Integrand size = 25, antiderivative size = 128 \[ \int \frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x} \, dx=-\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 (1+p)}+2 d e x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1-\frac {e^2 x^2}{d^2}\right )}{2 (1+p)} \]
-1/2*(-e^2*x^2+d^2)^(p+1)/(p+1)+2*d*e*x*(-e^2*x^2+d^2)^p*hypergeom([1/2, - p],[3/2],e^2*x^2/d^2)/((1-e^2*x^2/d^2)^p)-1/2*(-e^2*x^2+d^2)^(p+1)*hyperge om([1, p+1],[2+p],1-e^2*x^2/d^2)/(p+1)
Time = 0.20 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.80 \[ \int \frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x} \, dx=\frac {1}{2} \left (d^2-e^2 x^2\right )^p \left (4 d e x \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )-\frac {\left (d^2-e^2 x^2\right ) \left (1+\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1-\frac {e^2 x^2}{d^2}\right )\right )}{1+p}\right ) \]
((d^2 - e^2*x^2)^p*((4*d*e*x*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2 ])/(1 - (e^2*x^2)/d^2)^p - ((d^2 - e^2*x^2)*(1 + Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2]))/(1 + p)))/2
Time = 0.27 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {543, 27, 238, 237, 354, 90, 75}
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.
\(\displaystyle \int \frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x} \, dx\) |
\(\Big \downarrow \) 543 |
\(\displaystyle \int 2 d e \left (d^2-e^2 x^2\right )^pdx+\int \frac {\left (d^2-e^2 x^2\right )^p \left (d^2+e^2 x^2\right )}{x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 d e \int \left (d^2-e^2 x^2\right )^pdx+\int \frac {\left (d^2-e^2 x^2\right )^p \left (d^2+e^2 x^2\right )}{x}dx\) |
\(\Big \downarrow \) 238 |
\(\displaystyle 2 d e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \int \left (1-\frac {e^2 x^2}{d^2}\right )^pdx+\int \frac {\left (d^2-e^2 x^2\right )^p \left (d^2+e^2 x^2\right )}{x}dx\) |
\(\Big \downarrow \) 237 |
\(\displaystyle \int \frac {\left (d^2-e^2 x^2\right )^p \left (d^2+e^2 x^2\right )}{x}dx+2 d e x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {\left (d^2-e^2 x^2\right )^p \left (d^2+e^2 x^2\right )}{x^2}dx^2+2 d e x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {1}{2} \left (d^2 \int \frac {\left (d^2-e^2 x^2\right )^p}{x^2}dx^2-\frac {\left (d^2-e^2 x^2\right )^{p+1}}{p+1}\right )+2 d e x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )\) |
\(\Big \downarrow \) 75 |
\(\displaystyle 2 d e x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )+\frac {1}{2} \left (-\frac {\left (d^2-e^2 x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,1-\frac {e^2 x^2}{d^2}\right )}{p+1}-\frac {\left (d^2-e^2 x^2\right )^{p+1}}{p+1}\right )\) |
(2*d*e*x*(d^2 - e^2*x^2)^p*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2]) /(1 - (e^2*x^2)/d^2)^p + (-((d^2 - e^2*x^2)^(1 + p)/(1 + p)) - ((d^2 - e^2 *x^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2])/(1 + p))/2
3.3.55.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[- p, 1/2, 1/2 + 1, (-b)*(x^2/a)], x] /; FreeQ[{a, b, p}, x] && !IntegerQ[2*p ] && GtQ[a, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2) ^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[(1 + b*(x^2/a))^p, x], x] / ; FreeQ[{a, b, p}, x] && !IntegerQ[2*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_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Module[{k}, Int[x^m*Sum[Binomial[n, 2*k]*c^(n - 2*k)*d^(2*k)*x^(2*k), {k, 0, n/2}]*(a + b*x^2)^p, x] + Int[x^(m + 1)*Sum[Binomial[n, 2*k + 1]*c^ (n - 2*k - 1)*d^(2*k + 1)*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2)^p, x]] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[n, 1] && IntegerQ[m] && !IntegerQ[2*p] && !(EqQ[m, 1] && EqQ[b*c^2 + a*d^2, 0])
\[\int \frac {\left (e x +d \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{x}d x\]
\[ \int \frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x} \, dx=\int { \frac {{\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x} \,d x } \]
Time = 3.04 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.06 \[ \int \frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x} \, dx=- \frac {d^{2} e^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (1 - p\right )} + 2 d d^{2 p} e x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + e^{2} \left (\begin {cases} \frac {x^{2} \left (d^{2}\right )^{p}}{2} & \text {for}\: e^{2} = 0 \\- \frac {\begin {cases} \frac {\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (d^{2} - e^{2} x^{2} \right )} & \text {otherwise} \end {cases}}{2 e^{2}} & \text {otherwise} \end {cases}\right ) \]
-d**2*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)) + 2*d*d**(2*p)*e*x*hyper((1/2, -p), (3/2, ), e**2*x**2*exp_polar(2*I*pi)/d**2) + e**2*Piecewise((x**2*(d**2)**p/2, E q(e**2, 0)), (-Piecewise(((d**2 - e**2*x**2)**(p + 1)/(p + 1), Ne(p, -1)), (log(d**2 - e**2*x**2), True))/(2*e**2), True))
\[ \int \frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x} \, dx=\int { \frac {{\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x} \,d x } \]
\[ \int \frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x} \, dx=\int { \frac {{\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^p\,{\left (d+e\,x\right )}^2}{x} \,d x \]