\(\int \frac {(d+e x)^3 (d^2-e^2 x^2)^{5/2}}{x^{11}} \, dx\) [81]

Optimal result

Integrand size = 27, antiderivative size = 225 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{11}} \, dx=-\frac {33 e^8 \sqrt {d^2-e^2 x^2}}{256 d x^2}+\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {33 e^{10} \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^2} \]

[Out]

11/128*e^6*(-e^2*x^2+d^2)^(3/2)/d/x^4-11/160*e^4*(-e^2*x^2+d^2)^(5/2)/d/x^6-1/10*d*(-e^2*x^2+d^2)^(7/2)/x^10-1
/3*e*(-e^2*x^2+d^2)^(7/2)/x^9-33/80*e^2*(-e^2*x^2+d^2)^(7/2)/d/x^8-5/21*e^3*(-e^2*x^2+d^2)^(7/2)/d^2/x^7+33/25
6*e^10*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^2-33/256*e^8*(-e^2*x^2+d^2)^(1/2)/d/x^2

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1821, 849, 821, 272, 43, 65, 214} \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{11}} \, dx=\frac {33 e^{10} \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {33 e^8 \sqrt {d^2-e^2 x^2}}{256 d x^2}+\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7} \]

[In]

Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^11,x]

[Out]

(-33*e^8*Sqrt[d^2 - e^2*x^2])/(256*d*x^2) + (11*e^6*(d^2 - e^2*x^2)^(3/2))/(128*d*x^4) - (11*e^4*(d^2 - e^2*x^
2)^(5/2))/(160*d*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(10*x^10) - (e*(d^2 - e^2*x^2)^(7/2))/(3*x^9) - (33*e^2*(d^2
- e^2*x^2)^(7/2))/(80*d*x^8) - (5*e^3*(d^2 - e^2*x^2)^(7/2))/(21*d^2*x^7) + (33*e^10*ArcTanh[Sqrt[d^2 - e^2*x
^2]/d])/(256*d^2)

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])

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}}{10 x^{10}}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-30 d^4 e-33 d^3 e^2 x-10 d^2 e^3 x^2\right )}{x^{10}} \, dx}{10 d^2} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}+\frac {\int \frac {\left (297 d^5 e^2+150 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx}{90 d^4} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {\int \frac {\left (-1200 d^6 e^3-297 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx}{720 d^6} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {\left (33 e^4\right ) \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx}{80 d} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {\left (33 e^4\right ) \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{5/2}}{x^4} \, dx,x,x^2\right )}{160 d} \\ & = -\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}-\frac {\left (11 e^6\right ) \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{64 d} \\ & = \frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {\left (33 e^8\right ) \text {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{256 d} \\ & = -\frac {33 e^8 \sqrt {d^2-e^2 x^2}}{256 d x^2}+\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}-\frac {\left (33 e^{10}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{512 d} \\ & = -\frac {33 e^8 \sqrt {d^2-e^2 x^2}}{256 d x^2}+\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {\left (33 e^8\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 )}{256 d} \\ & = -\frac {33 e^8 \sqrt {d^2-e^2 x^2}}{256 d x^2}+\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {33 e^{10} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.83 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{11}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-2688 d^9-8960 d^8 e x-3024 d^7 e^2 x^2+20480 d^6 e^3 x^3+23352 d^5 e^4 x^4-7680 d^4 e^5 x^5-24570 d^3 e^6 x^6-10240 d^2 e^7 x^7+3465 d e^8 x^8+6400 e^9 x^9\right )}{26880 d^2 x^{10}}+\frac {33 \sqrt {d^2} e^{10} \log (x)}{256 d^3}-\frac {33 \sqrt {d^2} e^{10} \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{256 d^3} \]

[In]

Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^11,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-2688*d^9 - 8960*d^8*e*x - 3024*d^7*e^2*x^2 + 20480*d^6*e^3*x^3 + 23352*d^5*e^4*x^4 - 76
80*d^4*e^5*x^5 - 24570*d^3*e^6*x^6 - 10240*d^2*e^7*x^7 + 3465*d*e^8*x^8 + 6400*e^9*x^9))/(26880*d^2*x^10) + (3
3*Sqrt[d^2]*e^10*Log[x])/(256*d^3) - (33*Sqrt[d^2]*e^10*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/(256*d^3)

