\(\int \frac {(d+e x)^3 (d^2-e^2 x^2)^{5/2}}{x^{12}} \, dx\) [82]

Optimal result

Integrand size = 27, antiderivative size = 254 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{12}} \, dx=-\frac {19 e^9 \sqrt {d^2-e^2 x^2}}{256 d^2 x^2}+\frac {19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac {19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}+\frac {19 e^{11} \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^3} \]

[Out]

19/384*e^7*(-e^2*x^2+d^2)^(3/2)/d^2/x^4-19/480*e^5*(-e^2*x^2+d^2)^(5/2)/d^2/x^6-1/11*d*(-e^2*x^2+d^2)^(7/2)/x^
11-3/10*e*(-e^2*x^2+d^2)^(7/2)/x^10-37/99*e^2*(-e^2*x^2+d^2)^(7/2)/d/x^9-19/80*e^3*(-e^2*x^2+d^2)^(7/2)/d^2/x^
8-74/693*e^4*(-e^2*x^2+d^2)^(7/2)/d^3/x^7+19/256*e^11*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^3-19/256*e^9*(-e^2*x^2
+d^2)^(1/2)/d^2/x^2

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1821, 849, 821, 272, 43, 65, 214} \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{12}} \, dx=\frac {19 e^{11} \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^9 \sqrt {d^2-e^2 x^2}}{256 d^2 x^2}+\frac {19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac {19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7} \]

[In]

Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^12,x]

[Out]

(-19*e^9*Sqrt[d^2 - e^2*x^2])/(256*d^2*x^2) + (19*e^7*(d^2 - e^2*x^2)^(3/2))/(384*d^2*x^4) - (19*e^5*(d^2 - e^
2*x^2)^(5/2))/(480*d^2*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(11*x^11) - (3*e*(d^2 - e^2*x^2)^(7/2))/(10*x^10) - (3
7*e^2*(d^2 - e^2*x^2)^(7/2))/(99*d*x^9) - (19*e^3*(d^2 - e^2*x^2)^(7/2))/(80*d^2*x^8) - (74*e^4*(d^2 - e^2*x^2
)^(7/2))/(693*d^3*x^7) + (19*e^11*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(256*d^3)

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}}{11 x^{11}}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-33 d^4 e-37 d^3 e^2 x-11 d^2 e^3 x^2\right )}{x^{11}} \, dx}{11 d^2} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}+\frac {\int \frac {\left (370 d^5 e^2+209 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx}{110 d^4} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {\int \frac {\left (-1881 d^6 e^3-740 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx}{990 d^6} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}+\frac {\int \frac {\left (5920 d^7 e^4+1881 d^6 e^5 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx}{7920 d^8} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}+\frac {\left (19 e^5\right ) \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx}{80 d^2} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}+\frac {\left (19 e^5\right ) \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{5/2}}{x^4} \, dx,x,x^2\right )}{160 d^2} \\ & = -\frac {19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}-\frac {\left (19 e^7\right ) \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{192 d^2} \\ & = \frac {19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac {19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}+\frac {\left (19 e^9\right ) \text {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{256 d^2} \\ & = -\frac {19 e^9 \sqrt {d^2-e^2 x^2}}{256 d^2 x^2}+\frac {19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac {19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}-\frac {\left (19 e^{11}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{512 d^2} \\ & = -\frac {19 e^9 \sqrt {d^2-e^2 x^2}}{256 d^2 x^2}+\frac {19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac {19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}+\frac {\left (19 e^9\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^2} \\ & = -\frac {19 e^9 \sqrt {d^2-e^2 x^2}}{256 d^2 x^2}+\frac {19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac {19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}+\frac {19 e^{11} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.74 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{12}} \, dx=\frac {\frac {d \sqrt {d^2-e^2 x^2} \left (-80640 d^{10}-266112 d^9 e x-89600 d^8 e^2 x^2+587664 d^7 e^3 x^3+657920 d^6 e^4 x^4-201432 d^5 e^5 x^5-629760 d^4 e^6 x^6-251790 d^3 e^7 x^7+47360 d^2 e^8 x^8+65835 d e^9 x^9+94720 e^{10} x^{10}\right )}{x^{11}}+65835 \sqrt {d^2} e^{11} \log (x)-65835 \sqrt {d^2} e^{11} \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{887040 d^4} \]

[In]

Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^12,x]

[Out]

((d*Sqrt[d^2 - e^2*x^2]*(-80640*d^10 - 266112*d^9*e*x - 89600*d^8*e^2*x^2 + 587664*d^7*e^3*x^3 + 657920*d^6*e^
4*x^4 - 201432*d^5*e^5*x^5 - 629760*d^4*e^6*x^6 - 251790*d^3*e^7*x^7 + 47360*d^2*e^8*x^8 + 65835*d*e^9*x^9 + 9
4720*e^10*x^10))/x^11 + 65835*Sqrt[d^2]*e^11*Log[x] - 65835*Sqrt[d^2]*e^11*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]
])/(887040*d^4)

