Integrand size = 24, antiderivative size = 89 \[ \int x^{-3-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=-\frac {c x^{-2 (1+p)} \left (a+b x^2\right )^{1+p}}{2 a (1+p)}-\frac {d x^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{2 p} \] Output:
-1/2*c*(b*x^2+a)^(p+1)/a/(p+1)/(x^(2+2*p))-1/2*d*(b*x^2+a)^p*hypergeom([-p , -p],[-p+1],-b*x^2/a)/p/(x^(2*p))/((1+b*x^2/a)^p)
Time = 0.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int x^{-3-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\frac {1}{2} x^{-2 p} \left (a+b x^2\right )^p \left (-\frac {c \left (a+b x^2\right )}{a (1+p) x^2}-\frac {d \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{p}\right ) \] Input:
Integrate[x^(-3 - 2*p)*(a + b*x^2)^p*(c + d*x^2),x]
Output:
((a + b*x^2)^p*(-((c*(a + b*x^2))/(a*(1 + p)*x^2)) - (d*Hypergeometric2F1[ -p, -p, 1 - p, -((b*x^2)/a)])/(p*(1 + (b*x^2)/a)^p)))/(2*x^(2*p))
Time = 0.21 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {358, 279, 278}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^{-2 p-3} \left (c+d x^2\right ) \left (a+b x^2\right )^p \, dx\) |
\(\Big \downarrow \) 358 |
\(\displaystyle d \int x^{-2 p-1} \left (b x^2+a\right )^pdx-\frac {c x^{-2 (p+1)} \left (a+b x^2\right )^{p+1}}{2 a (p+1)}\) |
\(\Big \downarrow \) 279 |
\(\displaystyle d \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int x^{-2 p-1} \left (\frac {b x^2}{a}+1\right )^pdx-\frac {c x^{-2 (p+1)} \left (a+b x^2\right )^{p+1}}{2 a (p+1)}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -\frac {c x^{-2 (p+1)} \left (a+b x^2\right )^{p+1}}{2 a (p+1)}-\frac {d x^{-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{2 p}\) |
Input:
Int[x^(-3 - 2*p)*(a + b*x^2)^p*(c + d*x^2),x]
Output:
-1/2*(c*(a + b*x^2)^(1 + p))/(a*(1 + p)*x^(2*(1 + p))) - (d*(a + b*x^2)^p* Hypergeometric2F1[-p, -p, 1 - p, -((b*x^2)/a)])/(2*p*x^(2*p)*(1 + (b*x^2)/ a)^p)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[(c*x)^m* (1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && !(ILtQ[p, 0] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x_ Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + S imp[d/e^2 Int[(e*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e , m, p}, x] && NeQ[b*c - a*d, 0] && EqQ[Simplify[m + 2*p + 3], 0] && NeQ[m, -1]
\[\int x^{-3-2 p} \left (b \,x^{2}+a \right )^{p} \left (x^{2} d +c \right )d x\]
Input:
int(x^(-3-2*p)*(b*x^2+a)^p*(d*x^2+c),x)
Output:
int(x^(-3-2*p)*(b*x^2+a)^p*(d*x^2+c),x)
\[ \int x^{-3-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\int { {\left (d x^{2} + c\right )} {\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 3} \,d x } \] Input:
integrate(x^(-3-2*p)*(b*x^2+a)^p*(d*x^2+c),x, algorithm="fricas")
Output:
integral((d*x^2 + c)*(b*x^2 + a)^p*x^(-2*p - 3), x)
Result contains complex when optimal does not.
Time = 28.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.88 \[ \int x^{-3-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\frac {a^{p} c x^{- 2 p - 2} \left (1 + \frac {b x^{2}}{a}\right )^{p + 1} \Gamma \left (- p - 1\right )}{2 \Gamma \left (- p\right )} + \frac {a^{p} d x^{- 2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (1 - p\right )} \] Input:
integrate(x**(-3-2*p)*(b*x**2+a)**p*(d*x**2+c),x)
Output:
a**p*c*x**(-2*p - 2)*(1 + b*x**2/a)**(p + 1)*gamma(-p - 1)/(2*gamma(-p)) + a**p*d*gamma(-p)*hyper((-p, -p), (1 - p,), b*x**2*exp_polar(I*pi)/a)/(2*x **(2*p)*gamma(1 - p))
\[ \int x^{-3-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\int { {\left (d x^{2} + c\right )} {\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 3} \,d x } \] Input:
integrate(x^(-3-2*p)*(b*x^2+a)^p*(d*x^2+c),x, algorithm="maxima")
Output:
d*integrate(e^(p*log(b*x^2 + a) - 2*p*log(x))/x, x) - 1/2*(b*x^2 + a)*c*e^ (p*log(b*x^2 + a) - 2*p*log(x))/(a*(p + 1)*x^2)
\[ \int x^{-3-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\int { {\left (d x^{2} + c\right )} {\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 3} \,d x } \] Input:
integrate(x^(-3-2*p)*(b*x^2+a)^p*(d*x^2+c),x, algorithm="giac")
Output:
integrate((d*x^2 + c)*(b*x^2 + a)^p*x^(-2*p - 3), x)
Timed out. \[ \int x^{-3-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\int \frac {{\left (b\,x^2+a\right )}^p\,\left (d\,x^2+c\right )}{x^{2\,p+3}} \,d x \] Input:
int(((a + b*x^2)^p*(c + d*x^2))/x^(2*p + 3),x)
Output:
int(((a + b*x^2)^p*(c + d*x^2))/x^(2*p + 3), x)
\[ \int x^{-3-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\frac {-\left (b \,x^{2}+a \right )^{p} a c -\left (b \,x^{2}+a \right )^{p} b c \,x^{2}+2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} x}d x \right ) a d p \,x^{2}+2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} x}d x \right ) a d \,x^{2}}{2 x^{2 p} a \,x^{2} \left (p +1\right )} \] Input:
int(x^(-3-2*p)*(b*x^2+a)^p*(d*x^2+c),x)
Output:
( - (a + b*x**2)**p*a*c - (a + b*x**2)**p*b*c*x**2 + 2*x**(2*p)*int((a + b *x**2)**p/(x**(2*p)*x),x)*a*d*p*x**2 + 2*x**(2*p)*int((a + b*x**2)**p/(x** (2*p)*x),x)*a*d*x**2)/(2*x**(2*p)*a*x**2*(p + 1))