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.73

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-6400 e^{9} x^{9}-3465 d \,e^{8} x^{8}+10240 d^{2} e^{7} x^{7}+24570 d^{3} e^{6} x^{6}+7680 d^{4} e^{5} x^{5}-23352 d^{5} e^{4} x^{4}-20480 d^{6} e^{3} x^{3}+3024 x^{2} d^{7} e^{2}+8960 x \,d^{8} e +2688 d^{9}\right )}{26880 d^{2} x^{10}}+\frac {33 e^{10} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{256 d \sqrt {d^{2}}}\)	\(165\)
default	\(-\frac {e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 d^{2} x^{7}}+d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{10 d^{2} x^{10}}+\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 d^{2} x^{8}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )}{8 d^{2}}\right )}{10 d^{2}}\right )+3 d \,e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 d^{2} x^{8}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )}{8 d^{2}}\right )+3 d^{2} e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 d^{2} x^{9}}-\frac {2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 d^{4} x^{7}}\right )\)	\(568\)

[In]

int((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^11,x,method=_RETURNVERBOSE)

[Out]

-1/26880*(-e^2*x^2+d^2)^(1/2)*(-6400*e^9*x^9-3465*d*e^8*x^8+10240*d^2*e^7*x^7+24570*d^3*e^6*x^6+7680*d^4*e^5*x
^5-23352*d^5*e^4*x^4-20480*d^6*e^3*x^3+3024*d^7*e^2*x^2+8960*d^8*e*x+2688*d^9)/d^2/x^10+33/256/d*e^10/(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 = 0.33 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.68 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{11}} \, dx=-\frac {3465 \, e^{10} x^{10} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (6400 \, e^{9} x^{9} + 3465 \, d e^{8} x^{8} - 10240 \, d^{2} e^{7} x^{7} - 24570 \, d^{3} e^{6} x^{6} - 7680 \, d^{4} e^{5} x^{5} + 23352 \, d^{5} e^{4} x^{4} + 20480 \, d^{6} e^{3} x^{3} - 3024 \, d^{7} e^{2} x^{2} - 8960 \, d^{8} e x - 2688 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{26880 \, d^{2} x^{10}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^11,x, algorithm="fricas")

[Out]

-1/26880*(3465*e^10*x^10*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (6400*e^9*x^9 + 3465*d*e^8*x^8 - 10240*d^2*e^7*x
^7 - 24570*d^3*e^6*x^6 - 7680*d^4*e^5*x^5 + 23352*d^5*e^4*x^4 + 20480*d^6*e^3*x^3 - 3024*d^7*e^2*x^2 - 8960*d^
8*e*x - 2688*d^9)*sqrt(-e^2*x^2 + d^2))/(d^2*x^10)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 136.08 (sec) , antiderivative size = 2159, normalized size of antiderivative = 9.60 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{11}} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2)/x**11,x)

[Out]

d**7*Piecewise((-d**2/(10*e*x**11*sqrt(d**2/(e**2*x**2) - 1)) + 9*e/(80*x**9*sqrt(d**2/(e**2*x**2) - 1)) + e**
3/(480*d**2*x**7*sqrt(d**2/(e**2*x**2) - 1)) + 7*e**5/(1920*d**4*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 7*e**7/(76
8*d**6*x**3*sqrt(d**2/(e**2*x**2) - 1)) - 7*e**9/(256*d**8*x*sqrt(d**2/(e**2*x**2) - 1)) + 7*e**10*acosh(d/(e*
x))/(256*d**9), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(10*e*x**11*sqrt(-d**2/(e**2*x**2) + 1)) - 9*I*e/(80*x**9*
sqrt(-d**2/(e**2*x**2) + 1)) - I*e**3/(480*d**2*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - 7*I*e**5/(1920*d**4*x**5*s
qrt(-d**2/(e**2*x**2) + 1)) - 7*I*e**7/(768*d**6*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + 7*I*e**9/(256*d**8*x*sqrt
(-d**2/(e**2*x**2) + 1)) - 7*I*e**10*asin(d/(e*x))/(256*d**9), True)) + 3*d**6*e*Piecewise((-e*sqrt(d**2/(e**2
*x**2) - 1)/(9*x**8) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(63*d**2*x**6) + 2*e**5*sqrt(d**2/(e**2*x**2) - 1)/(105
*d**4*x**4) + 8*e**7*sqrt(d**2/(e**2*x**2) - 1)/(315*d**6*x**2) + 16*e**9*sqrt(d**2/(e**2*x**2) - 1)/(315*d**8
), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(9*x**8) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)
/(63*d**2*x**6) + 2*I*e**5*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**4*x**4) + 8*I*e**7*sqrt(-d**2/(e**2*x**2) + 1)/
(315*d**6*x**2) + 16*I*e**9*sqrt(-d**2/(e**2*x**2) + 1)/(315*d**8), True)) + d**5*e**2*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)) - 5*d**4*e**3*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(7*x**6) + e**3*sq
rt(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*s
qrt(-d**2/(e**2*x**2) + 1)/(105*d**6), True)) - 5*d**3*e**4*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)) +
d**2*e**5*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)) + 3*d*e**6*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)) + e**7*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))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.21 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{11}} \, dx=\frac {33 \, e^{10} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{256 \, d^{2}} - \frac {33 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{10}}{256 \, d^{3}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{10}}{256 \, d^{5}} - \frac {33 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{10}}{1280 \, d^{7}} - \frac {33 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{8}}{1280 \, d^{7} x^{2}} + \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{6}}{640 \, d^{5} x^{4}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{160 \, d^{3} x^{6}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{21 \, d^{2} x^{7}} - \frac {33 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{80 \, d x^{8}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{3 \, x^{9}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{10 \, x^{10}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^11,x, algorithm="maxima")

