\(\int \frac {(d+e x)^3 (d^2-e^2 x^2)^{5/2}}{x^9} \, dx\) [79]

Optimal result

Integrand size = 27, antiderivative size = 204 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx=-\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-e^8 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {125}{128} e^8 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]

[Out]

1/192*e^4*(64*e*x+125*d)*(-e^2*x^2+d^2)^(3/2)/x^4-1/240*e^2*(48*e*x+125*d)*(-e^2*x^2+d^2)^(5/2)/x^6-1/8*d*(-e^
2*x^2+d^2)^(7/2)/x^8-3/7*e*(-e^2*x^2+d^2)^(7/2)/x^7-e^8*arctan(e*x/(-e^2*x^2+d^2)^(1/2))+125/128*e^8*arctanh((
-e^2*x^2+d^2)^(1/2)/d)-1/128*e^6*(128*e*x+125*d)*(-e^2*x^2+d^2)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1821, 825, 858, 223, 209, 272, 65, 214} \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx=e^8 \left (-\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+\frac {125}{128} e^8 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4} \]

[In]

Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^9,x]

[Out]

-1/128*(e^6*(125*d + 128*e*x)*Sqrt[d^2 - e^2*x^2])/x^2 + (e^4*(125*d + 64*e*x)*(d^2 - e^2*x^2)^(3/2))/(192*x^4
) - (e^2*(125*d + 48*e*x)*(d^2 - e^2*x^2)^(5/2))/(240*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(8*x^8) - (3*e*(d^2 - e
^2*x^2)^(7/2))/(7*x^7) - e^8*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + (125*e^8*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/128

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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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 825

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

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 0]

Rule 1821

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])

Rubi steps \begin{align*} \text {integral}& = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-24 d^4 e-25 d^3 e^2 x-8 d^2 e^3 x^2\right )}{x^8} \, dx}{8 d^2} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}+\frac {\int \frac {\left (175 d^5 e^2+56 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx}{56 d^4} \\ & = -\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {\int \frac {\left (1750 d^7 e^4+672 d^6 e^5 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx}{672 d^6} \\ & = \frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}+\frac {\int \frac {\left (10500 d^9 e^6+5376 d^8 e^7 x\right ) \sqrt {d^2-e^2 x^2}}{x^3} \, dx}{5376 d^8} \\ & = -\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {\int \frac {21000 d^{11} e^8+21504 d^{10} e^9 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{21504 d^{10}} \\ & = -\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {1}{128} \left (125 d e^8\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-e^9 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {1}{256} \left (125 d e^8\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-e^9 \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right ) \\ & = -\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-e^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{128} \left (125 d 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 ) \\ & = -\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-e^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {125}{128} e^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.97 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-1680 d^7-5760 d^6 e x-1960 d^5 e^2 x^2+14592 d^4 e^3 x^3+17710 d^3 e^4 x^4-7424 d^2 e^5 x^5-27195 d e^6 x^6-14848 e^7 x^7\right )}{13440 x^8}+2 e^8 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )+\frac {125 \sqrt {d^2} e^8 \log (x)}{128 d}-\frac {125 \sqrt {d^2} e^8 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{128 d} \]

[In]

Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^9,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-1680*d^7 - 5760*d^6*e*x - 1960*d^5*e^2*x^2 + 14592*d^4*e^3*x^3 + 17710*d^3*e^4*x^4 - 74
24*d^2*e^5*x^5 - 27195*d*e^6*x^6 - 14848*e^7*x^7))/(13440*x^8) + 2*e^8*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^
2*x^2])] + (125*Sqrt[d^2]*e^8*Log[x])/(128*d) - (125*Sqrt[d^2]*e^8*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/(128*
d)

