\(\int \frac {(d+e x)^2 (d^2-e^2 x^2)^p}{x^3} \, dx\) [257]

Optimal result

Integrand size = 25, antiderivative size = 139 \[ \int \frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x^3} \, dx=-\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 x^2}-\frac {2 d e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e^2 (1-p) \left (d^2-e^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 (1+p)} \]

[Out]

-1/2*(-e^2*x^2+d^2)^(p+1)/x^2-2*d*e*(-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^2*(1-p)*(-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)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1821, 778, 372, 371, 272, 67} \[ \int \frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x^3} \, dx=-\frac {2 d e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e^2 (1-p) \left (d^2-e^2 x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 (p+1)}-\frac {\left (d^2-e^2 x^2\right )^{p+1}}{2 x^2} \]

[In]

Int[((d + e*x)^2*(d^2 - e^2*x^2)^p)/x^3,x]

[Out]

-1/2*(d^2 - e^2*x^2)^(1 + p)/x^2 - (2*d*e*(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^2*(1 - p)*(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))

Rule 67

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] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/
(1 + b*(x^n/a))^FracPart[p]), Int[(c*x)^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 778

Int[(x_)^(m_.)*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[f, Int[x^m*(a + c*x^2)^p, x]
, x] + Dist[g, Int[x^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, f, g, p}, x] && IntegerQ[m] &&  !IntegerQ[2
*p]

Rule 1821

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 x^2}-\frac {\int \frac {\left (-4 d^3 e-2 d^2 e^2 (1-p) x\right ) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx}{2 d^2} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 x^2}+(2 d e) \int \frac {\left (d^2-e^2 x^2\right )^p}{x^2} \, dx+\left (e^2 (1-p)\right ) \int \frac {\left (d^2-e^2 x^2\right )^p}{x} \, dx \\ & = -\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 x^2}+\frac {1}{2} \left (e^2 (1-p)\right ) \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^p}{x} \, dx,x,x^2\right )+\left (2 d e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int \frac {\left (1-\frac {e^2 x^2}{d^2}\right )^p}{x^2} \, dx \\ & = -\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 x^2}-\frac {2 d e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e^2 (1-p) \left (d^2-e^2 x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 (1+p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x^3} \, dx=\frac {e \left (d^2-e^2 x^2\right )^p \left (-\frac {4 d^3 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},\frac {e^2 x^2}{d^2}\right )}{x}+\frac {e \left (-d^2+e^2 x^2\right ) \left (\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1-\frac {e^2 x^2}{d^2}\right )+\operatorname {Hypergeometric2F1}\left (2,1+p,2+p,1-\frac {e^2 x^2}{d^2}\right )\right )}{1+p}\right )}{2 d^2} \]

[In]

Integrate[((d + e*x)^2*(d^2 - e^2*x^2)^p)/x^3,x]

[Out]

(e*(d^2 - e^2*x^2)^p*((-4*d^3*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)*(Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2] + Hypergeometric2F1[2, 1 + p, 2 + p, 1
- (e^2*x^2)/d^2]))/(1 + p)))/(2*d^2)

Maple [F]

\[\int \frac {\left (e x +d \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{x^{3}}d x\]

[In]

int((e*x+d)^2*(-e^2*x^2+d^2)^p/x^3,x)

[Out]

int((e*x+d)^2*(-e^2*x^2+d^2)^p/x^3,x)

Fricas [F]

\[ \int \frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x^3} \, dx=\int { \frac {{\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{3}} \,d x } \]

[In]

integrate((e*x+d)^2*(-e^2*x^2+d^2)^p/x^3,x, algorithm="fricas")

[Out]

integral((e^2*x^2 + 2*d*e*x + d^2)*(-e^2*x^2 + d^2)^p/x^3, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.58 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x^3} \, dx=- \frac {d^{2} e^{2 p} x^{2 p - 2} e^{i \pi p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (2 - p\right )} - \frac {2 d d^{2 p} e {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{x} - \frac {e^{2} e^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (1 - p\right )} \]

[In]

integrate((e*x+d)**2*(-e**2*x**2+d**2)**p/x**3,x)

[Out]

-d**2*e**(2*p)*x**(2*p - 2)*exp(I*pi*p)*gamma(1 - p)*hyper((-p, 1 - p), (2 - p,), d**2/(e**2*x**2))/(2*gamma(2
- p)) - 2*d*d**(2*p)*e*hyper((-1/2, -p), (1/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/x - e**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))

Maxima [F]

\[ \int \frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x^3} \, dx=\int { \frac {{\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{3}} \,d x } \]

[In]

integrate((e*x+d)^2*(-e^2*x^2+d^2)^p/x^3,x, algorithm="maxima")

[Out]

integrate((e*x + d)^2*(-e^2*x^2 + d^2)^p/x^3, x)

Giac [F]

\[ \int \frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x^3} \, dx=\int { \frac {{\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{3}} \,d x } \]

[In]

integrate((e*x+d)^2*(-e^2*x^2+d^2)^p/x^3,x, algorithm="giac")

[Out]

integrate((e*x + d)^2*(-e^2*x^2 + d^2)^p/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x^3} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^p\,{\left (d+e\,x\right )}^2}{x^3} \,d x \]

[In]

int(((d^2 - e^2*x^2)^p*(d + e*x)^2)/x^3,x)

[Out]

int(((d^2 - e^2*x^2)^p*(d + e*x)^2)/x^3, x)