3.32 \(\int \frac {(e x)^m (A+B x^n)}{(c+d x^n)^2} \, dx\)
Optimal. Leaf size=107 \[ \frac {(e x)^{m+1} (B c (m+1)-A d (m-n+1)) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {d x^n}{c}\right )}{c^2 d e (m+1) n}-\frac {(e x)^{m+1} (B c-A d)}{c d e n \left (c+d x^n\right )} \]
[Out]
-(-A*d+B*c)*(e*x)^(1+m)/c/d/e/n/(c+d*x^n)+(B*c*(1+m)-A*d*(1+m-n))*(e*x)^(1+m)*hypergeom([1, (1+m)/n],[(1+m+n)/
n],-d*x^n/c)/c^2/d/e/(1+m)/n
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 107, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used =
{457, 364} \[ \frac {(e x)^{m+1} (B c (m+1)-A d (m-n+1)) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {d x^n}{c}\right )}{c^2 d e (m+1) n}-\frac {(e x)^{m+1} (B c-A d)}{c d e n \left (c+d x^n\right )} \]
Antiderivative was successfully verified.
[In]
Int[((e*x)^m*(A + B*x^n))/(c + d*x^n)^2,x]
[Out]
-(((B*c - A*d)*(e*x)^(1 + m))/(c*d*e*n*(c + d*x^n))) + ((B*c*(1 + m) - A*d*(1 + m - n))*(e*x)^(1 + m)*Hypergeo
metric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c^2*d*e*(1 + m)*n)
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])
Rule 457
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d
)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b*e*n*(p + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b
*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] &&
LeQ[-1, m, -(n*(p + 1))]))
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx &=-\frac {(B c-A d) (e x)^{1+m}}{c d e n \left (c+d x^n\right )}+\frac {(B c (1+m)-A d (1+m-n)) \int \frac {(e x)^m}{c+d x^n} \, dx}{c d n}\\ &=-\frac {(B c-A d) (e x)^{1+m}}{c d e n \left (c+d x^n\right )}+\frac {(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 d e (1+m) n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 83, normalized size = 0.78 \[ \frac {x (e x)^m \left ((A d-B c) \, _2F_1\left (2,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {d x^n}{c}\right )+B c \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {d x^n}{c}\right )\right )}{c^2 d (m+1)} \]
Antiderivative was successfully verified.
[In]
Integrate[((e*x)^m*(A + B*x^n))/(c + d*x^n)^2,x]
[Out]
(x*(e*x)^m*(B*c*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + (-(B*c) + A*d)*Hypergeometric2F
1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(c^2*d*(1 + m))
________________________________________________________________________________________
fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{d^{2} x^{2 \, n} + 2 \, c d x^{n} + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(A+B*x^n)/(c+d*x^n)^2,x, algorithm="fricas")
[Out]
integral((B*x^n + A)*(e*x)^m/(d^2*x^(2*n) + 2*c*d*x^n + c^2), x)
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(A+B*x^n)/(c+d*x^n)^2,x, algorithm="giac")
[Out]
integrate((B*x^n + A)*(e*x)^m/(d*x^n + c)^2, x)
________________________________________________________________________________________
maple [F] time = 0.66, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{n}+A \right ) \left (e x \right )^{m}}{\left (d \,x^{n}+c \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m*(B*x^n+A)/(d*x^n+c)^2,x)
[Out]
int((e*x)^m*(B*x^n+A)/(d*x^n+c)^2,x)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (B c e^{m} - A d e^{m}\right )} x x^{m}}{c d^{2} n x^{n} + c^{2} d n} - {\left (A d e^{m} {\left (m - n + 1\right )} - B c e^{m} {\left (m + 1\right )}\right )} \int \frac {x^{m}}{c d^{2} n x^{n} + c^{2} d n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(A+B*x^n)/(c+d*x^n)^2,x, algorithm="maxima")
[Out]
-(B*c*e^m - A*d*e^m)*x*x^m/(c*d^2*n*x^n + c^2*d*n) - (A*d*e^m*(m - n + 1) - B*c*e^m*(m + 1))*integrate(x^m/(c*
d^2*n*x^n + c^2*d*n), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )}{{\left (c+d\,x^n\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((e*x)^m*(A + B*x^n))/(c + d*x^n)^2,x)
[Out]
int(((e*x)^m*(A + B*x^n))/(c + d*x^n)^2, x)
________________________________________________________________________________________
sympy [C] time = 17.42, size = 1897, normalized size = 17.73 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)**m*(A+B*x**n)/(c+d*x**n)**2,x)
[Out]
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*gam
ma(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*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*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c*(c*n**3*g
amma(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)*gamm
a(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*gamma
(m/n + 1 + 1/n)))) + 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*lerchph
i(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*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*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)*gamm
a(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_p
olar(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)*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))))
________________________________________________________________________________________