Integrand size = 20, antiderivative size = 176 \[ \int \frac {\left (a+b x^2\right )^p}{x (d+e x)} \, dx=-\frac {e x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2}+\frac {e^2 \left (a+b x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {e^2 \left (a+b x^2\right )}{b d^2+a e^2}\right )}{2 d \left (b d^2+a e^2\right ) (1+p)}-\frac {\left (a+b x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b x^2}{a}\right )}{2 a d (1+p)} \]
-e*x*(b*x^2+a)^p*AppellF1(1/2,1,-p,3/2,e^2*x^2/d^2,-b*x^2/a)/d^2/((1+b*x^2 /a)^p)+1/2*e^2*(b*x^2+a)^(p+1)*hypergeom([1, p+1],[2+p],e^2*(b*x^2+a)/(a*e ^2+b*d^2))/d/(a*e^2+b*d^2)/(p+1)-1/2*(b*x^2+a)^(p+1)*hypergeom([1, p+1],[2 +p],1+b*x^2/a)/a/d/(p+1)
Time = 0.26 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^p}{x (d+e x)} \, dx=\frac {\left (a+b x^2\right )^p \left (-\left (\frac {e \left (-\sqrt {-\frac {a}{b}}+x\right )}{d+e x}\right )^{-p} \left (\frac {e \left (\sqrt {-\frac {a}{b}}+x\right )}{d+e x}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {d-\sqrt {-\frac {a}{b}} e}{d+e x},\frac {d+\sqrt {-\frac {a}{b}} e}{d+e x}\right )+\left (1+\frac {a}{b x^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {a}{b x^2}\right )\right )}{2 d p} \]
((a + b*x^2)^p*(-(AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)]/(((e*(-Sqrt[-(a/b)] + x))/(d + e*x ))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p)) + Hypergeometric2F1[-p, -p, 1 - p, -(a/(b*x^2))]/(1 + a/(b*x^2))^p))/(2*d*p)
Time = 0.30 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {621, 334, 333, 354, 97, 75, 78}
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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^p}{x (d+e x)} \, dx\) |
\(\Big \downarrow \) 621 |
\(\displaystyle d \int \frac {\left (b x^2+a\right )^p}{x \left (d^2-e^2 x^2\right )}dx-e \int \frac {\left (b x^2+a\right )^p}{d^2-e^2 x^2}dx\) |
\(\Big \downarrow \) 334 |
\(\displaystyle d \int \frac {\left (b x^2+a\right )^p}{x \left (d^2-e^2 x^2\right )}dx-e \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int \frac {\left (\frac {b x^2}{a}+1\right )^p}{d^2-e^2 x^2}dx\) |
\(\Big \downarrow \) 333 |
\(\displaystyle d \int \frac {\left (b x^2+a\right )^p}{x \left (d^2-e^2 x^2\right )}dx-\frac {e x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2}\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} d \int \frac {\left (b x^2+a\right )^p}{x^2 \left (d^2-e^2 x^2\right )}dx^2-\frac {e x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2}\) |
\(\Big \downarrow \) 97 |
\(\displaystyle \frac {1}{2} d \left (\frac {e^2 \int \frac {\left (b x^2+a\right )^p}{d^2-e^2 x^2}dx^2}{d^2}+\frac {\int \frac {\left (b x^2+a\right )^p}{x^2}dx^2}{d^2}\right )-\frac {e x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle \frac {1}{2} d \left (\frac {e^2 \int \frac {\left (b x^2+a\right )^p}{d^2-e^2 x^2}dx^2}{d^2}-\frac {\left (a+b x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b x^2}{a}+1\right )}{a d^2 (p+1)}\right )-\frac {e x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2}\) |
\(\Big \downarrow \) 78 |
\(\displaystyle \frac {1}{2} d \left (\frac {e^2 \left (a+b x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{d^2 (p+1) \left (a e^2+b d^2\right )}-\frac {\left (a+b x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b x^2}{a}+1\right )}{a d^2 (p+1)}\right )-\frac {e x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2}\) |
-((e*x*(a + b*x^2)^p*AppellF1[1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2 ])/(d^2*(1 + (b*x^2)/a)^p)) + (d*((e^2*(a + b*x^2)^(1 + p)*Hypergeometric2 F1[1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)])/(d^2*(b*d^2 + a*e^ 2)*(1 + p)) - ((a + b*x^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*x^2)/a])/(a*d^2*(1 + p))))/2
3.5.13.3.1 Defintions of rubi rules used
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x )^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b *c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] && !IntegerQ[m] && IntegerQ[n]
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[b/(b*c - a*d) Int[(e + f*x)^p/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && !IntegerQ[p]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[ (1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && !(IntegerQ[p] || GtQ[a, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)), x_Symbol] :> Simp[c Int[x^m*((a + b*x^2)^p/(c^2 - d^2*x^2)), x], x] - Simp[d Int[ x^(m + 1)*((a + b*x^2)^p/(c^2 - d^2*x^2)), x], x] /; FreeQ[{a, b, c, d, m, p}, x]
\[\int \frac {\left (b \,x^{2}+a \right )^{p}}{x \left (e x +d \right )}d x\]
\[ \int \frac {\left (a+b x^2\right )^p}{x (d+e x)} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p}}{{\left (e x + d\right )} x} \,d x } \]
\[ \int \frac {\left (a+b x^2\right )^p}{x (d+e x)} \, dx=\int \frac {\left (a + b x^{2}\right )^{p}}{x \left (d + e x\right )}\, dx \]
\[ \int \frac {\left (a+b x^2\right )^p}{x (d+e x)} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p}}{{\left (e x + d\right )} x} \,d x } \]
\[ \int \frac {\left (a+b x^2\right )^p}{x (d+e x)} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p}}{{\left (e x + d\right )} x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^2\right )^p}{x (d+e x)} \, dx=\int \frac {{\left (b\,x^2+a\right )}^p}{x\,\left (d+e\,x\right )} \,d x \]