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.83

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (14848 e^{7} x^{7}+27195 d \,e^{6} x^{6}+7424 d^{2} e^{5} x^{5}-17710 d^{3} e^{4} x^{4}-14592 d^{4} e^{3} x^{3}+1960 d^{5} e^{2} x^{2}+5760 d^{6} e x +1680 d^{7}\right )}{13440 x^{8}}-\frac {e^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}+\frac {125 e^{8} d \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{128 \sqrt {d^{2}}}\)	\(170\)
default	\(e^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{5 d^{2} x^{5}}-\frac {2 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{3 d^{2} x^{3}}-\frac {4 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{d^{2}}\right )}{3 d^{2}}\right )}{5 d^{2}}\right )+d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 d^{2} x^{8}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{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 )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )}{8 d^{2}}\right )+3 d \,e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{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 )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )-\frac {3 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 x^{7}}\)	\(640\)

[In]

int((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/13440*(-e^2*x^2+d^2)^(1/2)*(14848*e^7*x^7+27195*d*e^6*x^6+7424*d^2*e^5*x^5-17710*d^3*e^4*x^4-14592*d^4*e^3*
x^3+1960*d^5*e^2*x^2+5760*d^6*e*x+1680*d^7)/x^8-e^9/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+125
/128*e^8*d/(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 = 0.31 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.80 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx=\frac {26880 \, e^{8} x^{8} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 13125 \, e^{8} x^{8} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (14848 \, e^{7} x^{7} + 27195 \, d e^{6} x^{6} + 7424 \, d^{2} e^{5} x^{5} - 17710 \, d^{3} e^{4} x^{4} - 14592 \, d^{4} e^{3} x^{3} + 1960 \, d^{5} e^{2} x^{2} + 5760 \, d^{6} e x + 1680 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{13440 \, x^{8}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^9,x, algorithm="fricas")

[Out]

1/13440*(26880*e^8*x^8*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - 13125*e^8*x^8*log(-(d - sqrt(-e^2*x^2 + d^2
))/x) - (14848*e^7*x^7 + 27195*d*e^6*x^6 + 7424*d^2*e^5*x^5 - 17710*d^3*e^4*x^4 - 14592*d^4*e^3*x^3 + 1960*d^5
*e^2*x^2 + 5760*d^6*e*x + 1680*d^7)*sqrt(-e^2*x^2 + d^2))/x^8

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 29.73 (sec) , antiderivative size = 1719, normalized size of antiderivative = 8.43 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2)/x**9,x)

[Out]

d**7*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)) + 3*d**6*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)/(10
5*d**4*x**2) + 8*I*e**7*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**6), True)) + d**5*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)) - 5*d**4*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/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)) - 5*d
**3*e**4*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/(e**2*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)) + d**2*e**5*Piecewise((-e*sqrt(d**2/(e**2*x**2) -
1)/(3*x**2) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(3*d**2), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x*
*2) + 1)/(3*x**2) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(3*d**2), True)) + 3*d*e**6*Piecewise((-e*sqrt(d**2/(e*
*2*x**2) - 1)/(2*x) + e**2*acosh(d/(e*x))/(2*d), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(2*e*x**3*sqrt(-d**2/(e**
2*x**2) + 1)) - I*e/(2*x*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**2*asin(d/(e*x))/(2*d), True)) + e**7*Piecewise((I
*d/(x*sqrt(-1 + e**2*x**2/d**2)) + I*e*acosh(e*x/d) - I*e**2*x/(d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d*
*2) > 1), (-d/(x*sqrt(1 - e**2*x**2/d**2)) - e*asin(e*x/d) + e**2*x/(d*sqrt(1 - e**2*x**2/d**2)), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (178) = 356\).

