\(\int \frac {(d+e x)^3 (d^2-e^2 x^2)^{5/2}}{x^{10}} \, dx\) [80]

Optimal result

Integrand size = 27, antiderivative size = 187 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {55 e^7 \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac {55 e^9 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d} \]

[Out]

55/192*e^5*(-e^2*x^2+d^2)^(3/2)/x^4-11/48*e^3*(-e^2*x^2+d^2)^(5/2)/x^6-1/9*d*(-e^2*x^2+d^2)^(7/2)/x^9-3/8*e*(-
e^2*x^2+d^2)^(7/2)/x^8-29/63*e^2*(-e^2*x^2+d^2)^(7/2)/d/x^7+55/128*e^9*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d-55/12
8*e^7*(-e^2*x^2+d^2)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1821, 821, 272, 43, 65, 214} \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx=\frac {55 e^9 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}-\frac {55 e^7 \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6} \]

[In]

Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^10,x]

[Out]

(-55*e^7*Sqrt[d^2 - e^2*x^2])/(128*x^2) + (55*e^5*(d^2 - e^2*x^2)^(3/2))/(192*x^4) - (11*e^3*(d^2 - e^2*x^2)^(
5/2))/(48*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(9*x^9) - (3*e*(d^2 - e^2*x^2)^(7/2))/(8*x^8) - (29*e^2*(d^2 - e^2*
x^2)^(7/2))/(63*d*x^7) + (55*e^9*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(128*d)

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 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}}{9 x^9}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-27 d^4 e-29 d^3 e^2 x-9 d^2 e^3 x^2\right )}{x^9} \, dx}{9 d^2} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}+\frac {\int \frac {\left (232 d^5 e^2+99 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx}{72 d^4} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac {1}{8} \left (11 e^3\right ) \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac {1}{16} \left (11 e^3\right ) \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{5/2}}{x^4} \, dx,x,x^2\right ) \\ & = -\frac {11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}-\frac {1}{96} \left (55 e^5\right ) \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right ) \\ & = \frac {55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac {1}{128} \left (55 e^7\right ) \text {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right ) \\ & = -\frac {55 e^7 \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}-\frac {1}{256} \left (55 e^9\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {55 e^7 \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac {1}{128} \left (55 e^7\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 {55 e^7 \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac {55 e^9 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.90 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-896 d^8-3024 d^7 e x-1024 d^6 e^2 x^2+7224 d^5 e^3 x^3+8448 d^4 e^4 x^4-3066 d^3 e^5 x^5-10240 d^2 e^6 x^6-4599 d e^7 x^7+3712 e^8 x^8\right )}{8064 d x^9}+\frac {55 e^9 \log (x)}{128 \sqrt {d^2}}-\frac {55 e^9 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{128 \sqrt {d^2}} \]

[In]

Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^10,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-896*d^8 - 3024*d^7*e*x - 1024*d^6*e^2*x^2 + 7224*d^5*e^3*x^3 + 8448*d^4*e^4*x^4 - 3066*
d^3*e^5*x^5 - 10240*d^2*e^6*x^6 - 4599*d*e^7*x^7 + 3712*e^8*x^8))/(8064*d*x^9) + (55*e^9*Log[x])/(128*Sqrt[d^2
]) - (55*e^9*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/(128*Sqrt[d^2])

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.81

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-3712 e^{8} x^{8}+4599 d \,e^{7} x^{7}+10240 d^{2} e^{6} x^{6}+3066 d^{3} e^{5} x^{5}-8448 d^{4} x^{4} e^{4}-7224 d^{5} e^{3} x^{3}+1024 d^{6} e^{2} x^{2}+3024 d^{7} e x +896 d^{8}\right )}{8064 x^{9} d}+\frac {55 e^{9} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{128 \sqrt {d^{2}}}\)	\(151\)
default	\(e^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )+d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 d^{2} x^{9}}-\frac {2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 d^{4} x^{7}}\right )-\frac {3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 d \,x^{7}}+3 d^{2} e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 d^{2} x^{8}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )}{8 d^{2}}\right )\)	\(504\)

[In]

int((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^10,x,method=_RETURNVERBOSE)

[Out]

-1/8064*(-e^2*x^2+d^2)^(1/2)*(-3712*e^8*x^8+4599*d*e^7*x^7+10240*d^2*e^6*x^6+3066*d^3*e^5*x^5-8448*d^4*e^4*x^4
-7224*d^5*e^3*x^3+1024*d^6*e^2*x^2+3024*d^7*e*x+896*d^8)/x^9/d+55/128*e^9/(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 = 142, normalized size of antiderivative = 0.76 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {3465 \, e^{9} x^{9} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (3712 \, e^{8} x^{8} - 4599 \, d e^{7} x^{7} - 10240 \, d^{2} e^{6} x^{6} - 3066 \, d^{3} e^{5} x^{5} + 8448 \, d^{4} e^{4} x^{4} + 7224 \, d^{5} e^{3} x^{3} - 1024 \, d^{6} e^{2} x^{2} - 3024 \, d^{7} e x - 896 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{8064 \, d x^{9}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^10,x, algorithm="fricas")

[Out]

