\(\int \frac {(d+e x)^3 \sqrt {d^2-e^2 x^2}}{x^5} \, dx\) [64]

Optimal result

Integrand size = 27, antiderivative size = 134 \[ \int \frac {(d+e x)^3 \sqrt {d^2-e^2 x^2}}{x^5} \, dx=-\frac {e^2 (13 d+8 e x) \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{x^3}-e^4 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {13}{8} e^4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]

[Out]

-1/4*d*(-e^2*x^2+d^2)^(3/2)/x^4-e*(-e^2*x^2+d^2)^(3/2)/x^3-e^4*arctan(e*x/(-e^2*x^2+d^2)^(1/2))+13/8*e^4*arcta
nh((-e^2*x^2+d^2)^(1/2)/d)-1/8*e^2*(8*e*x+13*d)*(-e^2*x^2+d^2)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1821, 825, 858, 223, 209, 272, 65, 214} \[ \int \frac {(d+e x)^3 \sqrt {d^2-e^2 x^2}}{x^5} \, dx=e^4 \left (-\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+\frac {13}{8} e^4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {e^2 (13 d+8 e x) \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \]

[In]

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

[Out]

-1/8*(e^2*(13*d + 8*e*x)*Sqrt[d^2 - e^2*x^2])/x^2 - (d*(d^2 - e^2*x^2)^(3/2))/(4*x^4) - (e*(d^2 - e^2*x^2)^(3/
2))/x^3 - e^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + (13*e^4*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/8

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

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

Rule 214

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

Rule 223

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 825

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

Rule 858

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

Rule 1821

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rubi steps \begin{align*} \text {integral}& = -\frac {d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {\int \frac {\sqrt {d^2-e^2 x^2} \left (-12 d^4 e-13 d^3 e^2 x-4 d^2 e^3 x^2\right )}{x^4} \, dx}{4 d^2} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{x^3}+\frac {\int \frac {\left (39 d^5 e^2+12 d^4 e^3 x\right ) \sqrt {d^2-e^2 x^2}}{x^3} \, dx}{12 d^4} \\ & = -\frac {e^2 (13 d+8 e x) \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{x^3}-\frac {\int \frac {78 d^7 e^4+48 d^6 e^5 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{48 d^6} \\ & = -\frac {e^2 (13 d+8 e x) \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{x^3}-\frac {1}{8} \left (13 d e^4\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-e^5 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {e^2 (13 d+8 e x) \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{x^3}-\frac {1}{16} \left (13 d e^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-e^5 \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right ) \\ & = -\frac {e^2 (13 d+8 e x) \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{x^3}-e^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{8} \left (13 d e^2\right ) \text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right ) \\ & = -\frac {e^2 (13 d+8 e x) \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{x^3}-e^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {13}{8} e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x)^3 \sqrt {d^2-e^2 x^2}}{x^5} \, dx=-\frac {d \sqrt {d^2-e^2 x^2} \left (2 d^2+8 d e x+11 e^2 x^2\right )}{8 x^4}-\frac {13}{4} e^4 \text {arctanh}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )+e \left (-e^2\right )^{3/2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right ) \]

[In]

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

[Out]

-1/8*(d*Sqrt[d^2 - e^2*x^2]*(2*d^2 + 8*d*e*x + 11*e^2*x^2))/x^4 - (13*e^4*ArcTanh[(Sqrt[-e^2]*x - Sqrt[d^2 - e
^2*x^2])/d])/4 + e*(-e^2)^(3/2)*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]]

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.87

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (11 e^{2} x^{2}+8 d e x +2 d^{2}\right ) d}{8 x^{4}}-\frac {e^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}+\frac {13 e^{4} d \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}}\)	\(116\)
default	\(d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4 d^{2} x^{4}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{2 d^{2}}\right )}{4 d^{2}}\right )+e^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{d^{2} x}-\frac {2 e^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{d^{2}}\right )+3 d \,e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{2 d^{2}}\right )-\frac {e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{x^{3}}\)	\(324\)

[In]

int((e*x+d)^3*(-e^2*x^2+d^2)^(1/2)/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/8*(-e^2*x^2+d^2)^(1/2)*(11*e^2*x^2+8*d*e*x+2*d^2)*d/x^4-e^5/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)
^(1/2))+13/8*e^4*d/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.83 \[ \int \frac {(d+e x)^3 \sqrt {d^2-e^2 x^2}}{x^5} \, dx=\frac {16 \, e^{4} x^{4} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 13 \, e^{4} x^{4} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (11 \, d e^{2} x^{2} + 8 \, d^{2} e x + 2 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{8 \, x^{4}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(1/2)/x^5,x, algorithm="fricas")

[Out]

