\(\int \frac {(d+e x) (d^2-e^2 x^2)^{3/2}}{x^9} \, dx\) [15]

Optimal result

Integrand size = 25, antiderivative size = 201 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^9} \, dx=\frac {3 e^6 \sqrt {d^2-e^2 x^2}}{128 d^3 x^2}-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac {3 e^8 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d^4} \]

[Out]

-1/64*e^4*(-e^2*x^2+d^2)^(3/2)/d^3/x^4-1/8*(-e^2*x^2+d^2)^(5/2)/d/x^8-1/7*e*(-e^2*x^2+d^2)^(5/2)/d^2/x^7-1/16*
e^2*(-e^2*x^2+d^2)^(5/2)/d^3/x^6-2/35*e^3*(-e^2*x^2+d^2)^(5/2)/d^4/x^5-3/128*e^8*arctanh((-e^2*x^2+d^2)^(1/2)/
d)/d^4+3/128*e^6*(-e^2*x^2+d^2)^(1/2)/d^3/x^2

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {849, 821, 272, 43, 65, 214} \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^9} \, dx=-\frac {3 e^8 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac {3 e^6 \sqrt {d^2-e^2 x^2}}{128 d^3 x^2}-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4} \]

[In]

Int[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^9,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 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 65

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 214

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

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 821

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 849

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])

Rubi steps \begin{align*} \text {integral}& = -\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 \text {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 ) \text {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 ) \text {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 ) \text {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] (verified)

Time = 0.33 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.82 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^9} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-560 d^7-640 d^6 e x+840 d^5 e^2 x^2+1024 d^4 e^3 x^3-70 d^3 e^4 x^4-128 d^2 e^5 x^5-105 d e^6 x^6-256 e^7 x^7\right )}{4480 d^4 x^8}-\frac {3 \sqrt {d^2} e^8 \log (x)}{128 d^5}+\frac {3 \sqrt {d^2} e^8 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{128 d^5} \]

[In]

Integrate[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^9,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))/(4480*d^4*x^8) - (3*Sqrt[d^2]*e^8*Log[x])/(128*d^5) + (3*Sqrt[d^2]*e^8*
Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/(128*d^5)

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.71

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (256 e^{7} x^{7}+105 d \,e^{6} x^{6}+128 d^{2} e^{5} x^{5}+70 d^{3} e^{4} x^{4}-1024 d^{4} e^{3} x^{3}-840 d^{5} e^{2} x^{2}+640 d^{6} e x +560 d^{7}\right )}{4480 x^{8} d^{4}}-\frac {3 e^{8} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{128 d^{3} \sqrt {d^{2}}}\)	\(143\)
default	\(e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{7 d^{2} x^{7}}-\frac {2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{35 d^{4} x^{5}}\right )+d \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{8 d^{2} x^{8}}+\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6 d^{2} x^{6}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{4 d^{2} x^{4}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{2 d^{2} x^{2}}-\frac {3 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )}{8 d^{2}}\right )\)	\(256\)

[In]

int((e*x+d)*(-e^2*x^2+d^2)^(3/2)/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/4480*(-e^2*x^2+d^2)^(1/2)*(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)/x^8/d^4-3/128/d^3*e^8/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))
/x)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.65 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^9} \, dx=\frac {105 \, e^{8} x^{8} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (256 \, e^{7} x^{7} + 105 \, d e^{6} x^{6} + 128 \, d^{2} e^{5} x^{5} + 70 \, d^{3} e^{4} x^{4} - 1024 \, d^{4} e^{3} x^{3} - 840 \, d^{5} e^{2} x^{2} + 640 \, d^{6} e x + 560 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{4480 \, d^{4} x^{8}} \]

[In]

integrate((e*x+d)*(-e^2*x^2+d^2)^(3/2)/x^9,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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 25.09 (sec) , antiderivative size = 1159, normalized size of antiderivative = 5.77 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^9} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)*(-e**2*x**2+d**2)**(3/2)/x**9,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/(e**2*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/(e**2*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*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*sqrt(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/(e**2*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), Ab
s(e**2*x**2/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))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.14 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^9} \, dx=-\frac {3 \, e^{8} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{128 \, d^{4}} + \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{8}}{128 \, d^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{8}}{128 \, d^{7}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}{128 \, d^{7} x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}}{64 \, d^{5} x^{4}} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}}{35 \, d^{4} x^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}}{16 \, d^{3} x^{6}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e}{7 \, d^{2} x^{7}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{8 \, d x^{8}} \]

[In]

integrate((e*x+d)*(-e^2*x^2+d^2)^(3/2)/x^9,x, algorithm="maxima")

[Out]

-3/128*e^8*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^4 + 3/128*sqrt(-e^2*x^2 + d^2)*e^8/d^5 + 1/12
8*(-e^2*x^2 + d^2)^(3/2)*e^8/d^7 + 1/128*(-e^2*x^2 + d^2)^(5/2)*e^6/(d^7*x^2) - 1/64*(-e^2*x^2 + d^2)^(5/2)*e^
4/(d^5*x^4) - 2/35*(-e^2*x^2 + d^2)^(5/2)*e^3/(d^4*x^5) - 1/16*(-e^2*x^2 + d^2)^(5/2)*e^2/(d^3*x^6) - 1/7*(-e^
2*x^2 + d^2)^(5/2)*e/(d^2*x^7) - 1/8*(-e^2*x^2 + d^2)^(5/2)/(d*x^8)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 463 vs. \(2 (173) = 346\).

