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

Optimal. Leaf size=219 \[ -\frac {7 b^5 \left (c x^2\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{128 a^{9/2} x^5}+\frac {7 b^4 c^2 \sqrt {a+b \sqrt {c x^2}}}{128 a^4 x}-\frac {7 b^3 \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{192 a^3 c x^7}+\frac {7 b^2 c \sqrt {a+b \sqrt {c x^2}}}{240 a^2 x^3}-\frac {b \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{40 a c^2 x^9}-\frac {\sqrt {a+b \sqrt {c x^2}}}{5 x^5} \]

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Rubi [A]  time = 0.09, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {368, 47, 51, 63, 208} \begin {gather*} \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 {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}-\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[c*x^2]]/x^6,x]

[Out]

-Sqrt[a + b*Sqrt[c*x^2]]/(5*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 47

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},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 51

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] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

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 208

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

Rule 368

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*} \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^6} \, dx &=\frac {\left (c x^2\right )^{5/2} \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}

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Mathematica [C]  time = 0.01, size = 63, normalized size = 0.29 \begin {gather*} \frac {2 b^5 \left (c x^2\right )^{5/2} \left (a+b \sqrt {c x^2}\right )^{3/2} \, _2F_1\left (\frac {3}{2},6;\frac {5}{2};\frac {\sqrt {c x^2} b}{a}+1\right )}{3 a^6 x^5} \end {gather*}

Antiderivative was successfully verified.

[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)

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IntegrateAlgebraic [B]  time = 1.82, size = 439, normalized size = 2.00 \begin {gather*} \frac {-\frac {7 b^5 c^{9/2} x^4 \sqrt {c x^2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{8 a^{9/2}}-\frac {7 b^5 c^5 x^5 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{8 a^{9/2}}+\frac {7 b^4 c^{9/2} x^4 \sqrt {a+b \sqrt {c x^2}}}{8 a^4}+\frac {7 b^4 c^4 x^3 \sqrt {c x^2} \sqrt {a+b \sqrt {c x^2}}}{8 a^4}-\frac {7 b^3 c^{7/2} x^2 \sqrt {c x^2} \sqrt {a+b \sqrt {c x^2}}}{12 a^3}-\frac {7 b^3 c^4 x^3 \sqrt {a+b \sqrt {c x^2}}}{12 a^3}+\frac {7 b^2 c^{7/2} x^2 \sqrt {a+b \sqrt {c x^2}}}{15 a^2}+\frac {7 b^2 c^3 x \sqrt {c x^2} \sqrt {a+b \sqrt {c x^2}}}{15 a^2}-\frac {2 b c^{5/2} \sqrt {c x^2} \sqrt {a+b \sqrt {c x^2}}}{5 a}-\frac {32}{5} c^{5/2} \sqrt {a+b \sqrt {c x^2}}-\frac {2 b c^3 x \sqrt {a+b \sqrt {c x^2}}}{5 a}}{\left (\sqrt {c x^2}+\sqrt {c} x\right )^5} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-32*c^(5/2)*Sqrt[a + b*Sqrt[c*x^2]])/5 - (2*b*c^3*x*Sqrt[a + b*Sqrt[c*x^2]])/(5*a) + (7*b^2*c^(7/2)*x^2*Sqrt
[a + b*Sqrt[c*x^2]])/(15*a^2) - (7*b^3*c^4*x^3*Sqrt[a + b*Sqrt[c*x^2]])/(12*a^3) + (7*b^4*c^(9/2)*x^4*Sqrt[a +
b*Sqrt[c*x^2]])/(8*a^4) - (2*b*c^(5/2)*Sqrt[c*x^2]*Sqrt[a + b*Sqrt[c*x^2]])/(5*a) + (7*b^2*c^3*x*Sqrt[c*x^2]*
Sqrt[a + b*Sqrt[c*x^2]])/(15*a^2) - (7*b^3*c^(7/2)*x^2*Sqrt[c*x^2]*Sqrt[a + b*Sqrt[c*x^2]])/(12*a^3) + (7*b^4*
c^4*x^3*Sqrt[c*x^2]*Sqrt[a + b*Sqrt[c*x^2]])/(8*a^4) - (7*b^5*c^5*x^5*ArcTanh[Sqrt[a + b*Sqrt[c*x^2]]/Sqrt[a]]
)/(8*a^(9/2)) - (7*b^5*c^(9/2)*x^4*Sqrt[c*x^2]*ArcTanh[Sqrt[a + b*Sqrt[c*x^2]]/Sqrt[a]])/(8*a^(9/2)))/(Sqrt[c]
*x + Sqrt[c*x^2])^5

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fricas [A]  time = 0.67, size = 314, normalized size = 1.43 \begin {gather*} \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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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giac [A]  time = 0.17, size = 156, normalized size = 0.71 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.02, size = 133, normalized size = 0.61 \begin {gather*} -\frac {105 \left (c \,x^{2}\right )^{\frac {5}{2}} a^{4} b^{5} \arctanh \left (\frac {\sqrt {a +\sqrt {c \,x^{2}}\, b}}{\sqrt {a}}\right )+105 \sqrt {a +\sqrt {c \,x^{2}}\, b}\, a^{\frac {17}{2}}+790 \left (a +\sqrt {c \,x^{2}}\, b \right )^{\frac {3}{2}} a^{\frac {15}{2}}-896 \left (a +\sqrt {c \,x^{2}}\, b \right )^{\frac {5}{2}} a^{\frac {13}{2}}+490 \left (a +\sqrt {c \,x^{2}}\, b \right )^{\frac {7}{2}} a^{\frac {11}{2}}-105 \left (a +\sqrt {c \,x^{2}}\, b \right )^{\frac {9}{2}} a^{\frac {9}{2}}}{1920 a^{\frac {17}{2}} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\sqrt {c x^{2}} b + a}}{x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {a+b\,\sqrt {c\,x^2}}}{x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b \sqrt {c x^{2}}}}{x^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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