3.116 \(\int \frac{(d^2-e^2 x^2)^{5/2}}{x^9 (d+e x)} \, dx\)

Optimal. Leaf size=201 \[ \frac{3 e^6 \sqrt{d^2-e^2 x^2}}{128 d^3 x^2}-\frac{e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}+\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{3 e^8 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{128 d^4} \]

[Out]

(3*e^6*Sqrt[d^2 - e^2*x^2])/(128*d^3*x^2) - (e^4*(d^2 - e^2*x^2)^(3/2))/(64*d^3*x^4) - (d^2 - e^2*x^2)^(5/2)/(
8*d*x^8) + (e*(d^2 - e^2*x^2)^(5/2))/(7*d^2*x^7) - (e^2*(d^2 - e^2*x^2)^(5/2))/(16*d^3*x^6) + (2*e^3*(d^2 - e^
2*x^2)^(5/2))/(35*d^4*x^5) - (3*e^8*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(128*d^4)

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Rubi [A]  time = 0.189864, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {850, 835, 807, 266, 47, 63, 208} \[ \frac{3 e^6 \sqrt{d^2-e^2 x^2}}{128 d^3 x^2}-\frac{e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}+\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{3 e^8 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{128 d^4} \]

Antiderivative was successfully verified.

[In]

Int[(d^2 - e^2*x^2)^(5/2)/(x^9*(d + e*x)),x]

[Out]

(3*e^6*Sqrt[d^2 - e^2*x^2])/(128*d^3*x^2) - (e^4*(d^2 - e^2*x^2)^(3/2))/(64*d^3*x^4) - (d^2 - e^2*x^2)^(5/2)/(
8*d*x^8) + (e*(d^2 - e^2*x^2)^(5/2))/(7*d^2*x^7) - (e^2*(d^2 - e^2*x^2)^(5/2))/(16*d^3*x^6) + (2*e^3*(d^2 - e^
2*x^2)^(5/2))/(35*d^4*x^5) - (3*e^8*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(128*d^4)

Rule 850

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

Rule 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 266

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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{x^9 (d+e x)} \, dx &=\int \frac{(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^9} \, dx\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{\int \frac{\left (8 d^2 e-3 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx}{8 d^2}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}+\frac{\int \frac{\left (21 d^3 e^2-16 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx}{56 d^4}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac{\int \frac{\left (96 d^4 e^3-21 d^3 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^6} \, dx}{336 d^6}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}+\frac{e^4 \int \frac{\left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx}{16 d^3}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}+\frac{e^4 \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{32 d^3}\\ &=-\frac{e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac{\left (3 e^6\right ) \operatorname{Subst}\left (\int \frac{\sqrt{d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{128 d^3}\\ &=\frac{3 e^6 \sqrt{d^2-e^2 x^2}}{128 d^3 x^2}-\frac{e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}+\frac{\left (3 e^8\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{256 d^3}\\ &=\frac{3 e^6 \sqrt{d^2-e^2 x^2}}{128 d^3 x^2}-\frac{e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac{\left (3 e^6\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{d^2}{e^2}-\frac{x^2}{e^2}} \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{128 d^3}\\ &=\frac{3 e^6 \sqrt{d^2-e^2 x^2}}{128 d^3 x^2}-\frac{e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac{3 e^8 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{128 d^4}\\ \end{align*}

Mathematica [A]  time = 0.236507, size = 139, normalized size = 0.69 \[ \frac{\sqrt{d^2-e^2 x^2} \left (840 d^5 e^2 x^2-1024 d^4 e^3 x^3-70 d^3 e^4 x^4+128 d^2 e^5 x^5+640 d^6 e x-560 d^7-105 d e^6 x^6+256 e^7 x^7\right )-105 e^8 x^8 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+105 e^8 x^8 \log (x)}{4480 d^4 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(d^2 - e^2*x^2)^(5/2)/(x^9*(d + e*x)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-560*d^7 + 640*d^6*e*x + 840*d^5*e^2*x^2 - 1024*d^4*e^3*x^3 - 70*d^3*e^4*x^4 + 128*d^2*e
^5*x^5 - 105*d*e^6*x^6 + 256*e^7*x^7) + 105*e^8*x^8*Log[x] - 105*e^8*x^8*Log[d + Sqrt[d^2 - e^2*x^2]])/(4480*d
^4*x^8)

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Maple [B]  time = 0.195, size = 571, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-e^2*x^2+d^2)^(5/2)/x^9/(e*x+d),x)

[Out]

