3.3.82 \(\int \frac {(d^2-e^2 x^2)^p}{x^2 (d+e x)^2} \, dx\) [282]

Optimal. Leaf size=137 \[ -\frac {\left (d^2-e^2 x^2\right )^{-1+p}}{x}+\frac {2 e^2 (2-p) x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1}{2},2-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^4}-\frac {e \left (d^2-e^2 x^2\right )^{-1+p} \, _2F_1\left (1,-1+p;p;1-\frac {e^2 x^2}{d^2}\right )}{d (1-p)} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {866, 1821, 778, 272, 67, 252, 251} \begin {gather*} -\frac {e \left (d^2-e^2 x^2\right )^{p-1} \, _2F_1\left (1,p-1;p;1-\frac {e^2 x^2}{d^2}\right )}{d (1-p)}-\frac {\left (d^2-e^2 x^2\right )^{p-1}}{x}+\frac {2 e^2 (2-p) x \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac {1}{2},2-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((d^2 - e^2*x^2)^(-1 + p)/x) + (2*e^2*(2 - p)*x*(d^2 - e^2*x^2)^p*Hypergeometric2F1[1/2, 2 - p, 3/2, (e^2*x^2
)/d^2])/(d^4*(1 - (e^2*x^2)/d^2)^p) - (e*(d^2 - e^2*x^2)^(-1 + p)*Hypergeometric2F1[1, -1 + p, p, 1 - (e^2*x^2
)/d^2])/(d*(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 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[p]), Int[(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 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 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 866

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(2*m)/a
^m, Int[(f + g*x)^n*((a + c*x^2)^(m + p)/(d - e*x)^m), x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f
- d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[f, 0] && ILtQ[m, -1] &&  !(IGtQ[n, 0] && ILtQ[m +
n, 0] &&  !GtQ[p, 1])

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

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Mathematica [A]
time = 0.46, size = 223, normalized size = 1.63 \begin {gather*} \frac {\left (d^2-e^2 x^2\right )^p \left (-\frac {4 d^2 \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 {2^{2+p} e (-d+e x) \left (1+\frac {e x}{d}\right )^{-p} \, _2F_1\left (1-p,1+p;2+p;\frac {d-e x}{2 d}\right )}{1+p}+\frac {2^p e (-d+e x) \left (1+\frac {e x}{d}\right )^{-p} \, _2F_1\left (2-p,1+p;2+p;\frac {d-e x}{2 d}\right )}{1+p}-\frac {4 d e \left (1-\frac {d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac {d^2}{e^2 x^2}\right )}{p}\right )}{4 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d^2 - e^2*x^2)^p*((-4*d^2*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) + (2^(2
+ p)*e*(-d + e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (2^p*e
*(-d + e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) - (4*d*e*Hyperg
eometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)])/(p*(1 - d^2/(e^2*x^2))^p)))/(4*d^4)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (-e^{2} x^{2}+d^{2}\right )^{p}}{x^{2} \left (e x +d \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-x^2*e^2 + d^2)^p/(x^4*e^2 + 2*d*x^3*e + d^2*x^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{x^{2} \left (d + e x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^p}{x^2\,{\left (d+e\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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