Time = 0.28 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.78 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx=-\frac {e^{9} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} + \frac {125}{128} \, e^{8} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{9} x}{d^{2}} - \frac {125 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{8}}{128 \, d} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{9} x}{3 \, d^{4}} - \frac {125 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{8}}{384 \, d^{3}} - \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{8}}{128 \, d^{5}} - \frac {8 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{7}}{15 \, d^{4} x} - \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{6}}{128 \, d^{5} x^{2}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{5}}{15 \, d^{4} x^{3}} + \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{192 \, d^{3} x^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{5 \, d^{2} x^{5}} - \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{48 \, d x^{6}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{7 \, x^{7}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{8 \, x^{8}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^9,x, algorithm="maxima")

[Out]

-e^9*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) + 125/128*e^8*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x)) -
sqrt(-e^2*x^2 + d^2)*e^9*x/d^2 - 125/128*sqrt(-e^2*x^2 + d^2)*e^8/d - 2/3*(-e^2*x^2 + d^2)^(3/2)*e^9*x/d^4 -
125/384*(-e^2*x^2 + d^2)^(3/2)*e^8/d^3 - 25/128*(-e^2*x^2 + d^2)^(5/2)*e^8/d^5 - 8/15*(-e^2*x^2 + d^2)^(5/2)*e
^7/(d^4*x) - 25/128*(-e^2*x^2 + d^2)^(7/2)*e^6/(d^5*x^2) + 2/15*(-e^2*x^2 + d^2)^(7/2)*e^5/(d^4*x^3) + 25/192*
(-e^2*x^2 + d^2)^(7/2)*e^4/(d^3*x^4) - 1/5*(-e^2*x^2 + d^2)^(7/2)*e^3/(d^2*x^5) - 25/48*(-e^2*x^2 + d^2)^(7/2)
*e^2/(d*x^6) - 3/7*(-e^2*x^2 + d^2)^(7/2)*e/x^7 - 1/8*(-e^2*x^2 + d^2)^(7/2)*d/x^8

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (178) = 356\).

Time = 0.31 (sec) , antiderivative size = 584, normalized size of antiderivative = 2.86 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx=\frac {{\left (105 \, e^{9} + \frac {720 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{7}}{x} + \frac {1120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{5}}{x^{2}} - \frac {3696 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} e^{3}}{x^{3}} - \frac {14280 \, {\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 {77280 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e^{3} x^{6}} + \frac {122640 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7}}{e^{5} x^{7}}\right )} e^{16} x^{8}}{215040 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8} {\left | e \right |}} - \frac {e^{9} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} + \frac {125 \, 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 \, {\left | e \right |}} - \frac {\frac {122640 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{13} {\left | e \right |}}{x} + \frac {77280 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{11} {\left | e \right |}}{x^{2}} - \frac {560 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} e^{9} {\left | e \right |}}{x^{3}} - \frac {14280 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} e^{7} {\left | e \right |}}{x^{4}} - \frac {3696 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} e^{5} {\left | e \right |}}{x^{5}} + \frac {1120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} e^{3} {\left | e \right |}}{x^{6}} + \frac {720 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7} e {\left | e \right |}}{x^{7}} + \frac {105 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8} {\left | e \right |}}{e x^{8}}}{215040 \, e^{8}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^9,x, algorithm="giac")

[Out]

1/215040*(105*e^9 + 720*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^7/x + 1120*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2
*e^5/x^2 - 3696*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*e^3/x^3 - 14280*(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) + 77280*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6/(e^3*x^6
) + 122640*(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*ab
s(e)) - e^9*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) + 125/128*e^9*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e
))/(e^2*abs(x)))/abs(e) - 1/215040*(122640*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^13*abs(e)/x + 77280*(d*e + sq
rt(-e^2*x^2 + d^2)*abs(e))^2*e^11*abs(e)/x^2 - 560*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*e^9*abs(e)/x^3 - 1428
0*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*e^7*abs(e)/x^4 - 3696*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5*e^5*abs(e)
/x^5 + 1120*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6*e^3*abs(e)/x^6 + 720*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7*e
*abs(e)/x^7 + 105*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^8*abs(e)/(e*x^8))/e^8

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^9} \,d x \]

[In]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^9,x)

[Out]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^9, x)