Integrand size = 26, antiderivative size = 456 \[ \int \frac {(e x)^{-3-2 p} \left (a+b x^2\right )^p}{(c+d x)^2} \, dx=-\frac {d^2 (e x)^{-2 p} \left (a+b x^2\right )^{1+p}}{2 a c^2 e^3 p \left (c^2-d^2 x^2\right )}+\frac {d^2 \left (b c^2+a d^2 (2+p)\right ) (e x)^{-2 p} \left (a+b x^2\right )^{1+p}}{2 a c^2 \left (b c^2+a d^2\right ) e^3 (1+p) \left (c^2-d^2 x^2\right )}-\frac {(e x)^{-2 (1+p)} \left (a+b x^2\right )^{1+p}}{2 a e (1+p) \left (c^2-d^2 x^2\right )}+\frac {2 d (e x)^{-1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2} (-1-2 p),-p,2,\frac {1}{2} (1-2 p),-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^3 e^2 (1+2 p)}-\frac {d^2 \left (2 b c^2+a d^2 (2+p)\right ) (e x)^{-2 p} \left (a+b x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,-p,1-p,\frac {\left (b+\frac {a d^2}{c^2}\right ) x^2}{a+b x^2}\right )}{2 c^4 \left (b c^2+a d^2\right ) e^3 p}+\frac {d^2 \left (b c^2+a d^2 (1+p)\right ) (e x)^{2-2 p} \left (a+b x^2\right )^{-1+p} \operatorname {Hypergeometric2F1}\left (2,1-p,2-p,\frac {\left (b+\frac {a d^2}{c^2}\right ) x^2}{a+b x^2}\right )}{2 c^6 e^5 (1-p) p} \] Output:
-1/2*d^2*(b*x^2+a)^(p+1)/a/c^2/e^3/p/((e*x)^(2*p))/(-d^2*x^2+c^2)+1/2*d^2* (b*c^2+a*d^2*(2+p))*(b*x^2+a)^(p+1)/a/c^2/(a*d^2+b*c^2)/e^3/(p+1)/((e*x)^( 2*p))/(-d^2*x^2+c^2)-1/2*(b*x^2+a)^(p+1)/a/e/(p+1)/((e*x)^(2*p+2))/(-d^2*x ^2+c^2)+2*d*(e*x)^(-1-2*p)*(b*x^2+a)^p*AppellF1(-1/2-p,2,-p,1/2-p,d^2*x^2/ c^2,-b*x^2/a)/c^3/e^2/(1+2*p)/((1+b*x^2/a)^p)-1/2*d^2*(2*b*c^2+a*d^2*(2+p) )*(b*x^2+a)^p*hypergeom([1, -p],[1-p],(b+a*d^2/c^2)*x^2/(b*x^2+a))/c^4/(a* d^2+b*c^2)/e^3/p/((e*x)^(2*p))+1/2*d^2*(b*c^2+a*d^2*(p+1))*(e*x)^(2-2*p)*( b*x^2+a)^(-1+p)*hypergeom([2, 1-p],[2-p],(b+a*d^2/c^2)*x^2/(b*x^2+a))/c^6/ e^5/(1-p)/p
\[ \int \frac {(e x)^{-3-2 p} \left (a+b x^2\right )^p}{(c+d x)^2} \, dx=\int \frac {(e x)^{-3-2 p} \left (a+b x^2\right )^p}{(c+d x)^2} \, dx \] Input:
Integrate[((e*x)^(-3 - 2*p)*(a + b*x^2)^p)/(c + d*x)^2,x]
Output:
Integrate[((e*x)^(-3 - 2*p)*(a + b*x^2)^p)/(c + d*x)^2, x]
Time = 0.80 (sec) , antiderivative size = 432, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {623, 622, 2009}
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.