-3/128*e^8/d^3/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)+1/7*e/d^4/x^7*(-e^2*x^2+d^2)^(7/2)
+3/640*e^8/d^9*(-e^2*x^2+d^2)^(5/2)+1/128*e^8/d^7*(-e^2*x^2+d^2)^(3/2)+3/128*e^8/d^5*(-e^2*x^2+d^2)^(1/2)-1/5*
e^8/d^9*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)-1/8/d^3/x^8*(-e^2*x^2+d^2)^(7/2)+1/5*e^3/d^6/x^5*(-e^2*x^2+d^2)^(
7/2)-13/64*e^4/d^7/x^4*(-e^2*x^2+d^2)^(7/2)-25/128*e^6/d^9/x^2*(-e^2*x^2+d^2)^(7/2)-1/4*e^9/d^8*(-(d/e+x)^2*e^
2+2*d*e*(d/e+x))^(3/2)*x-3/8*e^9/d^6*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x-3/8*e^9/d^4/(e^2)^(1/2)*arctan((e^
2)^(1/2)*x/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2))-3/16/d^5*e^2/x^6*(-e^2*x^2+d^2)^(7/2)+1/5*e^5/d^8/x^3*(-e^2*x
^2+d^2)^(7/2)+1/5*e^7/d^10/x*(-e^2*x^2+d^2)^(7/2)+1/5*e^9/d^10*x*(-e^2*x^2+d^2)^(5/2)+1/4*e^9/d^8*x*(-e^2*x^2+
d^2)^(3/2)+3/8*e^9/d^6*x*(-e^2*x^2+d^2)^(1/2)+3/8*e^9/d^4/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2
))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e^2*x^2+d^2)^(5/2)/x^9/(e*x+d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.85352, size = 292, normalized size = 1.45 \begin{align*} \frac{105 \, e^{8} x^{8} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (256 \, e^{7} x^{7} - 105 \, d e^{6} x^{6} + 128 \, d^{2} e^{5} x^{5} - 70 \, d^{3} e^{4} x^{4} - 1024 \, d^{4} e^{3} x^{3} + 840 \, d^{5} e^{2} x^{2} + 640 \, d^{6} e x - 560 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{4480 \, d^{4} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e^2*x^2+d^2)^(5/2)/x^9/(e*x+d),x, algorithm="fricas")

[Out]

1/4480*(105*e^8*x^8*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (256*e^7*x^7 - 105*d*e^6*x^6 + 128*d^2*e^5*x^5 - 70*d
^3*e^4*x^4 - 1024*d^4*e^3*x^3 + 840*d^5*e^2*x^2 + 640*d^6*e*x - 560*d^7)*sqrt(-e^2*x^2 + d^2))/(d^4*x^8)

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Sympy [C]  time = 30.168, size = 1171, normalized size = 5.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e**2*x**2+d**2)**(5/2)/x**9/(e*x+d),x)

[Out]

d**3*Piecewise((-d**2/(8*e*x**9*sqrt(d**2/(e**2*x**2) - 1)) + 7*e/(48*x**7*sqrt(d**2/(e**2*x**2) - 1)) + e**3/
(192*d**2*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 5*e**5/(384*d**4*x**3*sqrt(d**2/(e**2*x**2) - 1)) - 5*e**7/(128*d
**6*x*sqrt(d**2/(e**2*x**2) - 1)) + 5*e**8*acosh(d/(e*x))/(128*d**7), Abs(d**2)/(Abs(e**2)*Abs(x**2)) > 1), (I
*d**2/(8*e*x**9*sqrt(-d**2/(e**2*x**2) + 1)) - 7*I*e/(48*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**3/(192*d**2*
x**5*sqrt(-d**2/(e**2*x**2) + 1)) - 5*I*e**5/(384*d**4*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + 5*I*e**7/(128*d**6*
x*sqrt(-d**2/(e**2*x**2) + 1)) - 5*I*e**8*asin(d/(e*x))/(128*d**7), True)) - d**2*e*Piecewise((-e*sqrt(d**2/(e
**2*x**2) - 1)/(7*x**6) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(35*d**2*x**4) + 4*e**5*sqrt(d**2/(e**2*x**2) - 1)/(
105*d**4*x**2) + 8*e**7*sqrt(d**2/(e**2*x**2) - 1)/(105*d**6), Abs(d**2)/(Abs(e**2)*Abs(x**2)) > 1), (-I*e*sqr
t(-d**2/(e**2*x**2) + 1)/(7*x**6) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(35*d**2*x**4) + 4*I*e**5*sqrt(-d**2/(e
**2*x**2) + 1)/(105*d**4*x**2) + 8*I*e**7*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**6), True)) - d*e**2*Piecewise((-
d**2/(6*e*x**7*sqrt(d**2/(e**2*x**2) - 1)) + 5*e/(24*x**5*sqrt(d**2/(e**2*x**2) - 1)) + e**3/(48*d**2*x**3*sqr
t(d**2/(e**2*x**2) - 1)) - e**5/(16*d**4*x*sqrt(d**2/(e**2*x**2) - 1)) + e**6*acosh(d/(e*x))/(16*d**5), Abs(d*
*2)/(Abs(e**2)*Abs(x**2)) > 1), (I*d**2/(6*e*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - 5*I*e/(24*x**5*sqrt(-d**2/(e*
*2*x**2) + 1)) - I*e**3/(48*d**2*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**5/(16*d**4*x*sqrt(-d**2/(e**2*x**2)
+ 1)) - I*e**6*asin(d/(e*x))/(16*d**5), True)) + e**3*Piecewise((3*I*d**3*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*
x**5 + 15*e**2*x**7) - 4*I*d*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*I*e**6*x**
6*sqrt(-1 + e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - I*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)/(-15*d
**3*x**5 + 15*d*e**2*x**7), Abs(e**2*x**2)/Abs(d**2) > 1), (3*d**3*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 1
5*e**2*x**7) - 4*d*e**2*x**2*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*e**6*x**6*sqrt(1 - e*
*2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - e**4*x**4*sqrt(1 - e**2*x**2/d**2)/(-15*d**3*x**5 + 15*d*e
**2*x**7), True))

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e^2*x^2+d^2)^(5/2)/x^9/(e*x+d),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError