\(\int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^6} \, dx\) [2941]

Optimal result

Integrand size = 21, antiderivative size = 219 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^6} \, dx=-\frac {\sqrt {a+b \sqrt {c x^2}}}{5 x^5}+\frac {7 b^2 c \sqrt {a+b \sqrt {c x^2}}}{240 a^2 x^3}+\frac {7 b^4 c^2 \sqrt {a+b \sqrt {c x^2}}}{128 a^4 x}-\frac {b \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{40 a c^2 x^9}-\frac {7 b^3 \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{192 a^3 c x^7}-\frac {7 b^5 \left (c x^2\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{128 a^{9/2} x^5} \]

[Out]

-7/128*b^5*(c*x^2)^(5/2)*arctanh((a+b*(c*x^2)^(1/2))^(1/2)/a^(1/2))/a^(9/2)/x^5-1/5*(a+b*(c*x^2)^(1/2))^(1/2)/
x^5+7/240*b^2*c*(a+b*(c*x^2)^(1/2))^(1/2)/a^2/x^3+7/128*b^4*c^2*(a+b*(c*x^2)^(1/2))^(1/2)/a^4/x-1/40*b*(c*x^2)
^(5/2)*(a+b*(c*x^2)^(1/2))^(1/2)/a/c^2/x^9-7/192*b^3*(c*x^2)^(5/2)*(a+b*(c*x^2)^(1/2))^(1/2)/a^3/c/x^7

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {375, 43, 44, 65, 214} \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^6} \, dx=-\frac {7 b^5 \left (c x^2\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{128 a^{9/2} x^5}+\frac {7 b^4 c^2 \sqrt {a+b \sqrt {c x^2}}}{128 a^4 x}-\frac {7 b^3 \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{192 a^3 c x^7}+\frac {7 b^2 c \sqrt {a+b \sqrt {c x^2}}}{240 a^2 x^3}-\frac {b \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{40 a c^2 x^9}-\frac {\sqrt {a+b \sqrt {c x^2}}}{5 x^5} \]

[In]

Int[Sqrt[a + b*Sqrt[c*x^2]]/x^6,x]

[Out]

-1/5*Sqrt[a + b*Sqrt[c*x^2]]/x^5 + (7*b^2*c*Sqrt[a + b*Sqrt[c*x^2]])/(240*a^2*x^3) + (7*b^4*c^2*Sqrt[a + b*Sqr
t[c*x^2]])/(128*a^4*x) - (b*(c*x^2)^(5/2)*Sqrt[a + b*Sqrt[c*x^2]])/(40*a*c^2*x^9) - (7*b^3*(c*x^2)^(5/2)*Sqrt[
a + b*Sqrt[c*x^2]])/(192*a^3*c*x^7) - (7*b^5*(c*x^2)^(5/2)*ArcTanh[Sqrt[a + b*Sqrt[c*x^2]]/Sqrt[a]])/(128*a^(9
/2)*x^5)

