\(\int \frac {(d+e x)^3 (d^2-e^2 x^2)^{5/2}}{x^8} \, dx\) [78]

Optimal result

Integrand size = 27, antiderivative size = 206 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx=-\frac {3 e^6 (16 d-5 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3}-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}-3 d e^7 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {15}{16} d e^7 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]

[Out]

1/16*e^4*(5*e*x+16*d)*(-e^2*x^2+d^2)^(3/2)/x^3-1/40*e^2*(5*e*x+24*d)*(-e^2*x^2+d^2)^(5/2)/x^5-1/7*d*(-e^2*x^2+
d^2)^(7/2)/x^7-1/2*e*(-e^2*x^2+d^2)^(7/2)/x^6-3*d*e^7*arctan(e*x/(-e^2*x^2+d^2)^(1/2))-15/16*d*e^7*arctanh((-e
^2*x^2+d^2)^(1/2)/d)-3/16*e^6*(-5*e*x+16*d)*(-e^2*x^2+d^2)^(1/2)/x

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1821, 825, 827, 858, 223, 209, 272, 65, 214} \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx=-3 d e^7 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {15}{16} d e^7 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {3 e^6 (16 d-5 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3} \]

[In]

Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^8,x]

[Out]

(-3*e^6*(16*d - 5*e*x)*Sqrt[d^2 - e^2*x^2])/(16*x) + (e^4*(16*d + 5*e*x)*(d^2 - e^2*x^2)^(3/2))/(16*x^3) - (e^
2*(24*d + 5*e*x)*(d^2 - e^2*x^2)^(5/2))/(40*x^5) - (d*(d^2 - e^2*x^2)^(7/2))/(7*x^7) - (e*(d^2 - e^2*x^2)^(7/2
))/(2*x^6) - 3*d*e^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - (15*d*e^7*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/16

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 827

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

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}}{7 x^7}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-21 d^4 e-21 d^3 e^2 x-7 d^2 e^3 x^2\right )}{x^7} \, dx}{7 d^2} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}+\frac {\int \frac {\left (126 d^5 e^2+21 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^6} \, dx}{42 d^4} \\ & = -\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}-\frac {\int \frac {\left (1008 d^7 e^4+210 d^6 e^5 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx}{336 d^6} \\ & = \frac {e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3}-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}+\frac {\int \frac {\left (4032 d^9 e^6+1260 d^8 e^7 x\right ) \sqrt {d^2-e^2 x^2}}{x^2} \, dx}{1344 d^8} \\ & = -\frac {3 e^6 (16 d-5 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3}-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}-\frac {\int \frac {-2520 d^{10} e^7+8064 d^9 e^8 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{2688 d^8} \\ & = -\frac {3 e^6 (16 d-5 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3}-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}+\frac {1}{16} \left (15 d^2 e^7\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-\left (3 d e^8\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {3 e^6 (16 d-5 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3}-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}+\frac {1}{32} \left (15 d^2 e^7\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-\left (3 d e^8\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right ) \\ & = -\frac {3 e^6 (16 d-5 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3}-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}-3 d e^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{16} \left (15 d^2 e^5\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 {3 e^6 (16 d-5 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3}-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}-3 d e^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {15}{16} d e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-80 d^7-280 d^6 e x-96 d^5 e^2 x^2+770 d^4 e^3 x^3+992 d^3 e^4 x^4-525 d^2 e^5 x^5-2496 d e^6 x^6+560 e^7 x^7\right )}{560 x^7}+6 d e^7 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )-\frac {15}{16} \sqrt {d^2} e^7 \log (x)+\frac {15}{16} \sqrt {d^2} e^7 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right ) \]

[In]

Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^8,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-80*d^7 - 280*d^6*e*x - 96*d^5*e^2*x^2 + 770*d^4*e^3*x^3 + 992*d^3*e^4*x^4 - 525*d^2*e^5
*x^5 - 2496*d*e^6*x^6 + 560*e^7*x^7))/(560*x^7) + 6*d*e^7*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])] - (1
5*Sqrt[d^2]*e^7*Log[x])/16 + (15*Sqrt[d^2]*e^7*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/16