\(\displaystyle \int \frac {(e x)^{-2 p-3} \left (a+b x^2\right )^p}{(c+d x)^2} \, dx\) |
\(\Big \downarrow \) 623 |
\(\displaystyle x^{2 p+3} (e x)^{-2 p-3} \int \frac {x^{-2 p-3} \left (b x^2+a\right )^p}{(c+d x)^2}dx\) |
\(\Big \downarrow \) 622 |
\(\displaystyle x^{2 p+3} (e x)^{-2 p-3} \int \left (\frac {c^2 \left (b x^2+a\right )^p x^{-2 p-3}}{\left (c^2-d^2 x^2\right )^2}-\frac {2 c d \left (b x^2+a\right )^p x^{-2 p-2}}{\left (c^2-d^2 x^2\right )^2}+\frac {d^2 \left (b x^2+a\right )^p x^{-2 p-1}}{\left (d^2 x^2-c^2\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^{2 p+3} (e x)^{-2 p-3} \left (\frac {2 d x^{-2 p-1} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (-p-\frac {1}{2},-p,2,\frac {1}{2}-p,-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^3 (2 p+1)}-\frac {d^2 x^{-2 p} \left (a+b x^2\right )^{p+1}}{2 a c^2 p \left (c^2-d^2 x^2\right )}-\frac {x^{-2 (p+1)} \left (a+b x^2\right )^{p+1} \left (a d^2 (p+2)+b c^2\right )}{2 a c^2 (p+1) \left (a d^2+b c^2\right )}+\frac {d^2 x^{-2 (p+1)} \left (a+b x^2\right )^{p+1}}{2 \left (c^2-d^2 x^2\right ) \left (a d^2+b c^2\right )}+\frac {d^2 x^{2-2 p} \left (a+b x^2\right )^{p-1} \left (a d^2 (p+1)+b c^2\right ) \operatorname {Hypergeometric2F1}\left (2,1-p,2-p,\frac {\left (\frac {a d^2}{c^2}+b\right ) x^2}{b x^2+a}\right )}{2 c^6 (1-p) p}-\frac {d^2 x^{-2 p} \left (a+b x^2\right )^p \left (a d^2 (p+2)+2 b c^2\right ) \operatorname {Hypergeometric2F1}\left (1,-p,1-p,\frac {\left (\frac {a d^2}{c^2}+b\right ) x^2}{b x^2+a}\right )}{2 c^4 p \left (a d^2+b c^2\right )}\right )\) |
Input:
Int[((e*x)^(-3 - 2*p)*(a + b*x^2)^p)/(c + d*x)^2,x]
Output:
x^(3 + 2*p)*(e*x)^(-3 - 2*p)*(-1/2*((b*c^2 + a*d^2*(2 + p))*(a + b*x^2)^(1 + p))/(a*c^2*(b*c^2 + a*d^2)*(1 + p)*x^(2*(1 + p))) - (d^2*(a + b*x^2)^(1 + p))/(2*a*c^2*p*x^(2*p)*(c^2 - d^2*x^2)) + (d^2*(a + b*x^2)^(1 + p))/(2* (b*c^2 + a*d^2)*x^(2*(1 + p))*(c^2 - d^2*x^2)) + (2*d*x^(-1 - 2*p)*(a + b* x^2)^p*AppellF1[-1/2 - p, -p, 2, 1/2 - p, -((b*x^2)/a), (d^2*x^2)/c^2])/(c ^3*(1 + 2*p)*(1 + (b*x^2)/a)^p) - (d^2*(2*b*c^2 + a*d^2*(2 + p))*(a + b*x^ 2)^p*Hypergeometric2F1[1, -p, 1 - p, ((b + (a*d^2)/c^2)*x^2)/(a + b*x^2)]) /(2*c^4*(b*c^2 + a*d^2)*p*x^(2*p)) + (d^2*(b*c^2 + a*d^2*(1 + p))*x^(2 - 2 *p)*(a + b*x^2)^(-1 + p)*Hypergeometric2F1[2, 1 - p, 2 - p, ((b + (a*d^2)/ c^2)*x^2)/(a + b*x^2)])/(2*c^6*(1 - p)*p))
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Int[ExpandIntegrand[x^m*(a + b*x^2)^p, (c/(c^2 - d^2*x^2) - d*(x/(c^2 - d^2*x^2)))^(-n), x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[n, -1]
Int[((e_)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*x)^m/x^m Int[x^m*(c + d*x)^n*(a + b*x^2)^p, x], x] / ; FreeQ[{a, b, c, d, e, m, p}, x] && ILtQ[n, 0]
\[\int \frac {\left (e x \right )^{-3-2 p} \left (b \,x^{2}+a \right )^{p}}{\left (d x +c \right )^{2}}d x\]
Input:
int((e*x)^(-3-2*p)*(b*x^2+a)^p/(d*x+c)^2,x)
Output:
int((e*x)^(-3-2*p)*(b*x^2+a)^p/(d*x+c)^2,x)