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}& = \frac {\left (c x^2\right )^{5/2} \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^6} \, dx,x,\sqrt {c x^2}\right )}{x^5} \\ & = -\frac {\sqrt {a+b \sqrt {c x^2}}}{5 x^5}+\frac {\left (b \left (c x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{x^5 \sqrt {a+b x}} \, dx,x,\sqrt {c x^2}\right )}{10 x^5} \\ & = -\frac {\sqrt {a+b \sqrt {c x^2}}}{5 x^5}-\frac {b \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{40 a c^2 x^9}-\frac {\left (7 b^2 \left (c x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{x^4 \sqrt {a+b x}} \, dx,x,\sqrt {c x^2}\right )}{80 a x^5} \\ & = -\frac {\sqrt {a+b \sqrt {c x^2}}}{5 x^5}+\frac {7 b^2 c \sqrt {a+b \sqrt {c x^2}}}{240 a^2 x^3}-\frac {b \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{40 a c^2 x^9}+\frac {\left (7 b^3 \left (c x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,\sqrt {c x^2}\right )}{96 a^2 x^5} \\ & = -\frac {\sqrt {a+b \sqrt {c x^2}}}{5 x^5}+\frac {7 b^2 c \sqrt {a+b \sqrt {c x^2}}}{240 a^2 x^3}-\frac {b \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{40 a c^2 x^9}-\frac {7 b^3 \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{192 a^3 c x^7}-\frac {\left (7 b^4 \left (c x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\sqrt {c x^2}\right )}{128 a^3 x^5} \\ & = -\frac {\sqrt {a+b \sqrt {c x^2}}}{5 x^5}+\frac {7 b^2 c \sqrt {a+b \sqrt {c x^2}}}{240 a^2 x^3}+\frac {7 b^4 c^2 \sqrt {a+b \sqrt {c x^2}}}{128 a^4 x}-\frac {b \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{40 a c^2 x^9}-\frac {7 b^3 \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{192 a^3 c x^7}+\frac {\left (7 b^5 \left (c x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sqrt {c x^2}\right )}{256 a^4 x^5} \\ & = -\frac {\sqrt {a+b \sqrt {c x^2}}}{5 x^5}+\frac {7 b^2 c \sqrt {a+b \sqrt {c x^2}}}{240 a^2 x^3}+\frac {7 b^4 c^2 \sqrt {a+b \sqrt {c x^2}}}{128 a^4 x}-\frac {b \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{40 a c^2 x^9}-\frac {7 b^3 \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{192 a^3 c x^7}+\frac {\left (7 b^4 \left (c x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sqrt {c x^2}}\right )}{128 a^4 x^5} \\ & = -\frac {\sqrt {a+b \sqrt {c x^2}}}{5 x^5}+\frac {7 b^2 c \sqrt {a+b \sqrt {c x^2}}}{240 a^2 x^3}+\frac {7 b^4 c^2 \sqrt {a+b \sqrt {c x^2}}}{128 a^4 x}-\frac {b \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{40 a c^2 x^9}-\frac {7 b^3 \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{192 a^3 c x^7}-\frac {7 b^5 \left (c x^2\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{128 a^{9/2} x^5} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 1.35 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^6} \, dx=\frac {2 b^5 \left (c x^2\right )^{5/2} \left (a+b \sqrt {c x^2}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},6,\frac {5}{2},1+\frac {b \sqrt {c x^2}}{a}\right )}{3 a^6 x^5} \]

[In]

Integrate[Sqrt[a + b*Sqrt[c*x^2]]/x^6,x]

[Out]

(2*b^5*(c*x^2)^(5/2)*(a + b*Sqrt[c*x^2])^(3/2)*Hypergeometric2F1[3/2, 6, 5/2, 1 + (b*Sqrt[c*x^2])/a])/(3*a^6*x
^5)

Maple [A] (verified)

Time = 3.94 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.61

method	result	size
default	\(-\frac {105 a^{\frac {17}{2}} \sqrt {a +b \sqrt {c \,x^{2}}}+790 a^{\frac {15}{2}} \left (a +b \sqrt {c \,x^{2}}\right )^{\frac {3}{2}}-896 a^{\frac {13}{2}} \left (a +b \sqrt {c \,x^{2}}\right )^{\frac {5}{2}}+490 a^{\frac {11}{2}} \left (a +b \sqrt {c \,x^{2}}\right )^{\frac {7}{2}}-105 a^{\frac {9}{2}} \left (a +b \sqrt {c \,x^{2}}\right )^{\frac {9}{2}}+105 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \sqrt {c \,x^{2}}}}{\sqrt {a}}\right ) a^{4} b^{5} \left (c \,x^{2}\right )^{\frac {5}{2}}}{1920 x^{5} a^{\frac {17}{2}}}\)	\(133\)

[In]

int((a+b*(c*x^2)^(1/2))^(1/2)/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/1920*(105*a^(17/2)*(a+b*(c*x^2)^(1/2))^(1/2)+790*a^(15/2)*(a+b*(c*x^2)^(1/2))^(3/2)-896*a^(13/2)*(a+b*(c*x^
2)^(1/2))^(5/2)+490*a^(11/2)*(a+b*(c*x^2)^(1/2))^(7/2)-105*a^(9/2)*(a+b*(c*x^2)^(1/2))^(9/2)+105*arctanh((a+b*
(c*x^2)^(1/2))^(1/2)/a^(1/2))*a^4*b^5*(c*x^2)^(5/2))/x^5/a^(17/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^6} \, dx=\left [\frac {105 \, b^{5} c^{2} x^{5} \sqrt {\frac {c}{a}} \log \left (\frac {b c x^{2} - 2 \, \sqrt {\sqrt {c x^{2}} b + a} a x \sqrt {\frac {c}{a}} + 2 \, \sqrt {c x^{2}} a}{x^{2}}\right ) + 2 \, {\left (105 \, b^{4} c^{2} x^{4} + 56 \, a^{2} b^{2} c x^{2} - 384 \, a^{4} - 2 \, {\left (35 \, a b^{3} c x^{2} + 24 \, a^{3} b\right )} \sqrt {c x^{2}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{3840 \, a^{4} x^{5}}, -\frac {105 \, b^{5} c^{2} x^{5} \sqrt {-\frac {c}{a}} \arctan \left (-\frac {{\left (a b c x^{2} \sqrt {-\frac {c}{a}} - \sqrt {c x^{2}} a^{2} \sqrt {-\frac {c}{a}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{b^{2} c^{2} x^{3} - a^{2} c x}\right ) - {\left (105 \, b^{4} c^{2} x^{4} + 56 \, a^{2} b^{2} c x^{2} - 384 \, a^{4} - 2 \, {\left (35 \, a b^{3} c x^{2} + 24 \, a^{3} b\right )} \sqrt {c x^{2}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{1920 \, a^{4} x^{5}}\right ] \]