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.84

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-560 e^{7} x^{7}+2496 d \,e^{6} x^{6}+525 d^{2} e^{5} x^{5}-992 d^{3} e^{4} x^{4}-770 d^{4} e^{3} x^{3}+96 d^{5} e^{2} x^{2}+280 d^{6} e x +80 d^{7}\right )}{560 x^{7}}-\frac {3 d \,e^{8} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}-\frac {15 d^{2} e^{7} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{16 \sqrt {d^{2}}}\)	\(173\)
default	\(e^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )-\frac {d \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 x^{7}}+3 d^{2} e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )+3 d \,e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{5 d^{2} x^{5}}-\frac {2 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{3 d^{2} x^{3}}-\frac {4 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{d^{2}}\right )}{3 d^{2}}\right )}{5 d^{2}}\right )\)	\(580\)

[In]

int((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/560*(-e^2*x^2+d^2)^(1/2)*(-560*e^7*x^7+2496*d*e^6*x^6+525*d^2*e^5*x^5-992*d^3*e^4*x^4-770*d^4*e^3*x^3+96*d^
5*e^2*x^2+280*d^6*e*x+80*d^7)/x^7-3*d*e^8/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-15/16*d^2*e^7
/(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 = 0.27 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx=\frac {3360 \, d e^{7} x^{7} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 525 \, d e^{7} x^{7} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + 560 \, d e^{7} x^{7} + {\left (560 \, e^{7} x^{7} - 2496 \, d e^{6} x^{6} - 525 \, d^{2} e^{5} x^{5} + 992 \, d^{3} e^{4} x^{4} + 770 \, d^{4} e^{3} x^{3} - 96 \, d^{5} e^{2} x^{2} - 280 \, d^{6} e x - 80 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{560 \, x^{7}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^8,x, algorithm="fricas")

[Out]

1/560*(3360*d*e^7*x^7*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 525*d*e^7*x^7*log(-(d - sqrt(-e^2*x^2 + d^2)
)/x) + 560*d*e^7*x^7 + (560*e^7*x^7 - 2496*d*e^6*x^6 - 525*d^2*e^5*x^5 + 992*d^3*e^4*x^4 + 770*d^4*e^3*x^3 - 9
6*d^5*e^2*x^2 - 280*d^6*e*x - 80*d^7)*sqrt(-e^2*x^2 + d^2))/x^7

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 11.75 (sec) , antiderivative size = 1513, normalized size of antiderivative = 7.34 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2)/x**8,x)

[Out]

d**7*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(7*x**6) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(35*d**2*x**4) + 4*e*
*5*sqrt(d**2/(e**2*x**2) - 1)/(105*d**4*x**2) + 8*e**7*sqrt(d**2/(e**2*x**2) - 1)/(105*d**6), Abs(d**2/(e**2*x
**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(7*x**6) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(35*d**2*x**4) + 4
*I*e**5*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**4*x**2) + 8*I*e**7*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**6), True))
+ 3*d**6*e*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*s
qrt(-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**5*e**2*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**4*e**3*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), A
bs(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)) - 5*d**3*e**4
*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))
+ d**2*e**5*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)) + 3*d*e**6*Piecewise((I*d/(x*sqrt(-1 + e**2*x**2/d**2)) + I*e*acosh(e*x/d) - I*e**2*x/(d*s
qrt(-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)) + e**7*Piecewise((d**2/(e*x*sqrt(d**2/(e**2*x**2) - 1)) - d*acosh(d/(e*x
)) - e*x/sqrt(d**2/(e**2*x**2) - 1), Abs(d**2/(e**2*x**2)) > 1), (-I*d**2/(e*x*sqrt(-d**2/(e**2*x**2) + 1)) +
I*d*asin(d/(e*x)) + I*e*x/sqrt(-d**2/(e**2*x**2) + 1), True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.64 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx=-\frac {3 \, d e^{8} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} - \frac {15}{16} \, d e^{7} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{8} x}{d} + \frac {15}{16} \, \sqrt {-e^{2} x^{2} + d^{2}} e^{7} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{8} x}{d^{3}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{7}}{16 \, d^{2}} + \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{7}}{16 \, d^{4}} - \frac {8 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}{5 \, d^{3} x} + \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{5}}{16 \, d^{4} x^{2}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{5 \, d^{3} x^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{8 \, d^{2} x^{4}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{5 \, d x^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{2 \, x^{6}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{7 \, x^{7}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^8,x, algorithm="maxima")