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.69

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-94720 e^{10} x^{10}-65835 d \,e^{9} x^{9}-47360 d^{2} e^{8} x^{8}+251790 d^{3} e^{7} x^{7}+629760 d^{4} e^{6} x^{6}+201432 d^{5} e^{5} x^{5}-657920 d^{6} e^{4} x^{4}-587664 d^{7} e^{3} x^{3}+89600 d^{8} e^{2} x^{2}+266112 d^{9} e x +80640 d^{10}\right )}{887040 x^{11} d^{3}}+\frac {19 e^{11} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{256 d^{2} \sqrt {d^{2}}}\)	\(176\)
default	\(e^{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 )+d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{11 d^{2} x^{11}}+\frac {4 e^{2} \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 )}{11 d^{2}}\right )+3 d^{2} e \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}}}{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 )\)	\(626\)

[In]

int((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^12,x,method=_RETURNVERBOSE)

[Out]

-1/887040*(-e^2*x^2+d^2)^(1/2)*(-94720*e^10*x^10-65835*d*e^9*x^9-47360*d^2*e^8*x^8+251790*d^3*e^7*x^7+629760*d
^4*e^6*x^6+201432*d^5*e^5*x^5-657920*d^6*e^4*x^4-587664*d^7*e^3*x^3+89600*d^8*e^2*x^2+266112*d^9*e*x+80640*d^1
0)/x^11/d^3+19/256/d^2*e^11/(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 = 0.35 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.65 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{12}} \, dx=-\frac {65835 \, e^{11} x^{11} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (94720 \, e^{10} x^{10} + 65835 \, d e^{9} x^{9} + 47360 \, d^{2} e^{8} x^{8} - 251790 \, d^{3} e^{7} x^{7} - 629760 \, d^{4} e^{6} x^{6} - 201432 \, d^{5} e^{5} x^{5} + 657920 \, d^{6} e^{4} x^{4} + 587664 \, d^{7} e^{3} x^{3} - 89600 \, d^{8} e^{2} x^{2} - 266112 \, d^{9} e x - 80640 \, d^{10}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{887040 \, d^{3} x^{11}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^12,x, algorithm="fricas")

[Out]

-1/887040*(65835*e^11*x^11*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (94720*e^10*x^10 + 65835*d*e^9*x^9 + 47360*d^2
*e^8*x^8 - 251790*d^3*e^7*x^7 - 629760*d^4*e^6*x^6 - 201432*d^5*e^5*x^5 + 657920*d^6*e^4*x^4 + 587664*d^7*e^3*
x^3 - 89600*d^8*e^2*x^2 - 266112*d^9*e*x - 80640*d^10)*sqrt(-e^2*x^2 + d^2))/(d^3*x^11)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 146.79 (sec) , antiderivative size = 2397, normalized size of antiderivative = 9.44 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{12}} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2)/x**12,x)

[Out]

d**7*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(11*x**10) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(99*d**2*x**8) + 8*
e**5*sqrt(d**2/(e**2*x**2) - 1)/(693*d**4*x**6) + 16*e**7*sqrt(d**2/(e**2*x**2) - 1)/(1155*d**6*x**4) + 64*e**
9*sqrt(d**2/(e**2*x**2) - 1)/(3465*d**8*x**2) + 128*e**11*sqrt(d**2/(e**2*x**2) - 1)/(3465*d**10), Abs(d**2/(e
**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(11*x**10) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(99*d**2*x*
*8) + 8*I*e**5*sqrt(-d**2/(e**2*x**2) + 1)/(693*d**4*x**6) + 16*I*e**7*sqrt(-d**2/(e**2*x**2) + 1)/(1155*d**6*
x**4) + 64*I*e**9*sqrt(-d**2/(e**2*x**2) + 1)/(3465*d**8*x**2) + 128*I*e**11*sqrt(-d**2/(e**2*x**2) + 1)/(3465
*d**10), True)) + 3*d**6*e*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/(768*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*sqrt(-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)) + d**5*e**2*Piecew
ise((-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)) - 5*d**4*e**
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/(19
2*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**3*e**4*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**2*e**5*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)) + 3*d*e**6*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)) + e**7*P
iecewise((-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*s
qrt(-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))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{12}} \, dx=\frac {19 \, e^{11} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{256 \, d^{3}} - \frac {19 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{11}}{256 \, d^{4}} - \frac {19 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{11}}{768 \, d^{6}} - \frac {19 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{11}}{1280 \, d^{8}} - \frac {19 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{9}}{1280 \, d^{8} x^{2}} + \frac {19 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{7}}{1920 \, d^{6} x^{4}} - \frac {19 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{5}}{480 \, d^{4} x^{6}} - \frac {74 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{693 \, d^{3} x^{7}} - \frac {19 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{80 \, d^{2} x^{8}} - \frac {37 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{99 \, d x^{9}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{10 \, x^{10}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{11 \, x^{11}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^12,x, algorithm="maxima")