[Out]

33/256*e^10*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^2 - 33/256*sqrt(-e^2*x^2 + d^2)*e^10/d^3 - 1
1/256*(-e^2*x^2 + d^2)^(3/2)*e^10/d^5 - 33/1280*(-e^2*x^2 + d^2)^(5/2)*e^10/d^7 - 33/1280*(-e^2*x^2 + d^2)^(7/
2)*e^8/(d^7*x^2) + 11/640*(-e^2*x^2 + d^2)^(7/2)*e^6/(d^5*x^4) - 11/160*(-e^2*x^2 + d^2)^(7/2)*e^4/(d^3*x^6) -
5/21*(-e^2*x^2 + d^2)^(7/2)*e^3/(d^2*x^7) - 33/80*(-e^2*x^2 + d^2)^(7/2)*e^2/(d*x^8) - 1/3*(-e^2*x^2 + d^2)^(
7/2)*e/x^9 - 1/10*(-e^2*x^2 + d^2)^(7/2)*d/x^10

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 731 vs. \(2 (193) = 386\).

Time = 0.31 (sec) , antiderivative size = 731, normalized size of antiderivative = 3.25 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{11}} \, dx=\frac {{\left (42 \, e^{11} + \frac {280 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{9}}{x} + \frac {525 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{7}}{x^{2}} - \frac {600 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} e^{5}}{x^{3}} - \frac {3570 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} e^{3}}{x^{4}} - \frac {3360 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} e}{x^{5}} + \frac {5880 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e x^{6}} + \frac {16800 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7}}{e^{3} x^{7}} + \frac {10500 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8}}{e^{5} x^{8}} - \frac {31920 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{9}}{e^{7} x^{9}}\right )} e^{20} x^{10}}{430080 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{10} d^{2} {\left | e \right |}} + \frac {33 \, e^{11} \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 )}{256 \, d^{2} {\left | e \right |}} + \frac {\frac {31920 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{18} e^{17} {\left | e \right |}}{x} - \frac {10500 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{18} e^{15} {\left | e \right |}}{x^{2}} - \frac {16800 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{18} e^{13} {\left | e \right |}}{x^{3}} - \frac {5880 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{18} e^{11} {\left | e \right |}}{x^{4}} + \frac {3360 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{18} e^{9} {\left | e \right |}}{x^{5}} + \frac {3570 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} d^{18} e^{7} {\left | e \right |}}{x^{6}} + \frac {600 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7} d^{18} e^{5} {\left | e \right |}}{x^{7}} - \frac {525 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8} d^{18} e^{3} {\left | e \right |}}{x^{8}} - \frac {280 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{9} d^{18} e {\left | e \right |}}{x^{9}} - \frac {42 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{10} d^{18} {\left | e \right |}}{e x^{10}}}{430080 \, d^{20} e^{10}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^11,x, algorithm="giac")

[Out]

1/430080*(42*e^11 + 280*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^9/x + 525*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*
e^7/x^2 - 600*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*e^5/x^3 - 3570*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*e^3/x
^4 - 3360*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5*e/x^5 + 5880*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6/(e*x^6) + 1
6800*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7/(e^3*x^7) + 10500*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^8/(e^5*x^8) -
31920*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^9/(e^7*x^9))*e^20*x^10/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^10*d^2*
abs(e)) + 33/256*e^11*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/(d^2*abs(e)) + 1/43008
0*(31920*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^18*e^17*abs(e)/x - 10500*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*
d^18*e^15*abs(e)/x^2 - 16800*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^18*e^13*abs(e)/x^3 - 5880*(d*e + sqrt(-e^
2*x^2 + d^2)*abs(e))^4*d^18*e^11*abs(e)/x^4 + 3360*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5*d^18*e^9*abs(e)/x^5 +
3570*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6*d^18*e^7*abs(e)/x^6 + 600*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7*d^
18*e^5*abs(e)/x^7 - 525*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^8*d^18*e^3*abs(e)/x^8 - 280*(d*e + sqrt(-e^2*x^2 +
d^2)*abs(e))^9*d^18*e*abs(e)/x^9 - 42*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^10*d^18*abs(e)/(e*x^10))/(d^20*e^10
)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{11}} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^{11}} \,d x \]

[In]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^11,x)

[Out]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^11, x)