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

Optimal. Leaf size=59 \[ \frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2 c^{3/2}}+\frac {a x^2 \sqrt {c+\frac {d}{x^2}}}{2 c} \]

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Rubi [A]  time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {446, 78, 63, 208} \begin {gather*} \frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2 c^{3/2}}+\frac {a x^2 \sqrt {c+\frac {d}{x^2}}}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b/x^2)*x)/Sqrt[c + d/x^2],x]

[Out]

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

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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 79, normalized size = 1.34 \begin {gather*} \frac {\sqrt {c x^2+d} (2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {c x^2+d}}\right )+a \sqrt {c} x \left (c x^2+d\right )}{2 c^{3/2} x \sqrt {c+\frac {d}{x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.10, size = 67, normalized size = 1.14 \begin {gather*} \frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {\frac {c x^2+d}{x^2}}}{\sqrt {c}}\right )}{2 c^{3/2}}+\frac {a x^2 \sqrt {\frac {c x^2+d}{x^2}}}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b/x^2)*x)/Sqrt[c + d/x^2],x]

[Out]

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

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fricas [A]  time = 0.44, size = 146, normalized size = 2.47 \begin {gather*} \left [\frac {2 \, a c x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - {\left (2 \, b c - a d\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right )}{4 \, c^{2}}, \frac {a c x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - {\left (2 \, b c - a d\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right )}{2 \, c^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.26, size = 88, normalized size = 1.49 \begin {gather*} \frac {\sqrt {c x^{4} + d x^{2}} a}{2 \, c} - \frac {{\left (2 \, b c - a d\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )} \sqrt {c} - d \right |}\right )}{4 \, c^{\frac {3}{2}}} + \frac {2 \, b c \log \left ({\left | d \right |}\right ) - a d \log \left ({\left | d \right |}\right )}{4 \, c^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 90, normalized size = 1.53 \begin {gather*} \frac {\sqrt {c \,x^{2}+d}\, \left (-a c d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )+2 b \,c^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )+\sqrt {c \,x^{2}+d}\, a \,c^{\frac {3}{2}} x \right )}{2 \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, c^{\frac {5}{2}} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.20, size = 109, normalized size = 1.85 \begin {gather*} \frac {1}{4} \, a {\left (\frac {2 \, \sqrt {c + \frac {d}{x^{2}}} d}{{\left (c + \frac {d}{x^{2}}\right )} c - c^{2}} + \frac {d \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} - \frac {b \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{2 \, \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.08, size = 59, normalized size = 1.00 \begin {gather*} \frac {b\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {a\,x^2\,\sqrt {c+\frac {d}{x^2}}}{2\,c}-\frac {a\,d\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2\,c^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 84.41, size = 66, normalized size = 1.12 \begin {gather*} \frac {a \sqrt {d} x \sqrt {\frac {c x^{2}}{d} + 1}}{2 c} - \frac {a d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{2 c^{\frac {3}{2}}} + \frac {b \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{\sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*sqrt(d)*x*sqrt(c*x**2/d + 1)/(2*c) - a*d*asinh(sqrt(c)*x/sqrt(d))/(2*c**(3/2)) + b*asinh(sqrt(c)*x/sqrt(d))/
sqrt(c)

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