3.6 \(\int \frac{(e x)^m (A+B x^2) (c+d x^2)}{(a+b x^2)^2} \, dx\)
Optimal. Leaf size=171 \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) (A b (a d (m+1)+b (c-c m))+a B (b c (m+1)-a d (m+3)))}{2 a^2 b^2 e (m+1)}-\frac{d (e x)^{m+1} (A b (m+1)-a B (m+3))}{2 a b^2 e (m+1)}+\frac{\left (c+d x^2\right ) (e x)^{m+1} (A b-a B)}{2 a b e \left (a+b x^2\right )} \]
[Out]
-(d*(A*b*(1 + m) - a*B*(3 + m))*(e*x)^(1 + m))/(2*a*b^2*e*(1 + m)) + ((A*b - a*B)*(e*x)^(1 + m)*(c + d*x^2))/(
2*a*b*e*(a + b*x^2)) + ((a*B*(b*c*(1 + m) - a*d*(3 + m)) + A*b*(a*d*(1 + m) + b*(c - c*m)))*(e*x)^(1 + m)*Hype
rgeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(2*a^2*b^2*e*(1 + m))
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Rubi [A] time = 0.240919, antiderivative size = 171, 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 =
{577, 459, 364} \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) (A b (a d (m+1)+b (c-c m))+a B (b c (m+1)-a d (m+3)))}{2 a^2 b^2 e (m+1)}-\frac{d (e x)^{m+1} (A b (m+1)-a B (m+3))}{2 a b^2 e (m+1)}+\frac{\left (c+d x^2\right ) (e x)^{m+1} (A b-a B)}{2 a b e \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In]
Int[((e*x)^m*(A + B*x^2)*(c + d*x^2))/(a + b*x^2)^2,x]
[Out]
-(d*(A*b*(1 + m) - a*B*(3 + m))*(e*x)^(1 + m))/(2*a*b^2*e*(1 + m)) + ((A*b - a*B)*(e*x)^(1 + m)*(c + d*x^2))/(
2*a*b*e*(a + b*x^2)) + ((a*B*(b*c*(1 + m) - a*d*(3 + m)) + A*b*(a*d*(1 + m) + b*(c - c*m)))*(e*x)^(1 + m)*Hype
rgeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(2*a^2*b^2*e*(1 + m))
Rule 577
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}, x] &&
IGtQ[n, 0] && 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^2\right ) \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx &=\frac{(A b-a B) (e x)^{1+m} \left (c+d x^2\right )}{2 a b e \left (a+b x^2\right )}-\frac{\int \frac{(e x)^m \left (-c (A b (1-m)+a B (1+m))+d (A b (1+m)-a B (3+m)) x^2\right )}{a+b x^2} \, dx}{2 a b}\\ &=-\frac{d (A b (1+m)-a B (3+m)) (e x)^{1+m}}{2 a b^2 e (1+m)}+\frac{(A b-a B) (e x)^{1+m} \left (c+d x^2\right )}{2 a b e \left (a+b x^2\right )}+\frac{(a B (b c (1+m)-a d (3+m))+A b (a d (1+m)+b (c-c m))) \int \frac{(e x)^m}{a+b x^2} \, dx}{2 a b^2}\\ &=-\frac{d (A b (1+m)-a B (3+m)) (e x)^{1+m}}{2 a b^2 e (1+m)}+\frac{(A b-a B) (e x)^{1+m} \left (c+d x^2\right )}{2 a b e \left (a+b x^2\right )}+\frac{(a B (b c (1+m)-a d (3+m))+A b (a d (1+m)+b (c-c m))) (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{b x^2}{a}\right )}{2 a^2 b^2 e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.119886, size = 108, normalized size = 0.63 \[ \frac{x (e x)^m \left (a^2 B d+a \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) (-2 a B d+A b d+b B c)+(A b-a B) (b c-a d) \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )\right )}{a^2 b^2 (m+1)} \]
Antiderivative was successfully verified.