Time = 0.29 (sec) , antiderivative size = 463, normalized size of antiderivative = 2.30 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^9} \, dx=\frac {{\left (35 \, e^{9} + \frac {80 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{7}}{x} - \frac {112 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} e^{3}}{x^{3}} - \frac {280 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} e}{x^{4}} - \frac {560 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{e x^{5}} + \frac {1680 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7}}{e^{5} x^{7}}\right )} e^{16} x^{8}}{71680 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8} d^{4} {\left | e \right |}} - \frac {3 \, e^{9} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{128 \, d^{4} {\left | e \right |}} - \frac {\frac {1680 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{28} e^{13} {\left | e \right |}}{x} - \frac {560 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{28} e^{9} {\left | e \right |}}{x^{3}} - \frac {280 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{28} e^{7} {\left | e \right |}}{x^{4}} - \frac {112 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{28} e^{5} {\left | e \right |}}{x^{5}} + \frac {80 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7} d^{28} e {\left | e \right |}}{x^{7}} + \frac {35 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8} d^{28} {\left | e \right |}}{e x^{8}}}{71680 \, d^{32} e^{8}} \]

[In]

integrate((e*x+d)*(-e^2*x^2+d^2)^(3/2)/x^9,x, algorithm="giac")

[Out]

1/71680*(35*e^9 + 80*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^7/x - 112*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*e^3
/x^3 - 280*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*e/x^4 - 560*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5/(e*x^5) + 1
680*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7/(e^5*x^7))*e^16*x^8/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^8*d^4*abs(e
)) - 3/128*e^9*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/(d^4*abs(e)) - 1/71680*(1680*
(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^28*e^13*abs(e)/x - 560*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^28*e^9*ab
s(e)/x^3 - 280*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^28*e^7*abs(e)/x^4 - 112*(d*e + sqrt(-e^2*x^2 + d^2)*abs
(e))^5*d^28*e^5*abs(e)/x^5 + 80*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7*d^28*e*abs(e)/x^7 + 35*(d*e + sqrt(-e^2*
x^2 + d^2)*abs(e))^8*d^28*abs(e)/(e*x^8))/(d^32*e^8)

Mupad [B] (verification not implemented)

Time = 14.95 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.05 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^9} \, dx=\frac {3\,d^3\,\sqrt {d^2-e^2\,x^2}}{128\,x^8}-\frac {11\,d\,{\left (d^2-e^2\,x^2\right )}^{3/2}}{128\,x^8}-\frac {11\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{128\,d\,x^8}+\frac {3\,{\left (d^2-e^2\,x^2\right )}^{7/2}}{128\,d^3\,x^8}+\frac {8\,e^3\,\sqrt {d^2-e^2\,x^2}}{35\,x^5}-\frac {e^5\,\sqrt {d^2-e^2\,x^2}}{35\,d^2\,x^3}-\frac {2\,e^7\,\sqrt {d^2-e^2\,x^2}}{35\,d^4\,x}-\frac {d^2\,e\,\sqrt {d^2-e^2\,x^2}}{7\,x^7}+\frac {e^8\,\mathrm {atan}\left (\frac {\sqrt {d^2-e^2\,x^2}\,1{}\mathrm {i}}{d}\right )\,3{}\mathrm {i}}{128\,d^4} \]

[In]

int(((d^2 - e^2*x^2)^(3/2)*(d + e*x))/x^9,x)

[Out]

(3*d^3*(d^2 - e^2*x^2)^(1/2))/(128*x^8) - (11*d*(d^2 - e^2*x^2)^(3/2))/(128*x^8) - (11*(d^2 - e^2*x^2)^(5/2))/
(128*d*x^8) + (3*(d^2 - e^2*x^2)^(7/2))/(128*d^3*x^8) + (8*e^3*(d^2 - e^2*x^2)^(1/2))/(35*x^5) + (e^8*atan(((d
^2 - e^2*x^2)^(1/2)*1i)/d)*3i)/(128*d^4) - (e^5*(d^2 - e^2*x^2)^(1/2))/(35*d^2*x^3) - (2*e^7*(d^2 - e^2*x^2)^(
1/2))/(35*d^4*x) - (d^2*e*(d^2 - e^2*x^2)^(1/2))/(7*x^7)