3.491 \(\int \frac{(e x)^m (A+B x)}{(a+c x^2)^3} \, dx\)
Optimal. Leaf size=91 \[ \frac{A (e x)^{m+1} \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )}{a^3 e (m+1)}+\frac{B (e x)^{m+2} \, _2F_1\left (3,\frac{m+2}{2};\frac{m+4}{2};-\frac{c x^2}{a}\right )}{a^3 e^2 (m+2)} \]
[Out]
(A*(e*x)^(1 + m)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((c*x^2)/a)])/(a^3*e*(1 + m)) + (B*(e*x)^(2 + m)*
Hypergeometric2F1[3, (2 + m)/2, (4 + m)/2, -((c*x^2)/a)])/(a^3*e^2*(2 + m))
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Rubi [A] time = 0.0389993, antiderivative size = 91, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used =
{808, 364} \[ \frac{A (e x)^{m+1} \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )}{a^3 e (m+1)}+\frac{B (e x)^{m+2} \, _2F_1\left (3,\frac{m+2}{2};\frac{m+4}{2};-\frac{c x^2}{a}\right )}{a^3 e^2 (m+2)} \]
Antiderivative was successfully verified.
[In]
Int[((e*x)^m*(A + B*x))/(a + c*x^2)^3,x]
[Out]
(A*(e*x)^(1 + m)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((c*x^2)/a)])/(a^3*e*(1 + m)) + (B*(e*x)^(2 + m)*
Hypergeometric2F1[3, (2 + m)/2, (4 + m)/2, -((c*x^2)/a)])/(a^3*e^2*(2 + m))
Rule 808
Int[((e_.)*(x_))^(m_)*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[f, Int[(e*x)^m*(a + c*
x^2)^p, x], x] + Dist[g/e, Int[(e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, p}, x] && !Ration
alQ[m] && !IGtQ[p, 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 (A+B x)}{\left (a+c x^2\right )^3} \, dx &=A \int \frac{(e x)^m}{\left (a+c x^2\right )^3} \, dx+\frac{B \int \frac{(e x)^{1+m}}{\left (a+c x^2\right )^3} \, dx}{e}\\ &=\frac{A (e x)^{1+m} \, _2F_1\left (3,\frac{1+m}{2};\frac{3+m}{2};-\frac{c x^2}{a}\right )}{a^3 e (1+m)}+\frac{B (e x)^{2+m} \, _2F_1\left (3,\frac{2+m}{2};\frac{4+m}{2};-\frac{c x^2}{a}\right )}{a^3 e^2 (2+m)}\\ \end{align*}
Mathematica [A] time = 0.0264396, size = 82, normalized size = 0.9 \[ \frac{x (e x)^m \left (A (m+2) \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )+B (m+1) x \, _2F_1\left (3,\frac{m}{2}+1;\frac{m}{2}+2;-\frac{c x^2}{a}\right )\right )}{a^3 (m+1) (m+2)} \]
Antiderivative was successfully verified.