[In]

integrate((a+b*(c*x^2)^(1/2))^(1/2)/x^6,x, algorithm="fricas")

[Out]

[1/3840*(105*b^5*c^2*x^5*sqrt(c/a)*log((b*c*x^2 - 2*sqrt(sqrt(c*x^2)*b + a)*a*x*sqrt(c/a) + 2*sqrt(c*x^2)*a)/x
^2) + 2*(105*b^4*c^2*x^4 + 56*a^2*b^2*c*x^2 - 384*a^4 - 2*(35*a*b^3*c*x^2 + 24*a^3*b)*sqrt(c*x^2))*sqrt(sqrt(c
*x^2)*b + a))/(a^4*x^5), -1/1920*(105*b^5*c^2*x^5*sqrt(-c/a)*arctan(-(a*b*c*x^2*sqrt(-c/a) - sqrt(c*x^2)*a^2*s
qrt(-c/a))*sqrt(sqrt(c*x^2)*b + a)/(b^2*c^2*x^3 - a^2*c*x)) - (105*b^4*c^2*x^4 + 56*a^2*b^2*c*x^2 - 384*a^4 -
2*(35*a*b^3*c*x^2 + 24*a^3*b)*sqrt(c*x^2))*sqrt(sqrt(c*x^2)*b + a))/(a^4*x^5)]

Sympy [F]

\[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^6} \, dx=\int \frac {\sqrt {a + b \sqrt {c x^{2}}}}{x^{6}}\, dx \]

[In]

integrate((a+b*(c*x**2)**(1/2))**(1/2)/x**6,x)

[Out]

Integral(sqrt(a + b*sqrt(c*x**2))/x**6, x)

Maxima [F]

\[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^6} \, dx=\int { \frac {\sqrt {\sqrt {c x^{2}} b + a}}{x^{6}} \,d x } \]

[In]

integrate((a+b*(c*x^2)^(1/2))^(1/2)/x^6,x, algorithm="maxima")

[Out]

integrate(sqrt(sqrt(c*x^2)*b + a)/x^6, x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^6} \, dx=\frac {\frac {105 \, b^{6} c^{3} \arctan \left (\frac {\sqrt {b \sqrt {c} x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \frac {105 \, {\left (b \sqrt {c} x + a\right )}^{\frac {9}{2}} b^{6} c^{3} - 490 \, {\left (b \sqrt {c} x + a\right )}^{\frac {7}{2}} a b^{6} c^{3} + 896 \, {\left (b \sqrt {c} x + a\right )}^{\frac {5}{2}} a^{2} b^{6} c^{3} - 790 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} a^{3} b^{6} c^{3} - 105 \, \sqrt {b \sqrt {c} x + a} a^{4} b^{6} c^{3}}{a^{4} b^{5} c^{\frac {5}{2}} x^{5}}}{1920 \, b \sqrt {c}} \]

[In]

integrate((a+b*(c*x^2)^(1/2))^(1/2)/x^6,x, algorithm="giac")

[Out]

1/1920*(105*b^6*c^3*arctan(sqrt(b*sqrt(c)*x + a)/sqrt(-a))/(sqrt(-a)*a^4) + (105*(b*sqrt(c)*x + a)^(9/2)*b^6*c
^3 - 490*(b*sqrt(c)*x + a)^(7/2)*a*b^6*c^3 + 896*(b*sqrt(c)*x + a)^(5/2)*a^2*b^6*c^3 - 790*(b*sqrt(c)*x + a)^(
3/2)*a^3*b^6*c^3 - 105*sqrt(b*sqrt(c)*x + a)*a^4*b^6*c^3)/(a^4*b^5*c^(5/2)*x^5))/(b*sqrt(c))

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^6} \, dx=\int \frac {\sqrt {a+b\,\sqrt {c\,x^2}}}{x^6} \,d x \]

[In]

int((a + b*(c*x^2)^(1/2))^(1/2)/x^6,x)

[Out]

int((a + b*(c*x^2)^(1/2))^(1/2)/x^6, x)