3.40 \(\int \frac{(e x)^m (A+B x^2)}{(c+d x^2)^3} \, dx\)
Optimal. Leaf size=103 \[ \frac{(e x)^{m+1} (A d (3-m)+B c (m+1)) \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{4 c^3 d e (m+1)}-\frac{(e x)^{m+1} (B c-A d)}{4 c d e \left (c+d x^2\right )^2} \]
[Out]
-((B*c - A*d)*(e*x)^(1 + m))/(4*c*d*e*(c + d*x^2)^2) + ((A*d*(3 - m) + B*c*(1 + m))*(e*x)^(1 + m)*Hypergeometr
ic2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(4*c^3*d*e*(1 + m))
________________________________________________________________________________________
Rubi [A] time = 0.0474653, antiderivative size = 103, normalized size of antiderivative = 1.,
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} \[ \frac{(e x)^{m+1} (A d (3-m)+B c (m+1)) \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{4 c^3 d e (m+1)}-\frac{(e x)^{m+1} (B c-A d)}{4 c d e \left (c+d x^2\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[((e*x)^m*(A + B*x^2))/(c + d*x^2)^3,x]
[Out]
-((B*c - A*d)*(e*x)^(1 + m))/(4*c*d*e*(c + d*x^2)^2) + ((A*d*(3 - m) + B*c*(1 + m))*(e*x)^(1 + m)*Hypergeometr
ic2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(4*c^3*d*e*(1 + m))
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))]))
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 )^3} \, dx &=-\frac{(B c-A d) (e x)^{1+m}}{4 c d e \left (c+d x^2\right )^2}+\frac{(-A d (-3+m)+B c (1+m)) \int \frac{(e x)^m}{\left (c+d x^2\right )^2} \, dx}{4 c d}\\ &=-\frac{(B c-A d) (e x)^{1+m}}{4 c d e \left (c+d x^2\right )^2}+\frac{(A d (3-m)+B c (1+m)) (e x)^{1+m} \, _2F_1\left (2,\frac{1+m}{2};\frac{3+m}{2};-\frac{d x^2}{c}\right )}{4 c^3 d e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0613782, size = 81, normalized size = 0.79 \[ \frac{x (e x)^m \left ((A d-B c) \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )+B c \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )\right )}{c^3 d (m+1)} \]
Antiderivative was successfully verified.
[In]
Integrate[((e*x)^m*(A + B*x^2))/(c + d*x^2)^3,x]
[Out]
(x*(e*x)^m*(B*c*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] + (-(B*c) + A*d)*Hypergeometric2F1[3,
(1 + m)/2, (3 + m)/2, -((d*x^2)/c)]))/(c^3*d*(1 + m))
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Maple [F] time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( B{x}^{2}+A \right ) \left ( ex \right ) ^{m}}{ \left ( d{x}^{2}+c \right ) ^{3}}}\, 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)^3,x)
[Out]
int((e*x)^m*(B*x^2+A)/(d*x^2+c)^3,x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{{\left (d x^{2} + c\right )}^{3}}\,{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)^3,x, algorithm="maxima")
[Out]
integrate((B*x^2 + A)*(e*x)^m/(d*x^2 + c)^3, x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{d^{3} x^{6} + 3 \, c d^{2} x^{4} + 3 \, c^{2} d x^{2} + c^{3}}, 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)^3,x, algorithm="fricas")
[Out]
integral((B*x^2 + A)*(e*x)^m/(d^3*x^6 + 3*c*d^2*x^4 + 3*c^2*d*x^2 + c^3), x)
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Sympy [C] time = 144.324, size = 3172, normalized size = 30.8 \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)**3,x)
[Out]
A*(c**2*e**m*m**3*x*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2
+ 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) - 3*c**2*e**m*m**2*x*x**m*lerch
phi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(
m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) - 2*c**2*e**m*m**2*x*x**m*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2
+ 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) - c**2*e**m*m*x*x**m*lerchphi(
d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2
+ 3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) + 8*c**2*e**m*m*x*x**m*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2)
+ 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) + 3*c**2*e**m*x*x**m*lerchphi(d*x**2*
exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2)
+ 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) + 10*c**2*e**m*x*x**m*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c*
*4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) + 2*c*d*e**m*m**3*x**3*x**m*lerchphi(d*x**2*e
xp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) +
32*c**3*d**2*x**4*gamma(m/2 + 3/2)) - 6*c*d*e**m*m**2*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1
/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2
+ 3/2)) - 2*c*d*e**m*m**2*x**3*x**m*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3
/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) - 2*c*d*e**m*m*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 +
1/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m
/2 + 3/2)) + 4*c*d*e**m*m*x**3*x**m*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/
2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) + 6*c*d*e**m*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/
2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2
+ 3/2)) + 6*c*d*e**m*x**3*x**m*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) +
32*c**3*d**2*x**4*gamma(m/2 + 3/2)) + d**2*e**m*m**3*x**5*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2
)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2 +
3/2)) - 3*d**2*e**m*m**2*x**5*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*c**5
*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) - d**2*e**m*m*x**5*x
**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x*
*2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) + 3*d**2*e**m*x**5*x**m*lerchphi(d*x**2*exp_polar(I*
pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d*
*2*x**4*gamma(m/2 + 3/2))) + B*(c**2*e**m*m**3*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamm
a(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2))
+ 3*c**2*e**m*m**2*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(32*c**5*gamma
(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2)) - 2*c**2*e**m*m**2*x**3*x*
*m*gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2
+ 5/2)) - c**2*e**m*m*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(32*c**5*gam
ma(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2)) - 3*c**2*e**m*x**3*x**m*
lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*x**2*g
amma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2)) + 18*c**2*e**m*x**3*x**m*gamma(m/2 + 3/2)/(32*c**5*gamma
(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2)) + 2*c*d*e**m*m**3*x**5*x**
m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*x**2
*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2)) + 6*c*d*e**m*m**2*x**5*x**m*lerchphi(d*x**2*exp_polar(
I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32*c**3*
d**2*x**4*gamma(m/2 + 5/2)) - 2*c*d*e**m*m**2*x**5*x**m*gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d
*x**2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2)) - 2*c*d*e**m*m*x**5*x**m*lerchphi(d*x**2*exp_pola
r(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32*c**
3*d**2*x**4*gamma(m/2 + 5/2)) - 4*c*d*e**m*m*x**5*x**m*gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*
x**2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2)) - 6*c*d*e**m*x**5*x**m*lerchphi(d*x**2*exp_polar(I
*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32*c**3*d
**2*x**4*gamma(m/2 + 5/2)) + 6*c*d*e**m*x**5*x**m*gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*x**2*
gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2)) + d**2*e**m*m**3*x**7*x**m*lerchphi(d*x**2*exp_polar(I*
pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32*c**3*d*
*2*x**4*gamma(m/2 + 5/2)) + 3*d**2*e**m*m**2*x**7*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(
m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2)) -
d**2*e**m*m*x**7*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 +
5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2)) - 3*d**2*e**m*x**7*x**m*lerchphi(
d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2
+ 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2)))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{{\left (d x^{2} + c\right )}^{3}}\,{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)^3,x, algorithm="giac")
[Out]
integrate((B*x^2 + A)*(e*x)^m/(d*x^2 + c)^3, x)