Integrand size = 25, antiderivative size = 171 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x} \, dx=-\frac {3 d \left (d^2-e^2 x^2\right )^{1+p}}{2 (1+p)}-\frac {e x \left (d^2-e^2 x^2\right )^{1+p}}{3+2 p}+\frac {2 d^2 e (5+3 p) x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )}{3+2 p}-\frac {d \left (d^2-e^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1-\frac {e^2 x^2}{d^2}\right )}{2 (1+p)} \]
-3/2*d*(-e^2*x^2+d^2)^(p+1)/(p+1)-e*x*(-e^2*x^2+d^2)^(p+1)/(3+2*p)+2*d^2*e *(5+3*p)*x*(-e^2*x^2+d^2)^p*hypergeom([1/2, -p],[3/2],e^2*x^2/d^2)/(3+2*p) /((1-e^2*x^2/d^2)^p)-1/2*d*(-e^2*x^2+d^2)^(p+1)*hypergeom([1, p+1],[2+p],1 -e^2*x^2/d^2)/(p+1)
Time = 0.28 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x} \, dx=\frac {1}{6} \left (d^2-e^2 x^2\right )^p \left (-\frac {9 d \left (d^2-e^2 x^2\right )}{1+p}+18 d^2 e x \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )-\frac {3 d \left (d^2-e^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1-\frac {e^2 x^2}{d^2}\right )}{1+p}+2 e^3 x^3 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},\frac {e^2 x^2}{d^2}\right )\right ) \]
((d^2 - e^2*x^2)^p*((-9*d*(d^2 - e^2*x^2))/(1 + p) + (18*d^2*e*x*Hypergeom etric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p - (3*d*(d^2 - e^2*x^2)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2])/(1 + p) + (2*e^3*x^3*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2) /d^2)^p))/6
Time = 0.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {543, 299, 238, 237, 354, 27, 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.
| \(\displaystyle \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x} \, dx\) |
| \(\Big \downarrow \) 543 |
| \(\displaystyle \int \left (d^2-e^2 x^2\right )^p \left (x^2 e^3+3 d^2 e\right )dx+\int \frac {\left (d^2-e^2 x^2\right )^p \left (d^3+3 e^2 x^2 d\right )}{x}dx\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {2 d^2 e (3 p+5) \int \left (d^2-e^2 x^2\right )^pdx}{2 p+3}+\int \frac {\left (d^2-e^2 x^2\right )^p \left (d^3+3 e^2 x^2 d\right )}{x}dx-\frac {e x \left (d^2-e^2 x^2\right )^{p+1}}{2 p+3}\) |
| \(\Big \downarrow \) 238 |
| \(\displaystyle \frac {2 d^2 e (3 p+5) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \int \left (1-\frac {e^2 x^2}{d^2}\right )^pdx}{2 p+3}+\int \frac {\left (d^2-e^2 x^2\right )^p \left (d^3+3 e^2 x^2 d\right )}{x}dx-\frac {e x \left (d^2-e^2 x^2\right )^{p+1}}{2 p+3}\) |
| \(\Big \downarrow \) 237 |
| \(\displaystyle \int \frac {\left (d^2-e^2 x^2\right )^p \left (d^3+3 e^2 x^2 d\right )}{x}dx+\frac {2 d^2 e (3 p+5) x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )}{2 p+3}-\frac {e x \left (d^2-e^2 x^2\right )^{p+1}}{2 p+3}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {d \left (d^2-e^2 x^2\right )^p \left (d^2+3 e^2 x^2\right )}{x^2}dx^2+\frac {2 d^2 e (3 p+5) x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )}{2 p+3}-\frac {e x \left (d^2-e^2 x^2\right )^{p+1}}{2 p+3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} d \int \frac {\left (d^2-e^2 x^2\right )^p \left (d^2+3 e^2 x^2\right )}{x^2}dx^2+\frac {2 d^2 e (3 p+5) x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )}{2 p+3}-\frac {e x \left (d^2-e^2 x^2\right )^{p+1}}{2 p+3}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {1}{2} d \left (d^2 \int \frac {\left (d^2-e^2 x^2\right )^p}{x^2}dx^2-\frac {3 \left (d^2-e^2 x^2\right )^{p+1}}{p+1}\right )+\frac {2 d^2 e (3 p+5) x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )}{2 p+3}-\frac {e x \left (d^2-e^2 x^2\right )^{p+1}}{2 p+3}\) |
| \(\Big \downarrow \) 75 |
| \(\displaystyle \frac {2 d^2 e (3 p+5) x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )}{2 p+3}+\frac {1}{2} d \left (-\frac {\left (d^2-e^2 x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,1-\frac {e^2 x^2}{d^2}\right )}{p+1}-\frac {3 \left (d^2-e^2 x^2\right )^{p+1}}{p+1}\right )-\frac {e x \left (d^2-e^2 x^2\right )^{p+1}}{2 p+3}\) |
-((e*x*(d^2 - e^2*x^2)^(1 + p))/(3 + 2*p)) + (2*d^2*e*(5 + 3*p)*x*(d^2 - e ^2*x^2)^p*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2])/((3 + 2*p)*(1 - (e^2*x^2)/d^2)^p) + (d*((-3*(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
3.3.64.3.1 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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 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])
\[\int \frac {\left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{x}d x\]
\[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x} \, dx=\int { \frac {{\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x} \,d x } \]
Time = 3.81 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.04 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x} \, dx=- \frac {d^{3} e^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (1 - p\right )} + 3 d^{2} d^{2 p} e x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + 3 d e^{2} \left (\begin {cases} \frac {x^{2} \left (d^{2}\right )^{p}}{2} & \text {for}\: e^{2} = 0 \\- \frac {\begin {cases} \frac {\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (d^{2} - e^{2} x^{2} \right )} & \text {otherwise} \end {cases}}{2 e^{2}} & \text {otherwise} \end {cases}\right ) + \frac {d^{2 p} e^{3} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - p \\ \frac {5}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{3} \]
-d**3*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)) + 3*d**2*d**(2*p)*e*x*hyper((1/2, -p), (3 /2,), e**2*x**2*exp_polar(2*I*pi)/d**2) + 3*d*e**2*Piecewise((x**2*(d**2)* *p/2, Eq(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)) + d**(2*p)*e**3*x* *3*hyper((3/2, -p), (5/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/3
\[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x} \, dx=\int { \frac {{\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x} \,d x } \]
\[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x} \, dx=\int { \frac {{\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^p\,{\left (d+e\,x\right )}^3}{x} \,d x \]