[In]
Integrate[((e*x)^m*(A + B*x))/(a + c*x^2)^3,x]
[Out]
(x*(e*x)^m*(B*(1 + m)*x*Hypergeometric2F1[3, 1 + m/2, 2 + m/2, -((c*x^2)/a)] + A*(2 + m)*Hypergeometric2F1[3,
(1 + m)/2, (3 + m)/2, -((c*x^2)/a)]))/(a^3*(1 + m)*(2 + m))
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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( Bx+A \right ) }{ \left ( c{x}^{2}+a \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m*(B*x+A)/(c*x^2+a)^3,x)
[Out]
int((e*x)^m*(B*x+A)/(c*x^2+a)^3,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x + A\right )} \left (e x\right )^{m}}{{\left (c x^{2} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(B*x+A)/(c*x^2+a)^3,x, algorithm="maxima")
[Out]
integrate((B*x + A)*(e*x)^m/(c*x^2 + a)^3, x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B x + A\right )} \left (e x\right )^{m}}{c^{3} x^{6} + 3 \, a c^{2} x^{4} + 3 \, a^{2} c x^{2} + a^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(B*x+A)/(c*x^2+a)^3,x, algorithm="fricas")
[Out]
integral((B*x + A)*(e*x)^m/(c^3*x^6 + 3*a*c^2*x^4 + 3*a^2*c*x^2 + a^3), x)
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Sympy [C] time = 164.467, size = 2509, normalized size = 27.57 \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+A)/(c*x**2+a)**3,x)
[Out]
A*(a**2*e**m*m**3*x*x**m*lerchphi(c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2
+ 3/2) + 64*a**4*c*x**2*gamma(m/2 + 3/2) + 32*a**3*c**2*x**4*gamma(m/2 + 3/2)) - 3*a**2*e**m*m**2*x*x**m*lerch
phi(c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*c*x**2*gamma(
m/2 + 3/2) + 32*a**3*c**2*x**4*gamma(m/2 + 3/2)) - 2*a**2*e**m*m**2*x*x**m*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2
+ 3/2) + 64*a**4*c*x**2*gamma(m/2 + 3/2) + 32*a**3*c**2*x**4*gamma(m/2 + 3/2)) - a**2*e**m*m*x*x**m*lerchphi(
c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*c*x**2*gamma(m/2
+ 3/2) + 32*a**3*c**2*x**4*gamma(m/2 + 3/2)) + 8*a**2*e**m*m*x*x**m*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2)
+ 64*a**4*c*x**2*gamma(m/2 + 3/2) + 32*a**3*c**2*x**4*gamma(m/2 + 3/2)) + 3*a**2*e**m*x*x**m*lerchphi(c*x**2*
exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*c*x**2*gamma(m/2 + 3/2)
+ 32*a**3*c**2*x**4*gamma(m/2 + 3/2)) + 10*a**2*e**m*x*x**m*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a*
*4*c*x**2*gamma(m/2 + 3/2) + 32*a**3*c**2*x**4*gamma(m/2 + 3/2)) + 2*a*c*e**m*m**3*x**3*x**m*lerchphi(c*x**2*e
xp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*c*x**2*gamma(m/2 + 3/2) +
32*a**3*c**2*x**4*gamma(m/2 + 3/2)) - 6*a*c*e**m*m**2*x**3*x**m*lerchphi(c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1
/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*c*x**2*gamma(m/2 + 3/2) + 32*a**3*c**2*x**4*gamma(m/2
+ 3/2)) - 2*a*c*e**m*m**2*x**3*x**m*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*c*x**2*gamma(m/2 + 3
/2) + 32*a**3*c**2*x**4*gamma(m/2 + 3/2)) - 2*a*c*e**m*m*x**3*x**m*lerchphi(c*x**2*exp_polar(I*pi)/a, 1, m/2 +
1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*c*x**2*gamma(m/2 + 3/2) + 32*a**3*c**2*x**4*gamma(m
/2 + 3/2)) + 4*a*c*e**m*m*x**3*x**m*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*c*x**2*gamma(m/2 + 3/
