3.115 \(\int \frac{(d^2-e^2 x^2)^{5/2}}{x^8 (d+e x)} \, dx\)
Optimal. Leaf size=172 \[ -\frac{e^5 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}+\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac{e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^3} \]
[Out]
-(e^5*Sqrt[d^2 - e^2*x^2])/(16*d^2*x^2) + (e^3*(d^2 - e^2*x^2)^(3/2))/(24*d^2*x^4) - (d^2 - e^2*x^2)^(5/2)/(7*
d*x^7) + (e*(d^2 - e^2*x^2)^(5/2))/(6*d^2*x^6) - (2*e^2*(d^2 - e^2*x^2)^(5/2))/(35*d^3*x^5) + (e^7*ArcTanh[Sqr
t[d^2 - e^2*x^2]/d])/(16*d^3)
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Rubi [A] time = 0.153946, antiderivative size = 172, normalized size of antiderivative = 1.,
number of steps used = 9, 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} \[ -\frac{e^5 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}+\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac{e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^3} \]
Antiderivative was successfully verified.
[In]
Int[(d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)),x]
[Out]
-(e^5*Sqrt[d^2 - e^2*x^2])/(16*d^2*x^2) + (e^3*(d^2 - e^2*x^2)^(3/2))/(24*d^2*x^4) - (d^2 - e^2*x^2)^(5/2)/(7*
d*x^7) + (e*(d^2 - e^2*x^2)^(5/2))/(6*d^2*x^6) - (2*e^2*(d^2 - e^2*x^2)^(5/2))/(35*d^3*x^5) + (e^7*ArcTanh[Sqr
t[d^2 - e^2*x^2]/d])/(16*d^3)
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^8 (d+e x)} \, dx &=\int \frac{(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac{\int \frac{\left (7 d^2 e-2 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx}{7 d^2}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}+\frac{\int \frac{\left (12 d^3 e^2-7 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^6} \, dx}{42 d^4}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac{e^3 \int \frac{\left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx}{6 d^2}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac{e^3 \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{12 d^2}\\ &=\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac{e^5 \operatorname{Subst}\left (\int \frac{\sqrt{d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{16 d^2}\\ &=-\frac{e^5 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}+\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac{e^7 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{32 d^2}\\ &=-\frac{e^5 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}+\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac{e^5 \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 )}{16 d^2}\\ &=-\frac{e^5 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}+\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac{e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^3}\\ \end{align*}
Mathematica [A] time = 0.216027, size = 128, normalized size = 0.74 \[ \frac{\sqrt{d^2-e^2 x^2} \left (384 d^4 e^2 x^2-490 d^3 e^3 x^3-48 d^2 e^4 x^4+280 d^5 e x-240 d^6+105 d e^5 x^5-96 e^6 x^6\right )+105 e^7 x^7 \log \left (\sqrt{d^2-e^2 x^2}+d\right )-105 e^7 x^7 \log (x)}{1680 d^3 x^7} \]
Antiderivative was successfully verified.
[In]
Integrate[(d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)),x]
[Out]
(Sqrt[d^2 - e^2*x^2]*(-240*d^6 + 280*d^5*e*x + 384*d^4*e^2*x^2 - 490*d^3*e^3*x^3 - 48*d^2*e^4*x^4 + 105*d*e^5*
x^5 - 96*e^6*x^6) - 105*e^7*x^7*Log[x] + 105*e^7*x^7*Log[d + Sqrt[d^2 - e^2*x^2]])/(1680*d^3*x^7)
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Maple [B] time = 0.149, size = 546, normalized size = 3.2 \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^8/(e*x+d),x)
[Out]
-1/5*e^4/d^7/x^3*(-e^2*x^2+d^2)^(7/2)-1/5*e^6/d^9/x*(-e^2*x^2+d^2)^(7/2)-1/5*e^8/d^9*x*(-e^2*x^2+d^2)^(5/2)-1/
4*e^8/d^7*x*(-e^2*x^2+d^2)^(3/2)-3/8*e^8/d^5*x*(-e^2*x^2+d^2)^(1/2)-3/8*e^8/d^3/(e^2)^(1/2)*arctan((e^2)^(1/2)
*x/(-e^2*x^2+d^2)^(1/2))+1/6*e/d^4/x^6*(-e^2*x^2+d^2)^(7/2)+1/5*e^7/d^8*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)-1
/80*e^7/d^8*(-e^2*x^2+d^2)^(5/2)-1/48*e^7/d^6*(-e^2*x^2+d^2)^(3/2)-1/16*e^7/d^4*(-e^2*x^2+d^2)^(1/2)+1/16*e^7/
d^2/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-1/7/d^3/x^7*(-e^2*x^2+d^2)^(7/2)-1/5*e^2/d^5/
x^5*(-e^2*x^2+d^2)^(7/2)+5/24*e^3/d^6/x^4*(-e^2*x^2+d^2)^(7/2)+3/16*e^5/d^8/x^2*(-e^2*x^2+d^2)^(7/2)+1/4*e^8/d
^7*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2)*x+3/8*e^8/d^5*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x+3/8*e^8/d^3/(e^2)
^(1/2)*arctan((e^2)^(1/2)*x/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(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^8/(e*x+d),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 1.76293, size = 266, normalized size = 1.55 \begin{align*} -\frac{105 \, e^{7} x^{7} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (96 \, e^{6} x^{6} - 105 \, d e^{5} x^{5} + 48 \, d^{2} e^{4} x^{4} + 490 \, d^{3} e^{3} x^{3} - 384 \, d^{4} e^{2} x^{2} - 280 \, d^{5} e x + 240 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{1680 \, d^{3} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(5/2)/x^8/(e*x+d),x, algorithm="fricas")
[Out]
-1/1680*(105*e^7*x^7*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (96*e^6*x^6 - 105*d*e^5*x^5 + 48*d^2*e^4*x^4 + 490*d
^3*e^3*x^3 - 384*d^4*e^2*x^2 - 280*d^5*e*x + 240*d^6)*sqrt(-e^2*x^2 + d^2))/(d^3*x^7)
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Sympy [C] time = 26.6652, size = 1049, normalized size = 6.1 \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**8/(e*x+d),x)
[Out]
d**3*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*sqrt(-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**2*e*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*sqrt(d**2/(e**2*x**2) - 1)) - e**5/(16*d**4*x*sqrt(d**2/(e**2*x**2) - 1)) + e**6*a
cosh(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)) - d*e**2*Piecewise((3*I*d**3*sqr
t(-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 + 15*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)) + e**3*Piecewise((-d**2/(4*e*x**5*sqrt(d**2/(e**2*x**2) - 1
)) + 3*e/(8*x**3*sqrt(d**2/(e**2*x**2) - 1)) - e**3/(8*d**2*x*sqrt(d**2/(e**2*x**2) - 1)) + e**4*acosh(d/(e*x)
)/(8*d**3), Abs(d**2)/(Abs(e**2)*Abs(x**2)) > 1), (I*d**2/(4*e*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - 3*I*e/(8*x*
*3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**3/(8*d**2*x*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**4*asin(d/(e*x))/(8*d**3
), True))
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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^8/(e*x+d),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError