\(\int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^5} \, dx\) [2935]

Optimal result

Integrand size = 21, antiderivative size = 171 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^5} \, dx=-\frac {\sqrt {a+b \sqrt {c x^2}}}{4 x^4}+\frac {5 b^2 c \sqrt {a+b \sqrt {c x^2}}}{96 a^2 x^2}-\frac {b c^2 \sqrt {a+b \sqrt {c x^2}}}{24 a \left (c x^2\right )^{3/2}}-\frac {5 b^3 c^2 \sqrt {a+b \sqrt {c x^2}}}{64 a^3 \sqrt {c x^2}}+\frac {5 b^4 c^2 \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{64 a^{7/2}} \]

[Out]

5/64*b^4*c^2*arctanh((a+b*(c*x^2)^(1/2))^(1/2)/a^(1/2))/a^(7/2)-1/4*(a+b*(c*x^2)^(1/2))^(1/2)/x^4+5/96*b^2*c*(
a+b*(c*x^2)^(1/2))^(1/2)/a^2/x^2-1/24*b*c^2*(a+b*(c*x^2)^(1/2))^(1/2)/a/(c*x^2)^(3/2)-5/64*b^3*c^2*(a+b*(c*x^2
)^(1/2))^(1/2)/a^3/(c*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {375, 43, 44, 65, 214} \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^5} \, dx=\frac {5 b^4 c^2 \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{64 a^{7/2}}-\frac {5 b^3 c^2 \sqrt {a+b \sqrt {c x^2}}}{64 a^3 \sqrt {c x^2}}+\frac {5 b^2 c \sqrt {a+b \sqrt {c x^2}}}{96 a^2 x^2}-\frac {b c^2 \sqrt {a+b \sqrt {c x^2}}}{24 a \left (c x^2\right )^{3/2}}-\frac {\sqrt {a+b \sqrt {c x^2}}}{4 x^4} \]

[In]

Int[Sqrt[a + b*Sqrt[c*x^2]]/x^5,x]

[Out]

-1/4*Sqrt[a + b*Sqrt[c*x^2]]/x^4 + (5*b^2*c*Sqrt[a + b*Sqrt[c*x^2]])/(96*a^2*x^2) - (b*c^2*Sqrt[a + b*Sqrt[c*x
^2]])/(24*a*(c*x^2)^(3/2)) - (5*b^3*c^2*Sqrt[a + b*Sqrt[c*x^2]])/(64*a^3*Sqrt[c*x^2]) + (5*b^4*c^2*ArcTanh[Sqr
t[a + b*Sqrt[c*x^2]]/Sqrt[a]])/(64*a^(7/2))

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 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[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 375

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*((c*x^q
)^(1/q))^(m + 1)), Subst[Int[x^m*(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q
}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rubi steps \begin{align*} \text {integral}& = c^2 \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^5} \, dx,x,\sqrt {c x^2}\right ) \\ & = -\frac {\sqrt {a+b \sqrt {c x^2}}}{4 x^4}+\frac {1}{8} \left (b c^2\right ) \text {Subst}\left (\int \frac {1}{x^4 \sqrt {a+b x}} \, dx,x,\sqrt {c x^2}\right ) \\ & = -\frac {\sqrt {a+b \sqrt {c x^2}}}{4 x^4}-\frac {b c^2 \sqrt {a+b \sqrt {c x^2}}}{24 a \left (c x^2\right )^{3/2}}-\frac {\left (5 b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,\sqrt {c x^2}\right )}{48 a} \\ & = -\frac {\sqrt {a+b \sqrt {c x^2}}}{4 x^4}+\frac {5 b^2 c \sqrt {a+b \sqrt {c x^2}}}{96 a^2 x^2}-\frac {b c^2 \sqrt {a+b \sqrt {c x^2}}}{24 a \left (c x^2\right )^{3/2}}+\frac {\left (5 b^3 c^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\sqrt {c x^2}\right )}{64 a^2} \\ & = -\frac {\sqrt {a+b \sqrt {c x^2}}}{4 x^4}+\frac {5 b^2 c \sqrt {a+b \sqrt {c x^2}}}{96 a^2 x^2}-\frac {b c^2 \sqrt {a+b \sqrt {c x^2}}}{24 a \left (c x^2\right )^{3/2}}-\frac {5 b^3 c^2 \sqrt {a+b \sqrt {c x^2}}}{64 a^3 \sqrt {c x^2}}-\frac {\left (5 b^4 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sqrt {c x^2}\right )}{128 a^3} \\ & = -\frac {\sqrt {a+b \sqrt {c x^2}}}{4 x^4}+\frac {5 b^2 c \sqrt {a+b \sqrt {c x^2}}}{96 a^2 x^2}-\frac {b c^2 \sqrt {a+b \sqrt {c x^2}}}{24 a \left (c x^2\right )^{3/2}}-\frac {5 b^3 c^2 \sqrt {a+b \sqrt {c x^2}}}{64 a^3 \sqrt {c x^2}}-\frac {\left (5 b^3 c^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sqrt {c x^2}}\right )}{64 a^3} \\ & = -\frac {\sqrt {a+b \sqrt {c x^2}}}{4 x^4}+\frac {5 b^2 c \sqrt {a+b \sqrt {c x^2}}}{96 a^2 x^2}-\frac {b c^2 \sqrt {a+b \sqrt {c x^2}}}{24 a \left (c x^2\right )^{3/2}}-\frac {5 b^3 c^2 \sqrt {a+b \sqrt {c x^2}}}{64 a^3 \sqrt {c x^2}}+\frac {5 b^4 c^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{64 a^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.39 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^5} \, dx=-\frac {\sqrt {a+b \sqrt {c x^2}} \left (48 a^3-10 a b^2 c x^2+8 a^2 b \sqrt {c x^2}+15 b^3 \left (c x^2\right )^{3/2}\right )}{192 a^3 x^4}+\frac {5 b^4 c^2 \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{64 a^{7/2}} \]

