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

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

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Rubi [A]  time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {452, 335, 217, 206} \begin {gather*} \frac {b c-a d}{c d x \sqrt {c+\frac {d}{x^2}}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{d^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 452

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

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 71, normalized size = 1.20 \begin {gather*} \frac {\sqrt {d} (b c-a d)-b c \sqrt {c x^2+d} \tanh ^{-1}\left (\frac {\sqrt {c x^2+d}}{\sqrt {d}}\right )}{c d^{3/2} x \sqrt {c+\frac {d}{x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.12, size = 77, normalized size = 1.31 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (\frac {b c-a d}{c d \sqrt {c x^2+d}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {c x^2+d}}{\sqrt {d}}\right )}{d^{3/2}}\right )}{\sqrt {c x^2+d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.46, size = 195, normalized size = 3.31 \begin {gather*} \left [\frac {2 \, {\left (b c d - a d^{2}\right )} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + {\left (b c^{2} x^{2} + b c d\right )} \sqrt {d} \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right )}{2 \, {\left (c^{2} d^{2} x^{2} + c d^{3}\right )}}, \frac {{\left (b c d - a d^{2}\right )} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + {\left (b c^{2} x^{2} + b c d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right )}{c^{2} d^{2} x^{2} + c d^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is
real):Check [sign(t_nostep),sign(t_nostep+sqrt(d)/c*sign(t_nostep))]sym2poly/r2sym(const gen & e,const index_
m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.06, size = 79, normalized size = 1.34 \begin {gather*} -\frac {\left (c \,x^{2}+d \right ) \left (\sqrt {c \,x^{2}+d}\, b c d \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )+a \,d^{\frac {5}{2}}-b c \,d^{\frac {3}{2}}\right )}{\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} c \,d^{\frac {5}{2}} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.21, size = 80, normalized size = 1.36 \begin {gather*} \frac {1}{2} \, b {\left (\frac {\log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {2}{\sqrt {c + \frac {d}{x^{2}}} d x}\right )} - \frac {a}{\sqrt {c + \frac {d}{x^{2}}} c x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.11, size = 60, normalized size = 1.02 \begin {gather*} \frac {b}{d\,x\,\sqrt {c+\frac {d}{x^2}}}-\frac {a}{c\,x\,\sqrt {c+\frac {d}{x^2}}}-\frac {b\,\ln \left (\sqrt {c+\frac {d}{x^2}}+\frac {\sqrt {d}}{x}\right )}{d^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 11.71, size = 206, normalized size = 3.49 \begin {gather*} - \frac {a}{c \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + b \left (\frac {c d^{2} x^{2} \log {\left (\frac {c x^{2}}{d} \right )}}{2 c d^{\frac {7}{2}} x^{2} + 2 d^{\frac {9}{2}}} - \frac {2 c d^{2} x^{2} \log {\left (\sqrt {\frac {c x^{2}}{d} + 1} + 1 \right )}}{2 c d^{\frac {7}{2}} x^{2} + 2 d^{\frac {9}{2}}} + \frac {2 d^{3} \sqrt {\frac {c x^{2}}{d} + 1}}{2 c d^{\frac {7}{2}} x^{2} + 2 d^{\frac {9}{2}}} + \frac {d^{3} \log {\left (\frac {c x^{2}}{d} \right )}}{2 c d^{\frac {7}{2}} x^{2} + 2 d^{\frac {9}{2}}} - \frac {2 d^{3} \log {\left (\sqrt {\frac {c x^{2}}{d} + 1} + 1 \right )}}{2 c d^{\frac {7}{2}} x^{2} + 2 d^{\frac {9}{2}}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/(c*sqrt(d)*sqrt(c*x**2/d + 1)) + b*(c*d**2*x**2*log(c*x**2/d)/(2*c*d**(7/2)*x**2 + 2*d**(9/2)) - 2*c*d**2*x
**2*log(sqrt(c*x**2/d + 1) + 1)/(2*c*d**(7/2)*x**2 + 2*d**(9/2)) + 2*d**3*sqrt(c*x**2/d + 1)/(2*c*d**(7/2)*x**
2 + 2*d**(9/2)) + d**3*log(c*x**2/d)/(2*c*d**(7/2)*x**2 + 2*d**(9/2)) - 2*d**3*log(sqrt(c*x**2/d + 1) + 1)/(2*
c*d**(7/2)*x**2 + 2*d**(9/2)))

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