3.248 \(\int \frac {(\frac {c}{a+b x^2})^{3/2}}{x^3} \, dx\)

Optimal. Leaf size=104 \[ -\frac {3 b c \sqrt {\frac {c}{a+b x^2}}}{2 a^2}+\frac {3 b c \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c}{a+b x^2}} \tanh ^{-1}\left (\sqrt {\frac {b x^2}{a}+1}\right )}{2 a^2}-\frac {c \sqrt {\frac {c}{a+b x^2}}}{2 a x^2} \]

[Out]

-3/2*b*c*(c/(b*x^2+a))^(1/2)/a^2-1/2*c*(c/(b*x^2+a))^(1/2)/a/x^2+3/2*b*c*arctanh((1+b*x^2/a)^(1/2))*(c/(b*x^2+
a))^(1/2)*(1+b*x^2/a)^(1/2)/a^2

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Rubi [A]  time = 0.15, antiderivative size = 112, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6720, 266, 51, 63, 208} \[ -\frac {3 c \left (a+b x^2\right ) \sqrt {\frac {c}{a+b x^2}}}{2 a^2 x^2}+\frac {3 b c \sqrt {a+b x^2} \sqrt {\frac {c}{a+b x^2}} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}}+\frac {c \sqrt {\frac {c}{a+b x^2}}}{a x^2} \]

Antiderivative was successfully verified.

[In]

Int[(c/(a + b*x^2))^(3/2)/x^3,x]

[Out]

(c*Sqrt[c/(a + b*x^2)])/(a*x^2) - (3*c*Sqrt[c/(a + b*x^2)]*(a + b*x^2))/(2*a^2*x^2) + (3*b*c*Sqrt[c/(a + b*x^2
)]*Sqrt[a + b*x^2]*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*a^(5/2))

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 266

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 6720

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m)^FracPart[p])/v^(m*FracPart[p]), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rubi steps

\begin {align*} \int \frac {\left (\frac {c}{a+b x^2}\right )^{3/2}}{x^3} \, dx &=\left (c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \int \frac {1}{x^3 \left (a+b x^2\right )^{3/2}} \, dx\\ &=\frac {1}{2} \left (c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac {c \sqrt {\frac {c}{a+b x^2}}}{a x^2}+\frac {\left (3 c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac {c \sqrt {\frac {c}{a+b x^2}}}{a x^2}-\frac {3 c \sqrt {\frac {c}{a+b x^2}} \left (a+b x^2\right )}{2 a^2 x^2}-\frac {\left (3 b c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{4 a^2}\\ &=\frac {c \sqrt {\frac {c}{a+b x^2}}}{a x^2}-\frac {3 c \sqrt {\frac {c}{a+b x^2}} \left (a+b x^2\right )}{2 a^2 x^2}-\frac {\left (3 c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{2 a^2}\\ &=\frac {c \sqrt {\frac {c}{a+b x^2}}}{a x^2}-\frac {3 c \sqrt {\frac {c}{a+b x^2}} \left (a+b x^2\right )}{2 a^2 x^2}+\frac {3 b c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 40, normalized size = 0.38 \[ -\frac {b c \sqrt {\frac {c}{a+b x^2}} \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {b x^2}{a}+1\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c/(a + b*x^2))^(3/2)/x^3,x]

[Out]

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

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fricas [A]  time = 0.48, size = 175, normalized size = 1.68 \[ \left [\frac {3 \, b c x^{2} \sqrt {\frac {c}{a}} \log \left (-\frac {b c x^{2} + 2 \, a c + 2 \, {\left (a b x^{2} + a^{2}\right )} \sqrt {\frac {c}{b x^{2} + a}} \sqrt {\frac {c}{a}}}{x^{2}}\right ) - 2 \, {\left (3 \, b c x^{2} + a c\right )} \sqrt {\frac {c}{b x^{2} + a}}}{4 \, a^{2} x^{2}}, -\frac {3 \, b c x^{2} \sqrt {-\frac {c}{a}} \arctan \left (\frac {a \sqrt {\frac {c}{b x^{2} + a}} \sqrt {-\frac {c}{a}}}{c}\right ) + {\left (3 \, b c x^{2} + a c\right )} \sqrt {\frac {c}{b x^{2} + a}}}{2 \, a^{2} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c/(b*x^2+a))^(3/2)/x^3,x, algorithm="fricas")