[Out]

19/256*e^11*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^3 - 19/256*sqrt(-e^2*x^2 + d^2)*e^11/d^4 - 1
9/768*(-e^2*x^2 + d^2)^(3/2)*e^11/d^6 - 19/1280*(-e^2*x^2 + d^2)^(5/2)*e^11/d^8 - 19/1280*(-e^2*x^2 + d^2)^(7/
2)*e^9/(d^8*x^2) + 19/1920*(-e^2*x^2 + d^2)^(7/2)*e^7/(d^6*x^4) - 19/480*(-e^2*x^2 + d^2)^(7/2)*e^5/(d^4*x^6)
- 74/693*(-e^2*x^2 + d^2)^(7/2)*e^4/(d^3*x^7) - 19/80*(-e^2*x^2 + d^2)^(7/2)*e^3/(d^2*x^8) - 37/99*(-e^2*x^2 +
d^2)^(7/2)*e^2/(d*x^9) - 3/10*(-e^2*x^2 + d^2)^(7/2)*e/x^10 - 1/11*(-e^2*x^2 + d^2)^(7/2)*d/x^11

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 778 vs. \(2 (218) = 436\).

Time = 0.32 (sec) , antiderivative size = 778, normalized size of antiderivative = 3.06 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{12}} \, dx=\frac {{\left (630 \, e^{12} + \frac {4158 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{10}}{x} + \frac {8470 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{8}}{x^{2}} - \frac {3465 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} e^{6}}{x^{3}} - \frac {40590 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} e^{4}}{x^{4}} - \frac {57750 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} e^{2}}{x^{5}} + \frac {6930 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{x^{6}} + \frac {138600 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7}}{e^{2} x^{7}} + \frac {244860 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8}}{e^{4} x^{8}} + \frac {152460 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{9}}{e^{6} x^{9}} - \frac {568260 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{10}}{e^{8} x^{10}}\right )} e^{22} x^{11}}{14192640 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{11} d^{3} {\left | e \right |}} + \frac {19 \, e^{12} \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^{3} {\left | e \right |}} + \frac {\frac {568260 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{30} e^{20}}{x} - \frac {152460 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{30} e^{18}}{x^{2}} - \frac {244860 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{30} e^{16}}{x^{3}} - \frac {138600 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{30} e^{14}}{x^{4}} - \frac {6930 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{30} e^{12}}{x^{5}} + \frac {57750 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} d^{30} e^{10}}{x^{6}} + \frac {40590 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7} d^{30} e^{8}}{x^{7}} + \frac {3465 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8} d^{30} e^{6}}{x^{8}} - \frac {8470 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{9} d^{30} e^{4}}{x^{9}} - \frac {4158 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{10} d^{30} e^{2}}{x^{10}} - \frac {630 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{11} d^{30}}{x^{11}}}{14192640 \, d^{33} e^{10} {\left | e \right |}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^12,x, algorithm="giac")

[Out]

1/14192640*(630*e^12 + 4158*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^10/x + 8470*(d*e + sqrt(-e^2*x^2 + d^2)*abs(
e))^2*e^8/x^2 - 3465*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*e^6/x^3 - 40590*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))
^4*e^4/x^4 - 57750*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5*e^2/x^5 + 6930*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6/
x^6 + 138600*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7/(e^2*x^7) + 244860*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^8/(e
^4*x^8) + 152460*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^9/(e^6*x^9) - 568260*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^
10/(e^8*x^10))*e^22*x^11/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^11*d^3*abs(e)) + 19/256*e^12*log(1/2*abs(-2*d*e
- 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/(d^3*abs(e)) + 1/14192640*(568260*(d*e + sqrt(-e^2*x^2 + d^2)*a
bs(e))*d^30*e^20/x - 152460*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^30*e^18/x^2 - 244860*(d*e + sqrt(-e^2*x^2
+ d^2)*abs(e))^3*d^30*e^16/x^3 - 138600*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^30*e^14/x^4 - 6930*(d*e + sqrt
(-e^2*x^2 + d^2)*abs(e))^5*d^30*e^12/x^5 + 57750*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6*d^30*e^10/x^6 + 40590*(
d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7*d^30*e^8/x^7 + 3465*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^8*d^30*e^6/x^8 -
8470*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^9*d^30*e^4/x^9 - 4158*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^10*d^30*e^2
/x^10 - 630*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^11*d^30/x^11)/(d^33*e^10*abs(e))

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{12}} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^{12}} \,d x \]

[In]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^12,x)

[Out]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^12, x)