\[ \int \frac {(e x)^{-3-2 p} \left (a+b x^2\right )^p}{(c+d x)^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} \left (e x\right )^{-2 \, p - 3}}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate((e*x)^(-3-2*p)*(b*x^2+a)^p/(d*x+c)^2,x, algorithm="fricas")
Output:
integral((b*x^2 + a)^p*(e*x)^(-2*p - 3)/(d^2*x^2 + 2*c*d*x + c^2), x)
Timed out. \[ \int \frac {(e x)^{-3-2 p} \left (a+b x^2\right )^p}{(c+d x)^2} \, dx=\text {Timed out} \] Input:
integrate((e*x)**(-3-2*p)*(b*x**2+a)**p/(d*x+c)**2,x)
Output:
Timed out
\[ \int \frac {(e x)^{-3-2 p} \left (a+b x^2\right )^p}{(c+d x)^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} \left (e x\right )^{-2 \, p - 3}}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate((e*x)^(-3-2*p)*(b*x^2+a)^p/(d*x+c)^2,x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^p*(e*x)^(-2*p - 3)/(d*x + c)^2, x)
\[ \int \frac {(e x)^{-3-2 p} \left (a+b x^2\right )^p}{(c+d x)^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} \left (e x\right )^{-2 \, p - 3}}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate((e*x)^(-3-2*p)*(b*x^2+a)^p/(d*x+c)^2,x, algorithm="giac")
Output:
integrate((b*x^2 + a)^p*(e*x)^(-2*p - 3)/(d*x + c)^2, x)
Timed out. \[ \int \frac {(e x)^{-3-2 p} \left (a+b x^2\right )^p}{(c+d x)^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^p}{{\left (e\,x\right )}^{2\,p+3}\,{\left (c+d\,x\right )}^2} \,d x \] Input:
int((a + b*x^2)^p/((e*x)^(2*p + 3)*(c + d*x)^2),x)
Output:
int((a + b*x^2)^p/((e*x)^(2*p + 3)*(c + d*x)^2), x)
\[ \int \frac {(e x)^{-3-2 p} \left (a+b x^2\right )^p}{(c+d x)^2} \, dx=\text {too large to display} \] Input:
int((e*x)^(-3-2*p)*(b*x^2+a)^p/(d*x+c)^2,x)
Output:
( - 2*(a + b*x**2)**p*a**2*d**3*p**2 - 3*(a + b*x**2)**p*a**2*d**3*p - (a + b*x**2)**p*a**2*d**3 + 2*(a + b*x**2)**p*a*b*c**2*d*p + (a + b*x**2)**p* a*b*c**2*d + 2*(a + b*x**2)**p*a*b*c*d**2*p**2*x + 7*(a + b*x**2)**p*a*b*c *d**2*p*x + 4*(a + b*x**2)**p*a*b*c*d**2*x - 4*(a + b*x**2)**p*b**2*c**3*p *x - 4*(a + b*x**2)**p*b**2*c**3*x + 2*(a + b*x**2)**p*b**2*c**2*d*p*x**2 + (a + b*x**2)**p*b**2*c**2*d*x**2 + 2*(a + b*x**2)**p*b**2*c*d**2*p*x**3 + (a + b*x**2)**p*b**2*c*d**2*x**3 - 16*x**(2*p)*int((a + b*x**2)**p/(4*x* *(2*p)*a**2*c**2*d**2*p**3*x**2 + 8*x**(2*p)*a**2*c**2*d**2*p**2*x**2 + 5* x**(2*p)*a**2*c**2*d**2*p*x**2 + x**(2*p)*a**2*c**2*d**2*x**2 + 8*x**(2*p) *a**2*c*d**3*p**3*x**3 + 16*x**(2*p)*a**2*c*d**3*p**2*x**3 + 10*x**(2*p)*a **2*c*d**3*p*x**3 + 2*x**(2*p)*a**2*c*d**3*x**3 + 4*x**(2*p)*a**2*d**4*p** 3*x**4 + 8*x**(2*p)*a**2*d**4*p**2*x**4 + 5*x**(2*p)*a**2*d**4*p*x**4 + x* *(2*p)*a**2*d**4*x**4 - 4*x**(2*p)*a*b*c**4*p**2*x**2 - 4*x**(2*p)*a*b*c** 4*p*x**2 - x**(2*p)*a*b*c**4*x**2 - 8*x**(2*p)*a*b*c**3*d*p**2*x**3 - 8*x* *(2*p)*a*b*c**3*d*p*x**3 - 2*x**(2*p)*a*b*c**3*d*x**3 + 4*x**(2*p)*a*b*c** 2*d**2*p**3*x**4 + 4*x**(2*p)*a*b*c**2*d**2*p**2*x**4 + x**(2*p)*a*b*c**2* d**2*p*x**4 + 8*x**(2*p)*a*b*c*d**3*p**3*x**5 + 16*x**(2*p)*a*b*c*d**3*p** 2*x**5 + 10*x**(2*p)*a*b*c*d**3*p*x**5 + 2*x**(2*p)*a*b*c*d**3*x**5 + 4*x* *(2*p)*a*b*d**4*p**3*x**6 + 8*x**(2*p)*a*b*d**4*p**2*x**6 + 5*x**(2*p)*a*b *d**4*p*x**6 + x**(2*p)*a*b*d**4*x**6 - 4*x**(2*p)*b**2*c**4*p**2*x**4 ...