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

Optimal. Leaf size=42 \[ -\frac {b c-a d}{d^2 \sqrt {c+\frac {d}{x^2}}}-\frac {b \sqrt {c+\frac {d}{x^2}}}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {444, 43} \[ -\frac {b c-a d}{d^2 \sqrt {c+\frac {d}{x^2}}}-\frac {b \sqrt {c+\frac {d}{x^2}}}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 36, normalized size = 0.86 \[ \frac {a d x^2-b \left (2 c x^2+d\right )}{d^2 x^2 \sqrt {c+\frac {d}{x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.70, size = 46, normalized size = 1.10 \[ -\frac {{\left ({\left (2 \, b c - a d\right )} x^{2} + b d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c d^{2} x^{2} + d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.26, size = 37, normalized size = 0.88 \[ -\frac {\frac {{\left (2 \, b c - a d\right )} x^{2}}{d^{2}} + \frac {b}{d}}{\sqrt {c x^{4} + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 46, normalized size = 1.10 \[ \frac {\left (a d \,x^{2}-2 b c \,x^{2}-b d \right ) \left (c \,x^{2}+d \right )}{\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} d^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.46, size = 46, normalized size = 1.10 \[ -b {\left (\frac {\sqrt {c + \frac {d}{x^{2}}}}{d^{2}} + \frac {c}{\sqrt {c + \frac {d}{x^{2}}} d^{2}}\right )} + \frac {a}{\sqrt {c + \frac {d}{x^{2}}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.52, size = 46, normalized size = 1.10 \[ \frac {x\,\sqrt {c+\frac {d}{x^2}}\,\left (x^2\,\left (\frac {a}{d}-\frac {2\,b\,c}{d^2}\right )-\frac {b}{d}\right )}{c\,x^3+d\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 3.42, size = 68, normalized size = 1.62 \[ \begin {cases} \frac {a}{d \sqrt {c + \frac {d}{x^{2}}}} - \frac {2 b c}{d^{2} \sqrt {c + \frac {d}{x^{2}}}} - \frac {b}{d x^{2} \sqrt {c + \frac {d}{x^{2}}}} & \text {for}\: d \neq 0 \\\frac {- \frac {a}{2 x^{2}} - \frac {b}{4 x^{4}}}{c^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a/(d*sqrt(c + d/x**2)) - 2*b*c/(d**2*sqrt(c + d/x**2)) - b/(d*x**2*sqrt(c + d/x**2)), Ne(d, 0)), ((
-a/(2*x**2) - b/(4*x**4))/c**(3/2), True))

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