Integrand size = 23, antiderivative size = 108 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=-\frac {d \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e \left (d^2-e^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 (1+p)} \]
-d*(-e^2*x^2+d^2)^p*hypergeom([-1/2, -p],[1/2],e^2*x^2/d^2)/x/((1-e^2*x^2/ d^2)^p)-1/2*e*(-e^2*x^2+d^2)^(p+1)*hypergeom([1, p+1],[2+p],1-e^2*x^2/d^2) /d^2/(p+1)
Time = 0.18 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=-\frac {d \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e \left (d^2-e^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 (1+p)} \]
-((d*(d^2 - e^2*x^2)^p*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x *(1 - (e^2*x^2)/d^2)^p)) - (e*(d^2 - e^2*x^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2])/(2*d^2*(1 + p))
Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, 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.
\(\displaystyle \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx\) |
\(\Big \downarrow \) 542 |
\(\displaystyle d \int \frac {\left (d^2-e^2 x^2\right )^p}{x^2}dx+e \int \frac {\left (d^2-e^2 x^2\right )^p}{x}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle d \int \frac {\left (d^2-e^2 x^2\right )^p}{x^2}dx+\frac {1}{2} e \int \frac {\left (d^2-e^2 x^2\right )^p}{x^2}dx^2\) |
\(\Big \downarrow \) 75 |
\(\displaystyle d \int \frac {\left (d^2-e^2 x^2\right )^p}{x^2}dx-\frac {e \left (d^2-e^2 x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 (p+1)}\) |
\(\Big \downarrow \) 279 |
\(\displaystyle d \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \int \frac {\left (1-\frac {e^2 x^2}{d^2}\right )^p}{x^2}dx-\frac {e \left (d^2-e^2 x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 (p+1)}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -\frac {d \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e \left (d^2-e^2 x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 (p+1)}\) |
-((d*(d^2 - e^2*x^2)^p*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x *(1 - (e^2*x^2)/d^2)^p)) - (e*(d^2 - e^2*x^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2])/(2*d^2*(1 + p))
3.3.47.3.1 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]
\[\int \frac {\left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{p}}{x^{2}}d x\]
\[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{2}} \,d x } \]
Result contains complex when optimal does not.
Time = 1.79 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.76 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=- \frac {d d^{2 p} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{x} - \frac {e e^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (1 - p\right )} \]
-d*d**(2*p)*hyper((-1/2, -p), (1/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/x - e*e**(2*p)*x**(2*p)*exp(I*pi*p)*gamma(-p)*hyper((-p, -p), (1 - p,), d**2 /(e**2*x**2))/(2*gamma(1 - p))
\[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{2}} \,d x } \]
\[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^p\,\left (d+e\,x\right )}{x^2} \,d x \]