Integrand size = 18, antiderivative size = 91 \[ \int \frac {(d+e x) \left (a+b x^2\right )^p}{x^2} \, dx=-\frac {d \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b x^2}{a}\right )}{x}-\frac {e \left (a+b x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b x^2}{a}\right )}{2 a (1+p)} \]
-d*(b*x^2+a)^p*hypergeom([-1/2, -p],[1/2],-b*x^2/a)/x/((1+b*x^2/a)^p)-1/2* e*(b*x^2+a)^(p+1)*hypergeom([1, p+1],[2+p],1+b*x^2/a)/a/(p+1)
Time = 0.11 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x) \left (a+b x^2\right )^p}{x^2} \, dx=-\frac {d \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b x^2}{a}\right )}{x}-\frac {e \left (a+b x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b x^2}{a}\right )}{2 a (1+p)} \]
-((d*(a + b*x^2)^p*Hypergeometric2F1[-1/2, -p, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^p)) - (e*(a + b*x^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p , 1 + (b*x^2)/a])/(2*a*(1 + p))
Time = 0.21 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {542, 243, 75, 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.
\(\displaystyle \int \frac {(d+e x) \left (a+b x^2\right )^p}{x^2} \, dx\) |
\(\Big \downarrow \) 542 |
\(\displaystyle d \int \frac {\left (b x^2+a\right )^p}{x^2}dx+e \int \frac {\left (b x^2+a\right )^p}{x}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle d \int \frac {\left (b x^2+a\right )^p}{x^2}dx+\frac {1}{2} e \int \frac {\left (b x^2+a\right )^p}{x^2}dx^2\) |
\(\Big \downarrow \) 75 |
\(\displaystyle d \int \frac {\left (b x^2+a\right )^p}{x^2}dx-\frac {e \left (a+b x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b x^2}{a}+1\right )}{2 a (p+1)}\) |
\(\Big \downarrow \) 279 |
\(\displaystyle d \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int \frac {\left (\frac {b x^2}{a}+1\right )^p}{x^2}dx-\frac {e \left (a+b x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b x^2}{a}+1\right )}{2 a (p+1)}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -\frac {d \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b x^2}{a}\right )}{x}-\frac {e \left (a+b x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b x^2}{a}+1\right )}{2 a (p+1)}\) |
-((d*(a + b*x^2)^p*Hypergeometric2F1[-1/2, -p, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^p)) - (e*(a + b*x^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p , 1 + (b*x^2)/a])/(2*a*(1 + p))
3.4.88.3.1 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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c Int[x^m*(a + b*x^2)^p, x], x] + Simp[d Int[x^(m + 1)*(a + b*x^2 )^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && IntegerQ[m] && !IntegerQ[2*p]
\[\int \frac {\left (e x +d \right ) \left (b \,x^{2}+a \right )^{p}}{x^{2}}d x\]
\[ \int \frac {(d+e x) \left (a+b x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (e x + d\right )} {\left (b x^{2} + a\right )}^{p}}{x^{2}} \,d x } \]
Result contains complex when optimal does not.
Time = 3.95 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.75 \[ \int \frac {(d+e x) \left (a+b x^2\right )^p}{x^2} \, dx=- \frac {a^{p} d {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{x} - \frac {b^{p} e x^{2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 \Gamma \left (1 - p\right )} \]
-a**p*d*hyper((-1/2, -p), (1/2,), b*x**2*exp_polar(I*pi)/a)/x - b**p*e*x** (2*p)*gamma(-p)*hyper((-p, -p), (1 - p,), a*exp_polar(I*pi)/(b*x**2))/(2*g amma(1 - p))
\[ \int \frac {(d+e x) \left (a+b x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (e x + d\right )} {\left (b x^{2} + a\right )}^{p}}{x^{2}} \,d x } \]
\[ \int \frac {(d+e x) \left (a+b x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (e x + d\right )} {\left (b x^{2} + a\right )}^{p}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {(d+e x) \left (a+b x^2\right )^p}{x^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^p\,\left (d+e\,x\right )}{x^2} \,d x \]