\(\int \frac {x^m (c+d x^2)}{(a+b x^2)^2} \, dx\) [339]

Optimal result

Integrand size = 20, antiderivative size = 93 \[ \int \frac {x^m \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {(b c-a d) x^{1+m}}{2 a b \left (a+b x^2\right )}+\frac {(a d (1+m)+b (c-c m)) x^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{2 a^2 b (1+m)} \]

[Out]

1/2*(-a*d+b*c)*x^(1+m)/a/b/(b*x^2+a)+1/2*(a*d*(1+m)+b*(-c*m+c))*x^(1+m)*hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],-
b*x^2/a)/a^2/b/(1+m)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {468, 371} \[ \int \frac {x^m \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {x^{m+1} (a d (m+1)+b (c-c m)) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {b x^2}{a}\right )}{2 a^2 b (m+1)}+\frac {x^{m+1} (b c-a d)}{2 a b \left (a+b x^2\right )} \]

[In]

Int[(x^m*(c + d*x^2))/(a + b*x^2)^2,x]

[Out]

((b*c - a*d)*x^(1 + m))/(2*a*b*(a + b*x^2)) + ((a*d*(1 + m) + b*(c - c*m))*x^(1 + m)*Hypergeometric2F1[1, (1 +
m)/2, (3 + m)/2, -((b*x^2)/a)])/(2*a^2*b*(1 + m))

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 468

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*} \text {integral}& = \frac {(b c-a d) x^{1+m}}{2 a b \left (a+b x^2\right )}+\frac {(a d (1+m)+b (c-c m)) \int \frac {x^m}{a+b x^2} \, dx}{2 a b} \\ & = \frac {(b c-a d) x^{1+m}}{2 a b \left (a+b x^2\right )}+\frac {(a d (1+m)+b (c-c m)) 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 (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.86 \[ \int \frac {x^m \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {x^{1+m} \left (a d \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )+(b c-a d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )\right )}{a^2 b (1+m)} \]

[In]

Integrate[(x^m*(c + d*x^2))/(a + b*x^2)^2,x]

[Out]

(x^(1 + m)*(a*d*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + (b*c - a*d)*Hypergeometric2F1[2, (1
+ m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a^2*b*(1 + m))

Maple [F]

\[\int \frac {x^{m} \left (d \,x^{2}+c \right )}{\left (b \,x^{2}+a \right )^{2}}d x\]

[In]

int(x^m*(d*x^2+c)/(b*x^2+a)^2,x)

[Out]

int(x^m*(d*x^2+c)/(b*x^2+a)^2,x)

Fricas [F]

\[ \int \frac {x^m \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )} x^{m}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]

[In]

integrate(x^m*(d*x^2+c)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

integral((d*x^2 + c)*x^m/(b^2*x^4 + 2*a*b*x^2 + a^2), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 16.21 (sec) , antiderivative size = 906, normalized size of antiderivative = 9.74 \[ \int \frac {x^m \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(x**m*(d*x**2+c)/(b*x**2+a)**2,x)

[Out]

c*(-a*m**2*x**(m + 1)*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*m*x**(m + 1)*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*x**(m + 1)*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*x**(m + 1)*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*m**2*x**2*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*g
amma(m/2 + 1/2)/(8*a**3*gamma(m/2 + 3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/2)) + b*x**2*x**(m + 1)*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)))
+ d*(-a*m**2*x**(m + 3)*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*m*x**(m + 3)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*ga
mma(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*m*x**(m + 3)*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*x**(m + 3)*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*x**(m + 3)*ga
mma(m/2 + 3/2)/(8*a**3*gamma(m/2 + 5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2)) - b*m**2*x**2*x**(m + 3)*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*m*x**2*x**(m + 3)*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*x**2*x**(m + 3)*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)))

Maxima [F]

\[ \int \frac {x^m \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )} x^{m}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]

[In]

integrate(x^m*(d*x^2+c)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)*x^m/(b*x^2 + a)^2, x)

Giac [F]

\[ \int \frac {x^m \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )} x^{m}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]

[In]

integrate(x^m*(d*x^2+c)/(b*x^2+a)^2,x, algorithm="giac")

[Out]

integrate((d*x^2 + c)*x^m/(b*x^2 + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^m \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^m\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^2} \,d x \]

[In]

int((x^m*(c + d*x^2))/(a + b*x^2)^2,x)

[Out]

int((x^m*(c + d*x^2))/(a + b*x^2)^2, x)