[Out]

[1/4*(3*b*c*x^2*sqrt(c/a)*log(-(b*c*x^2 + 2*a*c + 2*(a*b*x^2 + a^2)*sqrt(c/(b*x^2 + a))*sqrt(c/a))/x^2) - 2*(3
*b*c*x^2 + a*c)*sqrt(c/(b*x^2 + a)))/(a^2*x^2), -1/2*(3*b*c*x^2*sqrt(-c/a)*arctan(a*sqrt(c/(b*x^2 + a))*sqrt(-
c/a)/c) + (3*b*c*x^2 + a*c)*sqrt(c/(b*x^2 + a)))/(a^2*x^2)]

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giac [A]  time = 0.33, size = 103, normalized size = 0.99 \[ -\frac {1}{2} \, c {\left (\frac {3 \, b c \arctan \left (\frac {\sqrt {b c x^{2} + a c}}{\sqrt {-a c}}\right )}{\sqrt {-a c} a^{2}} + \frac {2 \, a b c^{2} - 3 \, {\left (b c x^{2} + a c\right )} b c}{{\left (\sqrt {b c x^{2} + a c} a c - {\left (b c x^{2} + a c\right )}^{\frac {3}{2}}\right )} a^{2}}\right )} \mathrm {sgn}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c/(b*x^2+a))^(3/2)/x^3,x, algorithm="giac")

[Out]

-1/2*c*(3*b*c*arctan(sqrt(b*c*x^2 + a*c)/sqrt(-a*c))/(sqrt(-a*c)*a^2) + (2*a*b*c^2 - 3*(b*c*x^2 + a*c)*b*c)/((
sqrt(b*c*x^2 + a*c)*a*c - (b*c*x^2 + a*c)^(3/2))*a^2))*sgn(b*x^2 + a)

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maple [A]  time = 0.01, size = 81, normalized size = 0.78 \[ \frac {\left (\frac {c}{b \,x^{2}+a}\right )^{\frac {3}{2}} \left (b \,x^{2}+a \right ) \left (3 \sqrt {b \,x^{2}+a}\, a b \,x^{2} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )-3 a^{\frac {3}{2}} b \,x^{2}-a^{\frac {5}{2}}\right )}{2 a^{\frac {7}{2}} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/(b*x^2+a)*c)^(3/2)/x^3,x)

[Out]

1/2*(1/(b*x^2+a)*c)^(3/2)*(b*x^2+a)*(3*(b*x^2+a)^(1/2)*ln(2*(a+(b*x^2+a)^(1/2)*a^(1/2))/x)*x^2*a*b-3*a^(3/2)*x
^2*b-a^(5/2))/a^(7/2)/x^2

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maxima [A]  time = 1.99, size = 121, normalized size = 1.16 \[ -\frac {1}{4} \, b c {\left (\frac {2 \, c \sqrt {\frac {c}{b x^{2} + a}}}{a^{2} c - \frac {a^{3} c}{b x^{2} + a}} + \frac {3 \, c \log \left (\frac {a \sqrt {\frac {c}{b x^{2} + a}} - \sqrt {a c}}{a \sqrt {\frac {c}{b x^{2} + a}} + \sqrt {a c}}\right )}{\sqrt {a c} a^{2}} + \frac {4 \, \sqrt {\frac {c}{b x^{2} + a}}}{a^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c/(b*x^2+a))^(3/2)/x^3,x, algorithm="maxima")

[Out]

-1/4*b*c*(2*c*sqrt(c/(b*x^2 + a))/(a^2*c - a^3*c/(b*x^2 + a)) + 3*c*log((a*sqrt(c/(b*x^2 + a)) - sqrt(a*c))/(a
*sqrt(c/(b*x^2 + a)) + sqrt(a*c)))/(sqrt(a*c)*a^2) + 4*sqrt(c/(b*x^2 + a))/a^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {c}{b\,x^2+a}\right )}^{3/2}}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c/(a + b*x^2))^(3/2)/x^3,x)

[Out]

int((c/(a + b*x^2))^(3/2)/x^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {c}{a + b x^{2}}\right )^{\frac {3}{2}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c/(b*x**2+a))**(3/2)/x**3,x)

[Out]

Integral((c/(a + b*x**2))**(3/2)/x**3, x)

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