[In]
Integrate[((e*x)^m*(A + B*x^2)*(c + d*x^2))/(a + b*x^2)^2,x]
[Out]
(x*(e*x)^m*(a^2*B*d + a*(b*B*c + A*b*d - 2*a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + (
A*b - a*B)*(b*c - a*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a^2*b^2*(1 + m))
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Maple [F] time = 0.041, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( B{x}^{2}+A \right ) \left ( d{x}^{2}+c \right ) }{ \left ( b{x}^{2}+a \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m*(B*x^2+A)*(d*x^2+c)/(b*x^2+a)^2,x)
[Out]
int((e*x)^m*(B*x^2+A)*(d*x^2+c)/(b*x^2+a)^2,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )}{\left (d x^{2} + c\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(B*x^2+A)*(d*x^2+c)/(b*x^2+a)^2,x, algorithm="maxima")
[Out]
integrate((B*x^2 + A)*(d*x^2 + c)*(e*x)^m/(b*x^2 + a)^2, x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B d x^{4} +{\left (B c + A d\right )} x^{2} + A c\right )} \left (e x\right )^{m}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(B*x^2+A)*(d*x^2+c)/(b*x^2+a)^2,x, algorithm="fricas")
[Out]
integral((B*d*x^4 + (B*c + A*d)*x^2 + A*c)*(e*x)^m/(b^2*x^4 + 2*a*b*x^2 + a^2), x)
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Sympy [C] time = 82.7212, size = 2076, normalized size = 12.14 \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*(B*x**2+A)*(d*x**2+c)/(b*x**2+a)**2,x)
[Out]
A*c*(-a*e**m*m**2*x*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*a**3*gamma(m/2 +
3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/2)) + 2*a*e**m*m*x*x**m*gamma(m/2 + 1/2)/(8*a**3*gamma(m/2 + 3/2) + 8*a**2
*b*x**2*gamma(m/2 + 3/2)) + a*e**m*x*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8
*a**3*gamma(m/2 + 3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/2)) + 2*a*e**m*x*x**m*gamma(m/2 + 1/2)/(8*a**3*gamma(m/2
+ 3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/2)) - b*e**m*m**2*x**3*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1
/2)*gamma(m/2 + 1/2)/(8*a**3*gamma(m/2 + 3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/2)) + b*e**m*x**3*x**m*lerchphi(b*
x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*a**3*gamma(m/2 + 3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/
2))) + A*d*(-a*e**m*m**2*x**3*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*a**3*g
amma(m/2 + 5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2)) - 4*a*e**m*m*x**3*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1,
m/2 + 3/2)*gamma(m/2 + 3/2)/(8*a**3*gamma(m/2 + 5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2)) + 2*a*e**m*m*x**3*x**m
*gamma(m/2 + 3/2)/(8*a**3*gamma(m/2 + 5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2)) - 3*a*e**m*x**3*x**m*lerchphi(b*x
**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*a**3*gamma(m/2 + 5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2
)) + 6*a*e**m*x**3*x**m*gamma(m/2 + 3/2)/(8*a**3*gamma(m/2 + 5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2)) - b*e**m*m
**2*x**5*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*a**3*gamma(m/2 + 5/2) + 8*a
**2*b*x**2*gamma(m/2 + 5/2)) - 4*b*e**m*m*x**5*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2
+ 3/2)/(8*a**3*gamma(m/2 + 5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2)) - 3*b*e**m*x**5*x**m*lerchphi(b*x**2*exp_po
lar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*a**3*gamma(m/2 + 5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2))) + B*c*
(-a*e**m*m**2*x**3*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*a**3*gamma(m/2 +
5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2)) - 4*a*e**m*m*x**3*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)
*gamma(m/2 + 3/2)/(8*a**3*gamma(m/2 + 5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2)) + 2*a*e**m*m*x**3*x**m*gamma(m/2
+ 3/2)/(8*a**3*gamma(m/2 + 5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2)) - 3*a*e**m*x**3*x**m*lerchphi(b*x**2*exp_pol
ar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*a**3*gamma(m/2 + 5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2)) + 6*a*e*
*m*x**3*x**m*gamma(m/2 + 3/2)/(8*a**3*gamma(m/2 + 5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2)) - b*e**m*m**2*x**5*x*
*m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*a**3*gamma(m/2 + 5/2) + 8*a**2*b*x**2*
gamma(m/2 + 5/2)) - 4*b*e**m*m*x**5*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*
a**3*gamma(m/2 + 5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2)) - 3*b*e**m*x**5*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a
, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*a**3*gamma(m/2 + 5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2))) + B*d*(-a*e**m*m*
*2*x**5*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 5/2)*gamma(m/2 + 5/2)/(8*a**3*gamma(m/2 + 7/2) + 8*a*
*2*b*x**2*gamma(m/2 + 7/2)) - 8*a*e**m*m*x**5*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 5/2)*gamma(m/2
+ 5/2)/(8*a**3*gamma(m/2 + 7/2) + 8*a**2*b*x**2*gamma(m/2 + 7/2)) + 2*a*e**m*m*x**5*x**m*gamma(m/2 + 5/2)/(8*a
**3*gamma(m/2 + 7/2) + 8*a**2*b*x**2*gamma(m/2 + 7/2)) - 15*a*e**m*x**5*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a
, 1, m/2 + 5/2)*gamma(m/2 + 5/2)/(8*a**3*gamma(m/2 + 7/2) + 8*a**2*b*x**2*gamma(m/2 + 7/2)) + 10*a*e**m*x**5*x
**m*gamma(m/2 + 5/2)/(8*a**3*gamma(m/2 + 7/2) + 8*a**2*b*x**2*gamma(m/2 + 7/2)) - b*e**m*m**2*x**7*x**m*lerchp
hi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 5/2)*gamma(m/2 + 5/2)/(8*a**3*gamma(m/2 + 7/2) + 8*a**2*b*x**2*gamma(m/2
+ 7/2)) - 8*b*e**m*m*x**7*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 5/2)*gamma(m/2 + 5/2)/(8*a**3*gamm
a(m/2 + 7/2) + 8*a**2*b*x**2*gamma(m/2 + 7/2)) - 15*b*e**m*x**7*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2
+ 5/2)*gamma(m/2 + 5/2)/(8*a**3*gamma(m/2 + 7/2) + 8*a**2*b*x**2*gamma(m/2 + 7/2)))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )}{\left (d x^{2} + c\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(B*x^2+A)*(d*x^2+c)/(b*x^2+a)^2,x, algorithm="giac")
[Out]
integrate((B*x^2 + A)*(d*x^2 + c)*(e*x)^m/(b*x^2 + a)^2, x)