2) + 32*a**3*c**2*x**4*gamma(m/2 + 3/2)) + 6*a*c*e**m*x**3*x**m*lerchphi(c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/
2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*c*x**2*gamma(m/2 + 3/2) + 32*a**3*c**2*x**4*gamma(m/2
+ 3/2)) + 6*a*c*e**m*x**3*x**m*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*c*x**2*gamma(m/2 + 3/2) +
32*a**3*c**2*x**4*gamma(m/2 + 3/2)) + c**2*e**m*m**3*x**5*x**m*lerchphi(c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2
)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*c*x**2*gamma(m/2 + 3/2) + 32*a**3*c**2*x**4*gamma(m/2 +
3/2)) - 3*c**2*e**m*m**2*x**5*x**m*lerchphi(c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5
*gamma(m/2 + 3/2) + 64*a**4*c*x**2*gamma(m/2 + 3/2) + 32*a**3*c**2*x**4*gamma(m/2 + 3/2)) - c**2*e**m*m*x**5*x
**m*lerchphi(c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*c*x*
*2*gamma(m/2 + 3/2) + 32*a**3*c**2*x**4*gamma(m/2 + 3/2)) + 3*c**2*e**m*x**5*x**m*lerchphi(c*x**2*exp_polar(I*
pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*c*x**2*gamma(m/2 + 3/2) + 32*a**3*c*
*2*x**4*gamma(m/2 + 3/2))) + B*(a**2*e**m*m**3*x**2*x**m*lerchphi(c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1)*gamma(
m/2 + 1)/(32*a**5*gamma(m/2 + 2) + 64*a**4*c*x**2*gamma(m/2 + 2) + 32*a**3*c**2*x**4*gamma(m/2 + 2)) - 2*a**2*
e**m*m**2*x**2*x**m*gamma(m/2 + 1)/(32*a**5*gamma(m/2 + 2) + 64*a**4*c*x**2*gamma(m/2 + 2) + 32*a**3*c**2*x**4
*gamma(m/2 + 2)) - 4*a**2*e**m*m*x**2*x**m*lerchphi(c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1)*gamma(m/2 + 1)/(32*a
**5*gamma(m/2 + 2) + 64*a**4*c*x**2*gamma(m/2 + 2) + 32*a**3*c**2*x**4*gamma(m/2 + 2)) + 4*a**2*e**m*m*x**2*x*
*m*gamma(m/2 + 1)/(32*a**5*gamma(m/2 + 2) + 64*a**4*c*x**2*gamma(m/2 + 2) + 32*a**3*c**2*x**4*gamma(m/2 + 2))
+ 16*a**2*e**m*x**2*x**m*gamma(m/2 + 1)/(32*a**5*gamma(m/2 + 2) + 64*a**4*c*x**2*gamma(m/2 + 2) + 32*a**3*c**2
*x**4*gamma(m/2 + 2)) + 2*a*c*e**m*m**3*x**4*x**m*lerchphi(c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1)*gamma(m/2 + 1
)/(32*a**5*gamma(m/2 + 2) + 64*a**4*c*x**2*gamma(m/2 + 2) + 32*a**3*c**2*x**4*gamma(m/2 + 2)) - 2*a*c*e**m*m**
2*x**4*x**m*gamma(m/2 + 1)/(32*a**5*gamma(m/2 + 2) + 64*a**4*c*x**2*gamma(m/2 + 2) + 32*a**3*c**2*x**4*gamma(m
/2 + 2)) - 8*a*c*e**m*m*x**4*x**m*lerchphi(c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1)*gamma(m/2 + 1)/(32*a**5*gamma
(m/2 + 2) + 64*a**4*c*x**2*gamma(m/2 + 2) + 32*a**3*c**2*x**4*gamma(m/2 + 2)) + 8*a*c*e**m*x**4*x**m*gamma(m/2
+ 1)/(32*a**5*gamma(m/2 + 2) + 64*a**4*c*x**2*gamma(m/2 + 2) + 32*a**3*c**2*x**4*gamma(m/2 + 2)) + c**2*e**m*
m**3*x**6*x**m*lerchphi(c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1)*gamma(m/2 + 1)/(32*a**5*gamma(m/2 + 2) + 64*a**4
*c*x**2*gamma(m/2 + 2) + 32*a**3*c**2*x**4*gamma(m/2 + 2)) - 4*c**2*e**m*m*x**6*x**m*lerchphi(c*x**2*exp_polar
(I*pi)/a, 1, m/2 + 1)*gamma(m/2 + 1)/(32*a**5*gamma(m/2 + 2) + 64*a**4*c*x**2*gamma(m/2 + 2) + 32*a**3*c**2*x*
*4*gamma(m/2 + 2)))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x + A\right )} \left (e x\right )^{m}}{{\left (c x^{2} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(B*x+A)/(c*x^2+a)^3,x, algorithm="giac")
[Out]
integrate((B*x + A)*(e*x)^m/(c*x^2 + a)^3, x)