A*a*c**(m/n + 1/n)*c**(-m/n - 1 - 1/n)*e**m*m*x**(m + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamm
a(m/n + 1/n)/(n**2*gamma(m/n + 1 + 1/n)) + A*a*c**(m/n + 1/n)*c**(-m/n - 1 - 1/n)*e**m*x**(m + 1)*lerchphi(d*x
**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(n**2*gamma(m/n + 1 + 1/n)) + A*b*c**(-m/n - 2 - 1/n)*c*
*(m/n + 1 + 1/n)*e**m*m*x**(m + n + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/
n)/(n**2*gamma(m/n + 2 + 1/n)) + A*b*c**(-m/n - 2 - 1/n)*c**(m/n + 1 + 1/n)*e**m*x**(m + n + 1)*lerchphi(d*x**
n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(n*gamma(m/n + 2 + 1/n)) + A*b*c**(-m/n - 2 - 1/n)
*c**(m/n + 1 + 1/n)*e**m*x**(m + n + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1
/n)/(n**2*gamma(m/n + 2 + 1/n)) + B*a*c**(-m/n - 2 - 1/n)*c**(m/n + 1 + 1/n)*e**m*m*x**(m + n + 1)*lerchphi(d*
x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(n**2*gamma(m/n + 2 + 1/n)) + B*a*c**(-m/n - 2
- 1/n)*c**(m/n + 1 + 1/n)*e**m*x**(m + n + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n +
1 + 1/n)/(n*gamma(m/n + 2 + 1/n)) + B*a*c**(-m/n - 2 - 1/n)*c**(m/n + 1 + 1/n)*e**m*x**(m + n + 1)*lerchphi(d
*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(n**2*gamma(m/n + 2 + 1/n)) + B*b*c**(-m/n - 3
- 1/n)*c**(m/n + 2 + 1/n)*e**m*m*x**(m + 2*n + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 2 + 1/n)*gamma(
m/n + 2 + 1/n)/(n**2*gamma(m/n + 3 + 1/n)) + 2*B*b*c**(-m/n - 3 - 1/n)*c**(m/n + 2 + 1/n)*e**m*x**(m + 2*n + 1
)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(n*gamma(m/n + 3 + 1/n)) + B*b*c**
(-m/n - 3 - 1/n)*c**(m/n + 2 + 1/n)*e**m*x**(m + 2*n + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 2 + 1/n)
*gamma(m/n + 2 + 1/n)/(n**2*gamma(m/n + 3 + 1/n))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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