[Out]

-3*d*e^8*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) - 15/16*d*e^7*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x
)) - 3*sqrt(-e^2*x^2 + d^2)*e^8*x/d + 15/16*sqrt(-e^2*x^2 + d^2)*e^7 - 2*(-e^2*x^2 + d^2)^(3/2)*e^8*x/d^3 + 5/
16*(-e^2*x^2 + d^2)^(3/2)*e^7/d^2 + 3/16*(-e^2*x^2 + d^2)^(5/2)*e^7/d^4 - 8/5*(-e^2*x^2 + d^2)^(5/2)*e^6/(d^3*
x) + 3/16*(-e^2*x^2 + d^2)^(7/2)*e^5/(d^4*x^2) + 2/5*(-e^2*x^2 + d^2)^(7/2)*e^4/(d^3*x^3) - 1/8*(-e^2*x^2 + d^
2)^(7/2)*e^3/(d^2*x^4) - 3/5*(-e^2*x^2 + d^2)^(7/2)*e^2/(d*x^5) - 1/2*(-e^2*x^2 + d^2)^(7/2)*e/x^6 - 1/7*(-e^2
*x^2 + d^2)^(7/2)*d/x^7

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (180) = 360\).

Time = 0.31 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.63 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx=\frac {{\left (5 \, d e^{8} + \frac {35 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d e^{6}}{x} + \frac {49 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d e^{4}}{x^{2}} - \frac {245 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d e^{2}}{x^{3}} - \frac {875 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d}{x^{4}} + \frac {455 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d}{e^{2} x^{5}} + \frac {9065 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} d}{e^{4} x^{6}}\right )} e^{14} x^{7}}{4480 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7} {\left | e \right |}} - \frac {3 \, d e^{8} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} - \frac {15 \, d e^{8} \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 )}{16 \, {\left | e \right |}} + \sqrt {-e^{2} x^{2} + d^{2}} e^{7} - \frac {\frac {9065 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d e^{12}}{x} + \frac {455 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d e^{10}}{x^{2}} - \frac {875 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d e^{8}}{x^{3}} - \frac {245 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d e^{6}}{x^{4}} + \frac {49 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d e^{4}}{x^{5}} + \frac {35 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} d e^{2}}{x^{6}} + \frac {5 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7} d}{x^{7}}}{4480 \, e^{6} {\left | e \right |}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^8,x, algorithm="giac")

[Out]

1/4480*(5*d*e^8 + 35*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d*e^6/x + 49*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d*
e^4/x^2 - 245*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d*e^2/x^3 - 875*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d/x^
4 + 455*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5*d/(e^2*x^5) + 9065*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6*d/(e^4*
x^6))*e^14*x^7/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7*abs(e)) - 3*d*e^8*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - 1
5/16*d*e^8*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/abs(e) + sqrt(-e^2*x^2 + d^2)*e^7
- 1/4480*(9065*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d*e^12/x + 455*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d*e^1
0/x^2 - 875*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d*e^8/x^3 - 245*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d*e^6/
x^4 + 49*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5*d*e^4/x^5 + 35*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6*d*e^2/x^6
+ 5*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7*d/x^7)/(e^6*abs(e))

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^8} \,d x \]

[In]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^8,x)

[Out]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^8, x)