1/8*(16*e^4*x^4*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - 13*e^4*x^4*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (1
1*d*e^2*x^2 + 8*d^2*e*x + 2*d^3)*sqrt(-e^2*x^2 + d^2))/x^4

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.42 (sec) , antiderivative size = 544, normalized size of antiderivative = 4.06 \[ \int \frac {(d+e x)^3 \sqrt {d^2-e^2 x^2}}{x^5} \, dx=d^{3} \left (\begin {cases} - \frac {d^{2}}{4 e x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {3 e}{8 x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{3}}{8 d^{2} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{4} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{4 e x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {3 i e}{8 x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{3}}{8 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{4} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {otherwise} \end {cases}\right ) + 3 d^{2} e \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 x^{2}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 x^{2}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2}} & \text {otherwise} \end {cases}\right ) + 3 d e^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{2 x} + \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{2 e x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e}{2 x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d} & \text {otherwise} \end {cases}\right ) + e^{3} \left (\begin {cases} \frac {i d}{x \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + i e \operatorname {acosh}{\left (\frac {e x}{d} \right )} - \frac {i e^{2} x}{d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {d}{x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - e \operatorname {asin}{\left (\frac {e x}{d} \right )} + \frac {e^{2} x}{d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) \]

[In]

integrate((e*x+d)**3*(-e**2*x**2+d**2)**(1/2)/x**5,x)

[Out]

d**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), 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)) + 3*d**2*e*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(
3*x**2) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(3*d**2), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) +
1)/(3*x**2) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(3*d**2), True)) + 3*d*e**2*Piecewise((-e*sqrt(d**2/(e**2*x*
*2) - 1)/(2*x) + e**2*acosh(d/(e*x))/(2*d), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(2*e*x**3*sqrt(-d**2/(e**2*x**
2) + 1)) - I*e/(2*x*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**2*asin(d/(e*x))/(2*d), True)) + e**3*Piecewise((I*d/(x
*sqrt(-1 + e**2*x**2/d**2)) + I*e*acosh(e*x/d) - I*e**2*x/(d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) >
1), (-d/(x*sqrt(1 - e**2*x**2/d**2)) - e*asin(e*x/d) + e**2*x/(d*sqrt(1 - e**2*x**2/d**2)), True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.28 \[ \int \frac {(d+e x)^3 \sqrt {d^2-e^2 x^2}}{x^5} \, dx=-\frac {e^{5} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} + \frac {13}{8} \, e^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {13 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{4}}{8 \, d} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{3}}{x} - \frac {13 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{8 \, d x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e}{x^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{4 \, x^{4}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(1/2)/x^5,x, algorithm="maxima")

[Out]

-e^5*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) + 13/8*e^4*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x)) - 13
/8*sqrt(-e^2*x^2 + d^2)*e^4/d - sqrt(-e^2*x^2 + d^2)*e^3/x - 13/8*(-e^2*x^2 + d^2)^(3/2)*e^2/(d*x^2) - (-e^2*x
^2 + d^2)^(3/2)*e/x^3 - 1/4*(-e^2*x^2 + d^2)^(3/2)*d/x^4

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (118) = 236\).

Time = 0.30 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.43 \[ \int \frac {(d+e x)^3 \sqrt {d^2-e^2 x^2}}{x^5} \, dx=\frac {{\left (e^{5} + \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{3}}{x} + \frac {24 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e}{x^{2}} + \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e x^{3}}\right )} e^{8} x^{4}}{64 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} {\left | e \right |}} - \frac {e^{5} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} + \frac {13 \, e^{5} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{8 \, {\left | e \right |}} - \frac {\frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{5} {\left | e \right |}}{x} + \frac {24 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{3} {\left | e \right |}}{x^{2}} + \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} e {\left | e \right |}}{x^{3}} + \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} {\left | e \right |}}{e x^{4}}}{64 \, e^{4}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(1/2)/x^5,x, algorithm="giac")

[Out]

1/64*(e^5 + 8*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^3/x + 24*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*e/x^2 + 8*(
d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3/(e*x^3))*e^8*x^4/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*abs(e)) - e^5*arc
sin(e*x/d)*sgn(d)*sgn(e)/abs(e) + 13/8*e^5*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/a
bs(e) - 1/64*(8*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^5*abs(e)/x + 24*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*e^
3*abs(e)/x^2 + 8*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*e*abs(e)/x^3 + (d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*ab
s(e)/(e*x^4))/e^4

Mupad [F(-1)]

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

[In]

int(((d^2 - e^2*x^2)^(1/2)*(d + e*x)^3)/x^5,x)

[Out]

int(((d^2 - e^2*x^2)^(1/2)*(d + e*x)^3)/x^5, x)