3.170 \(\int \frac{(d^2-e^2 x^2)^{5/2}}{x^8 (d+e x)^2} \, dx\)

Optimal. Leaf size=198 \[ \frac{e^5 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2}-\frac{22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}+\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac{11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}-\frac{e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^4} \]

[Out]

(e^5*Sqrt[d^2 - e^2*x^2])/(8*d^3*x^2) - (d^2 - e^2*x^2)^(3/2)/(7*x^7) + (e*(d^2 - e^2*x^2)^(3/2))/(3*d*x^6) -
(11*e^2*(d^2 - e^2*x^2)^(3/2))/(35*d^2*x^5) + (e^3*(d^2 - e^2*x^2)^(3/2))/(4*d^3*x^4) - (22*e^4*(d^2 - e^2*x^2
)^(3/2))/(105*d^4*x^3) - (e^7*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(8*d^4)

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Rubi [A]  time = 0.235818, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {852, 1807, 835, 807, 266, 47, 63, 208} \[ \frac{e^5 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2}-\frac{22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}+\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac{11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}-\frac{e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^4} \]

Antiderivative was successfully verified.

[In]

Int[(d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)^2),x]

[Out]

(e^5*Sqrt[d^2 - e^2*x^2])/(8*d^3*x^2) - (d^2 - e^2*x^2)^(3/2)/(7*x^7) + (e*(d^2 - e^2*x^2)^(3/2))/(3*d*x^6) -
(11*e^2*(d^2 - e^2*x^2)^(3/2))/(35*d^2*x^5) + (e^3*(d^2 - e^2*x^2)^(3/2))/(4*d^3*x^4) - (22*e^4*(d^2 - e^2*x^2
)^(3/2))/(105*d^4*x^3) - (e^7*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(8*d^4)

Rule 852

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 1807

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])

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)^2} \, dx &=\int \frac{(d-e x)^2 \sqrt{d^2-e^2 x^2}}{x^8} \, dx\\ &=-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}-\frac{\int \frac{\left (14 d^3 e-11 d^2 e^2 x\right ) \sqrt{d^2-e^2 x^2}}{x^7} \, dx}{7 d^2}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}+\frac{\int \frac{\left (66 d^4 e^2-42 d^3 e^3 x\right ) \sqrt{d^2-e^2 x^2}}{x^6} \, dx}{42 d^4}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac{11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}-\frac{\int \frac{\left (210 d^5 e^3-132 d^4 e^4 x\right ) \sqrt{d^2-e^2 x^2}}{x^5} \, dx}{210 d^6}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac{11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}+\frac{\int \frac{\left (528 d^6 e^4-210 d^5 e^5 x\right ) \sqrt{d^2-e^2 x^2}}{x^4} \, dx}{840 d^8}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac{11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac{22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\frac{e^5 \int \frac{\sqrt{d^2-e^2 x^2}}{x^3} \, dx}{4 d^3}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac{11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac{22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\frac{e^5 \operatorname{Subst}\left (\int \frac{\sqrt{d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{8 d^3}\\ &=\frac{e^5 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac{11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac{22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}+\frac{e^7 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{16 d^3}\\ &=\frac{e^5 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac{11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac{22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\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 )}{8 d^3}\\ &=\frac{e^5 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac{11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac{22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\frac{e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^4}\\ \end{align*}

Mathematica [A]  time = 0.261762, size = 128, normalized size = 0.65 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-144 d^4 e^2 x^2-70 d^3 e^3 x^3+88 d^2 e^4 x^4+280 d^5 e x-120 d^6-105 d e^5 x^5+176 e^6 x^6\right )-105 e^7 x^7 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+105 e^7 x^7 \log (x)}{840 d^4 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)^2),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-120*d^6 + 280*d^5*e*x - 144*d^4*e^2*x^2 - 70*d^3*e^3*x^3 + 88*d^2*e^4*x^4 - 105*d*e^5*x
^5 + 176*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]])/(840*d^4*x^7)

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Maple [B]  time = 0.154, size = 591, normalized size = 3. \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)^2,x)

[Out]

-1/7/d^4/x^7*(-e^2*x^2+d^2)^(7/2)-19/15/d^8*e^4/x^3*(-e^2*x^2+d^2)^(7/2)-29/15/d^10*e^6/x*(-e^2*x^2+d^2)^(7/2)
-29/15/d^10*e^8*x*(-e^2*x^2+d^2)^(5/2)+1/3/d^5*e/x^6*(-e^2*x^2+d^2)^(7/2)-1/8/d^3*e^7/(d^2)^(1/2)*ln((2*d^2+2*
(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)+29/15/d^9*e^7*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)+1/40/d^9*e^7*(-e^2*x^2
+d^2)^(5/2)+1/24/d^7*e^7*(-e^2*x^2+d^2)^(3/2)+1/8/d^5*e^7*(-e^2*x^2+d^2)^(1/2)-29/12/d^8*e^8*x*(-e^2*x^2+d^2)^
(3/2)-29/8/d^6*e^8*x*(-e^2*x^2+d^2)^(1/2)-29/8/d^4*e^8/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-
3/5/d^6*e^2/x^5*(-e^2*x^2+d^2)^(7/2)+11/12/d^7*e^3/x^4*(-e^2*x^2+d^2)^(7/2)+13/8/d^9*e^5/x^2*(-e^2*x^2+d^2)^(7
/2)+29/12/d^8*e^8*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2)*x+29/8/d^6*e^8*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x+2
9/8/d^4*e^8/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2))+1/3/d^9*e^5/(d/e+x)^2*(-(d/
e+x)^2*e^2+2*d*e*(d/e+x))^(7/2)

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.70271, size = 263, normalized size = 1.33 \begin{align*} \frac{105 \, e^{7} x^{7} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (176 \, e^{6} x^{6} - 105 \, d e^{5} x^{5} + 88 \, d^{2} e^{4} x^{4} - 70 \, d^{3} e^{3} x^{3} - 144 \, d^{4} e^{2} x^{2} + 280 \, d^{5} e x - 120 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{840 \, d^{4} 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)^2,x, algorithm="fricas")

[Out]

1/840*(105*e^7*x^7*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (176*e^6*x^6 - 105*d*e^5*x^5 + 88*d^2*e^4*x^4 - 70*d^3
*e^3*x^3 - 144*d^4*e^2*x^2 + 280*d^5*e*x - 120*d^6)*sqrt(-e^2*x^2 + d^2))/(d^4*x^7)

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Sympy [C]  time = 18.2054, size = 843, normalized size = 4.26 \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)**2,x)

[Out]

d**2*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)) - 2*d*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*ac
osh(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**2*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*sq
rt(-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))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError