3.4.1 \(\int \frac {(d^2-e^2 x^2)^p}{x (d+e x)^4} \, dx\) [301]

Optimal. Leaf size=204 \[ \frac {4 d^2 \left (d^2-e^2 x^2\right )^{-3+p}}{3-p}-\frac {4 d e x \left (d^2-e^2 x^2\right )^{-3+p}}{5-2 p}-\frac {\left (d^2-e^2 x^2\right )^{-2+p}}{2 (2-p)}-\frac {8 e (2-p) x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1}{2},4-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^5 (5-2 p)}+\frac {\left (d^2-e^2 x^2\right )^{-2+p} \, _2F_1\left (1,-2+p;-1+p;1-\frac {e^2 x^2}{d^2}\right )}{2 (2-p)} \]

[Out]

4*d^2*(-e^2*x^2+d^2)^(-3+p)/(3-p)-4*d*e*x*(-e^2*x^2+d^2)^(-3+p)/(5-2*p)-1/2*(-e^2*x^2+d^2)^(-2+p)/(2-p)-8*e*(2
-p)*x*(-e^2*x^2+d^2)^p*hypergeom([1/2, 4-p],[3/2],e^2*x^2/d^2)/d^5/(5-2*p)/((1-e^2*x^2/d^2)^p)+1/2*(-e^2*x^2+d
^2)^(-2+p)*hypergeom([1, -2+p],[-1+p],1-e^2*x^2/d^2)/(2-p)

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Rubi [A]
time = 0.13, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {866, 1666, 1265, 965, 80, 67, 396, 252, 251} \begin {gather*} \frac {\left (d^2-e^2 x^2\right )^{p-2} \, _2F_1\left (1,p-2;p-1;1-\frac {e^2 x^2}{d^2}\right )}{2 (2-p)}-\frac {4 d e x \left (d^2-e^2 x^2\right )^{p-3}}{5-2 p}+\frac {4 d^2 \left (d^2-e^2 x^2\right )^{p-3}}{3-p}-\frac {\left (d^2-e^2 x^2\right )^{p-2}}{2 (2-p)}-\frac {8 e (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},4-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^5 (5-2 p)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*d^2*(d^2 - e^2*x^2)^(-3 + p))/(3 - p) - (4*d*e*x*(d^2 - e^2*x^2)^(-3 + p))/(5 - 2*p) - (d^2 - e^2*x^2)^(-2
+ p)/(2*(2 - p)) - (8*e*(2 - p)*x*(d^2 - e^2*x^2)^p*Hypergeometric2F1[1/2, 4 - p, 3/2, (e^2*x^2)/d^2])/(d^5*(5
- 2*p)*(1 - (e^2*x^2)/d^2)^p) + ((d^2 - e^2*x^2)^(-2 + p)*Hypergeometric2F1[1, -2 + p, -1 + p, 1 - (e^2*x^2)/
d^2])/(2*(2 - 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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c
, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimplerQ[p, 1]

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 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

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 965

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Simp[c^p*(d + e*x)^(m + 2*p)*((f + g*x)^(n + 1)/(g*e^(2*p)*(m + n + 2*p + 1))), x] + Dist[1/(g*e^(2*p)*(m +
n + 2*p + 1)), Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^(2*p)*(a + b*x + c*x^2)^p - c^p*
(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p)*(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x
] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && NeQ[m + n + 2*
p + 1, 0] && (IntegerQ[n] ||  !IntegerQ[m])

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 1666

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[x^m*Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2)^p, x] + Int[x^(m + 1)*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2)^p, x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2] && IGtQ[m, -2] &&  !
IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^p}{x (d+e x)^4} \, dx &=\int \frac {(d-e x)^4 \left (d^2-e^2 x^2\right )^{-4+p}}{x} \, dx\\ &=\int \left (d^2-e^2 x^2\right )^{-4+p} \left (-4 d^3 e-4 d e^3 x^2\right ) \, dx+\int \frac {\left (d^2-e^2 x^2\right )^{-4+p} \left (d^4+6 d^2 e^2 x^2+e^4 x^4\right )}{x} \, dx\\ &=-\frac {4 d e x \left (d^2-e^2 x^2\right )^{-3+p}}{5-2 p}+\frac {1}{2} \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{-4+p} \left (d^4+6 d^2 e^2 x+e^4 x^2\right )}{x} \, dx,x,x^2\right )-\frac {\left (8 d^3 e (2-p)\right ) \int \left (d^2-e^2 x^2\right )^{-4+p} \, dx}{5-2 p}\\ &=-\frac {4 d e x \left (d^2-e^2 x^2\right )^{-3+p}}{5-2 p}-\frac {\left (d^2-e^2 x^2\right )^{-2+p}}{2 (2-p)}-\frac {\text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{-4+p} \left (-d^4 e^4 (2-p)-7 d^2 e^6 (2-p) x\right )}{x} \, dx,x,x^2\right )}{2 e^4 (2-p)}-\frac {\left (8 e (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 )^{-4+p} \, dx}{d^5 (5-2 p)}\\ &=\frac {4 d^2 \left (d^2-e^2 x^2\right )^{-3+p}}{3-p}-\frac {4 d e x \left (d^2-e^2 x^2\right )^{-3+p}}{5-2 p}-\frac {\left (d^2-e^2 x^2\right )^{-2+p}}{2 (2-p)}-\frac {8 e (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},4-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^5 (5-2 p)}+\frac {1}{2} d^2 \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{-3+p}}{x} \, dx,x,x^2\right )\\ &=\frac {4 d^2 \left (d^2-e^2 x^2\right )^{-3+p}}{3-p}-\frac {4 d e x \left (d^2-e^2 x^2\right )^{-3+p}}{5-2 p}-\frac {\left (d^2-e^2 x^2\right )^{-2+p}}{2 (2-p)}-\frac {8 e (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},4-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^5 (5-2 p)}+\frac {\left (d^2-e^2 x^2\right )^{-2+p} \, _2F_1\left (1,-2+p;-1+p;1-\frac {e^2 x^2}{d^2}\right )}{2 (2-p)}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(417\) vs. \(2(204)=408\).
time = 0.45, size = 417, normalized size = 2.04 \begin {gather*} \frac {2^{-4+p} \left (1-\frac {d^2}{e^2 x^2}\right )^{-p} \left (1+\frac {e x}{d}\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (8 p \left (1-\frac {d^2}{e^2 x^2}\right )^p (d-e x) \, _2F_1\left (1-p,1+p;2+p;\frac {d-e x}{2 d}\right )+4 p \left (1-\frac {d^2}{e^2 x^2}\right )^p (d-e x) \, _2F_1\left (2-p,1+p;2+p;\frac {d-e x}{2 d}\right )+2 d p \left (1-\frac {d^2}{e^2 x^2}\right )^p \, _2F_1\left (3-p,1+p;2+p;\frac {d-e x}{2 d}\right )-2 e p \left (1-\frac {d^2}{e^2 x^2}\right )^p x \, _2F_1\left (3-p,1+p;2+p;\frac {d-e x}{2 d}\right )+d p \left (1-\frac {d^2}{e^2 x^2}\right )^p \, _2F_1\left (4-p,1+p;2+p;\frac {d-e x}{2 d}\right )-e p \left (1-\frac {d^2}{e^2 x^2}\right )^p x \, _2F_1\left (4-p,1+p;2+p;\frac {d-e x}{2 d}\right )+8 d \left (\frac {1}{2}+\frac {e x}{2 d}\right )^p \, _2F_1\left (-p,-p;1-p;\frac {d^2}{e^2 x^2}\right )+8 d p \left (\frac {1}{2}+\frac {e x}{2 d}\right )^p \, _2F_1\left (-p,-p;1-p;\frac {d^2}{e^2 x^2}\right )\right )}{d^5 p (1+p)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2^(-4 + p)*(d^2 - e^2*x^2)^p*(8*p*(1 - d^2/(e^2*x^2))^p*(d - e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d -
e*x)/(2*d)] + 4*p*(1 - d^2/(e^2*x^2))^p*(d - e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + 2
*d*p*(1 - d^2/(e^2*x^2))^p*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] - 2*e*p*(1 - d^2/(e^2*x^2))
^p*x*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + d*p*(1 - d^2/(e^2*x^2))^p*Hypergeometric2F1[4 -
p, 1 + p, 2 + p, (d - e*x)/(2*d)] - e*p*(1 - d^2/(e^2*x^2))^p*x*Hypergeometric2F1[4 - p, 1 + p, 2 + p, (d - e
*x)/(2*d)] + 8*d*(1/2 + (e*x)/(2*d))^p*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)] + 8*d*p*(1/2 + (e*x)/(2
*d))^p*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)]))/(d^5*p*(1 + p)*(1 - d^2/(e^2*x^2))^p*(1 + (e*x)/d)^p)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (-e^{2} x^{2}+d^{2}\right )^{p}}{x \left (e x +d \right )^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((-e^2*x^2+d^2)^p/x/(e*x+d)^4,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/(e*x+d)^4,x, algorithm="maxima")

[Out]

integrate((-x^2*e^2 + d^2)^p/((x*e + d)^4*x), 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/(e*x+d)^4,x, algorithm="fricas")

[Out]

integral((-x^2*e^2 + d^2)^p/(x^5*e^4 + 4*d*x^4*e^3 + 6*d^2*x^3*e^2 + 4*d^3*x^2*e + d^4*x), 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 \left (d + e x\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e**2*x**2+d**2)**p/x/(e*x+d)**4,x)

[Out]

Integral((-(-d + e*x)*(d + e*x))**p/(x*(d + e*x)**4), 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/(e*x+d)^4,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^p}{x\,{\left (d+e\,x\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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