-1/8064*(3465*e^9*x^9*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (3712*e^8*x^8 - 4599*d*e^7*x^7 - 10240*d^2*e^6*x^6
- 3066*d^3*e^5*x^5 + 8448*d^4*e^4*x^4 + 7224*d^5*e^3*x^3 - 1024*d^6*e^2*x^2 - 3024*d^7*e*x - 896*d^8)*sqrt(-e^
2*x^2 + d^2))/(d*x^9)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 31.65 (sec) , antiderivative size = 1889, normalized size of antiderivative = 10.10 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2)/x**10,x)

[Out]

d**7*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*sq
rt(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)) +
3*d**6*e*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*sqr
t(-d**2/(e**2*x**2) + 1)) - 5*I*e**8*asin(d/(e*x))/(128*d**7), True)) + d**5*e**2*Piecewise((-e*sqrt(d**2/(e**
2*x**2) - 1)/(7*x**6) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(35*d**2*x**4) + 4*e**5*sqrt(d**2/(e**2*x**2) - 1)/(10
5*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)) - 5*d**4*e**3*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*as
in(d/(e*x))/(16*d**5), True)) - 5*d**3*e**4*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
)) + d**2*e**5*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*sq
rt(-d**2/(e**2*x**2) + 1)) - I*e**4*asin(d/(e*x))/(8*d**3), True)) + 3*d*e**6*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)) + e**7*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))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.32 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx=\frac {55 \, e^{9} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{128 \, d} - \frac {55 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{9}}{128 \, d^{2}} - \frac {55 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{9}}{384 \, d^{4}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{9}}{128 \, d^{6}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{7}}{128 \, d^{6} x^{2}} + \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{5}}{192 \, d^{4} x^{4}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{48 \, d^{2} x^{6}} - \frac {29 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{63 \, d x^{7}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{8 \, x^{8}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{9 \, x^{9}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^10,x, algorithm="maxima")

[Out]

55/128*e^9*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d - 55/128*sqrt(-e^2*x^2 + d^2)*e^9/d^2 - 55/38
4*(-e^2*x^2 + d^2)^(3/2)*e^9/d^4 - 11/128*(-e^2*x^2 + d^2)^(5/2)*e^9/d^6 - 11/128*(-e^2*x^2 + d^2)^(7/2)*e^7/(
d^6*x^2) + 11/192*(-e^2*x^2 + d^2)^(7/2)*e^5/(d^4*x^4) - 11/48*(-e^2*x^2 + d^2)^(7/2)*e^3/(d^2*x^6) - 29/63*(-
e^2*x^2 + d^2)^(7/2)*e^2/(d*x^7) - 3/8*(-e^2*x^2 + d^2)^(7/2)*e/x^8 - 1/9*(-e^2*x^2 + d^2)^(7/2)*d/x^9

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 648 vs. \(2 (159) = 318\).

Time = 0.32 (sec) , antiderivative size = 648, normalized size of antiderivative = 3.47 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx=\frac {{\left (28 \, e^{10} + \frac {189 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{8}}{x} + \frac {324 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{6}}{x^{2}} - \frac {672 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} e^{4}}{x^{3}} - \frac {3024 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} e^{2}}{x^{4}} - \frac {1512 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{x^{5}} + \frac {9744 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e^{2} x^{6}} + \frac {18144 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7}}{e^{4} x^{7}} - \frac {16632 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8}}{e^{6} x^{8}}\right )} e^{18} x^{9}}{129024 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{9} d {\left | e \right |}} + \frac {55 \, e^{10} \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 {\left | e \right |}} + \frac {\frac {16632 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{8} e^{16}}{x} - \frac {18144 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{8} e^{14}}{x^{2}} - \frac {9744 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{8} e^{12}}{x^{3}} + \frac {1512 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{8} e^{10}}{x^{4}} + \frac {3024 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{8} e^{8}}{x^{5}} + \frac {672 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} d^{8} e^{6}}{x^{6}} - \frac {324 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7} d^{8} e^{4}}{x^{7}} - \frac {189 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8} d^{8} e^{2}}{x^{8}} - \frac {28 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{9} d^{8}}{x^{9}}}{129024 \, d^{9} e^{8} {\left | e \right |}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^10,x, algorithm="giac")

[Out]

1/129024*(28*e^10 + 189*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^8/x + 324*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*
e^6/x^2 - 672*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*e^4/x^3 - 3024*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*e^2/x
^4 - 1512*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5/x^5 + 9744*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6/(e^2*x^6) + 1
8144*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7/(e^4*x^7) - 16632*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^8/(e^6*x^8))*
e^18*x^9/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^9*d*abs(e)) + 55/128*e^10*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 +
d^2)*abs(e))/(e^2*abs(x)))/(d*abs(e)) + 1/129024*(16632*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^8*e^16/x - 1814
4*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^8*e^14/x^2 - 9744*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^8*e^12/x^3
+ 1512*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^8*e^10/x^4 + 3024*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5*d^8*e^
8/x^5 + 672*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6*d^8*e^6/x^6 - 324*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7*d^8*
e^4/x^7 - 189*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^8*d^8*e^2/x^8 - 28*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^9*d^8
/x^9)/(d^9*e^8*abs(e))

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^{10}} \,d x \]

[In]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^10,x)

[Out]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^10, x)