[In]

Integrate[Sqrt[a + b*Sqrt[c*x^2]]/x^5,x]

[Out]

-1/192*(Sqrt[a + b*Sqrt[c*x^2]]*(48*a^3 - 10*a*b^2*c*x^2 + 8*a^2*b*Sqrt[c*x^2] + 15*b^3*(c*x^2)^(3/2)))/(a^3*x
^4) + (5*b^4*c^2*ArcTanh[Sqrt[a + b*Sqrt[c*x^2]]/Sqrt[a]])/(64*a^(7/2))

Maple [A] (verified)

Time = 4.04 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.67

method	result	size
default	\(-\frac {15 a^{\frac {7}{2}} \left (a +b \sqrt {c \,x^{2}}\right )^{\frac {7}{2}}-15 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \sqrt {c \,x^{2}}}}{\sqrt {a}}\right ) a^{3} b^{4} c^{2} x^{4}-55 a^{\frac {9}{2}} \left (a +b \sqrt {c \,x^{2}}\right )^{\frac {5}{2}}+73 a^{\frac {11}{2}} \left (a +b \sqrt {c \,x^{2}}\right )^{\frac {3}{2}}+15 a^{\frac {13}{2}} \sqrt {a +b \sqrt {c \,x^{2}}}}{192 a^{\frac {13}{2}} x^{4}}\)	\(114\)

[In]

int((a+b*(c*x^2)^(1/2))^(1/2)/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/192*(15*a^(7/2)*(a+b*(c*x^2)^(1/2))^(7/2)-15*arctanh((a+b*(c*x^2)^(1/2))^(1/2)/a^(1/2))*a^3*b^4*c^2*x^4-55*
a^(9/2)*(a+b*(c*x^2)^(1/2))^(5/2)+73*a^(11/2)*(a+b*(c*x^2)^(1/2))^(3/2)+15*a^(13/2)*(a+b*(c*x^2)^(1/2))^(1/2))
/a^(13/2)/x^4

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^5} \, dx=\left [\frac {15 \, \sqrt {a} b^{4} c^{2} x^{4} \log \left (\frac {b c x^{2} + 2 \, \sqrt {c x^{2}} \sqrt {\sqrt {c x^{2}} b + a} \sqrt {a} + 2 \, \sqrt {c x^{2}} a}{x^{2}}\right ) + 2 \, {\left (10 \, a^{2} b^{2} c x^{2} - 48 \, a^{4} - {\left (15 \, a b^{3} c x^{2} + 8 \, a^{3} b\right )} \sqrt {c x^{2}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{384 \, a^{4} x^{4}}, -\frac {15 \, \sqrt {-a} b^{4} c^{2} x^{4} \arctan \left (\frac {\sqrt {\sqrt {c x^{2}} b + a} \sqrt {-a}}{a}\right ) - {\left (10 \, a^{2} b^{2} c x^{2} - 48 \, a^{4} - {\left (15 \, a b^{3} c x^{2} + 8 \, a^{3} b\right )} \sqrt {c x^{2}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{192 \, a^{4} x^{4}}\right ] \]

[In]

integrate((a+b*(c*x^2)^(1/2))^(1/2)/x^5,x, algorithm="fricas")

[Out]

[1/384*(15*sqrt(a)*b^4*c^2*x^4*log((b*c*x^2 + 2*sqrt(c*x^2)*sqrt(sqrt(c*x^2)*b + a)*sqrt(a) + 2*sqrt(c*x^2)*a)
/x^2) + 2*(10*a^2*b^2*c*x^2 - 48*a^4 - (15*a*b^3*c*x^2 + 8*a^3*b)*sqrt(c*x^2))*sqrt(sqrt(c*x^2)*b + a))/(a^4*x
^4), -1/192*(15*sqrt(-a)*b^4*c^2*x^4*arctan(sqrt(sqrt(c*x^2)*b + a)*sqrt(-a)/a) - (10*a^2*b^2*c*x^2 - 48*a^4 -
(15*a*b^3*c*x^2 + 8*a^3*b)*sqrt(c*x^2))*sqrt(sqrt(c*x^2)*b + a))/(a^4*x^4)]

Sympy [F]

\[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^5} \, dx=\int \frac {\sqrt {a + b \sqrt {c x^{2}}}}{x^{5}}\, dx \]

[In]

integrate((a+b*(c*x**2)**(1/2))**(1/2)/x**5,x)

[Out]

Integral(sqrt(a + b*sqrt(c*x**2))/x**5, x)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^5} \, dx=-\frac {1}{384} \, {\left (\frac {15 \, b^{4} \log \left (\frac {\sqrt {\sqrt {c x^{2}} b + a} - \sqrt {a}}{\sqrt {\sqrt {c x^{2}} b + a} + \sqrt {a}}\right )}{a^{\frac {7}{2}}} + \frac {2 \, {\left (15 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {7}{2}} b^{4} - 55 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {5}{2}} a b^{4} + 73 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {3}{2}} a^{2} b^{4} + 15 \, \sqrt {\sqrt {c x^{2}} b + a} a^{3} b^{4}\right )}}{{\left (\sqrt {c x^{2}} b + a\right )}^{4} a^{3} - 4 \, {\left (\sqrt {c x^{2}} b + a\right )}^{3} a^{4} + 6 \, {\left (\sqrt {c x^{2}} b + a\right )}^{2} a^{5} - 4 \, {\left (\sqrt {c x^{2}} b + a\right )} a^{6} + a^{7}}\right )} c^{2} \]

[In]

integrate((a+b*(c*x^2)^(1/2))^(1/2)/x^5,x, algorithm="maxima")

[Out]

-1/384*(15*b^4*log((sqrt(sqrt(c*x^2)*b + a) - sqrt(a))/(sqrt(sqrt(c*x^2)*b + a) + sqrt(a)))/a^(7/2) + 2*(15*(s
qrt(c*x^2)*b + a)^(7/2)*b^4 - 55*(sqrt(c*x^2)*b + a)^(5/2)*a*b^4 + 73*(sqrt(c*x^2)*b + a)^(3/2)*a^2*b^4 + 15*s
qrt(sqrt(c*x^2)*b + a)*a^3*b^4)/((sqrt(c*x^2)*b + a)^4*a^3 - 4*(sqrt(c*x^2)*b + a)^3*a^4 + 6*(sqrt(c*x^2)*b +
a)^2*a^5 - 4*(sqrt(c*x^2)*b + a)*a^6 + a^7))*c^2

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^5} \, dx=-\frac {\frac {15 \, b^{5} c^{\frac {5}{2}} \arctan \left (\frac {\sqrt {b \sqrt {c} x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} + \frac {15 \, {\left (b \sqrt {c} x + a\right )}^{\frac {7}{2}} b^{5} c^{\frac {5}{2}} - 55 \, {\left (b \sqrt {c} x + a\right )}^{\frac {5}{2}} a b^{5} c^{\frac {5}{2}} + 73 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} a^{2} b^{5} c^{\frac {5}{2}} + 15 \, \sqrt {b \sqrt {c} x + a} a^{3} b^{5} c^{\frac {5}{2}}}{a^{3} b^{4} c^{2} x^{4}}}{192 \, b \sqrt {c}} \]

[In]

integrate((a+b*(c*x^2)^(1/2))^(1/2)/x^5,x, algorithm="giac")

[Out]

-1/192*(15*b^5*c^(5/2)*arctan(sqrt(b*sqrt(c)*x + a)/sqrt(-a))/(sqrt(-a)*a^3) + (15*(b*sqrt(c)*x + a)^(7/2)*b^5
*c^(5/2) - 55*(b*sqrt(c)*x + a)^(5/2)*a*b^5*c^(5/2) + 73*(b*sqrt(c)*x + a)^(3/2)*a^2*b^5*c^(5/2) + 15*sqrt(b*s
qrt(c)*x + a)*a^3*b^5*c^(5/2))/(a^3*b^4*c^2*x^4))/(b*sqrt(c))

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^5} \, dx=\int \frac {\sqrt {a+b\,\sqrt {c\,x^2}}}{x^5} \,d x \]

[In]

int((a + b*(c*x^2)^(1/2))^(1/2)/x^5,x)

[Out]

int((a + b*(c*x^2)^(1/2))^(1/2)/x^5, x)