3.7.31 \(\int \frac {a+\frac {b}{x^2}}{(c+\frac {d}{x^2})^{3/2} x^5} \, dx\)

Optimal. Leaf size=68 \[ \frac {\sqrt {c+\frac {d}{x^2}} (2 b c-a d)}{d^3}+\frac {c (b c-a d)}{d^3 \sqrt {c+\frac {d}{x^2}}}-\frac {b \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d^3} \]

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Rubi [A]  time = 0.05, antiderivative size = 68, 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 = {446, 77} \begin {gather*} \frac {\sqrt {c+\frac {d}{x^2}} (2 b c-a d)}{d^3}+\frac {c (b c-a d)}{d^3 \sqrt {c+\frac {d}{x^2}}}-\frac {b \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)/((c + d/x^2)^(3/2)*x^5),x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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

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Mathematica [A]  time = 0.03, size = 60, normalized size = 0.88 \begin {gather*} \frac {b \left (8 c^2 x^4+4 c d x^2-d^2\right )-3 a d x^2 \left (2 c x^2+d\right )}{3 d^3 x^4 \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^5),x]

[Out]

(-3*a*d*x^2*(d + 2*c*x^2) + b*(-d^2 + 4*c*d*x^2 + 8*c^2*x^4))/(3*d^3*Sqrt[c + d/x^2]*x^4)

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IntegrateAlgebraic [A]  time = 0.08, size = 75, normalized size = 1.10 \begin {gather*} \frac {\sqrt {\frac {c x^2+d}{x^2}} \left (-6 a c d x^4-3 a d^2 x^2+8 b c^2 x^4+4 b c d x^2-b d^2\right )}{3 d^3 x^2 \left (c x^2+d\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b/x^2)/((c + d/x^2)^(3/2)*x^5),x]

[Out]

(Sqrt[(d + c*x^2)/x^2]*(-(b*d^2) + 4*b*c*d*x^2 - 3*a*d^2*x^2 + 8*b*c^2*x^4 - 6*a*c*d*x^4))/(3*d^3*x^2*(d + c*x
^2))

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fricas [A]  time = 0.42, size = 73, normalized size = 1.07 \begin {gather*} \frac {{\left (2 \, {\left (4 \, b c^{2} - 3 \, a c d\right )} x^{4} - b d^{2} + {\left (4 \, b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{3 \, {\left (c d^{3} x^{4} + d^{4} x^{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^5,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + \frac {b}{x^{2}}}{{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} x^{5}}\,{d 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^5,x, algorithm="giac")

[Out]

integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^5), x)

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maple [A]  time = 0.05, size = 69, normalized size = 1.01 \begin {gather*} -\frac {\left (6 a c d \,x^{4}-8 b \,c^{2} x^{4}+3 a \,d^{2} x^{2}-4 b c d \,x^{2}+b \,d^{2}\right ) \left (c \,x^{2}+d \right )}{3 \left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} d^{3} x^{6}} \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^5,x)

[Out]

-1/3*(6*a*c*d*x^4-8*b*c^2*x^4+3*a*d^2*x^2-4*b*c*d*x^2+b*d^2)*(c*x^2+d)/((c*x^2+d)/x^2)^(3/2)/d^3/x^6

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maxima [A]  time = 0.61, size = 81, normalized size = 1.19 \begin {gather*} -\frac {1}{3} \, b {\left (\frac {{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}}}{d^{3}} - \frac {6 \, \sqrt {c + \frac {d}{x^{2}}} c}{d^{3}} - \frac {3 \, c^{2}}{\sqrt {c + \frac {d}{x^{2}}} d^{3}}\right )} - a {\left (\frac {\sqrt {c + \frac {d}{x^{2}}}}{d^{2}} + \frac {c}{\sqrt {c + \frac {d}{x^{2}}} d^{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^5,x, algorithm="maxima")

[Out]

-1/3*b*((c + d/x^2)^(3/2)/d^3 - 6*sqrt(c + d/x^2)*c/d^3 - 3*c^2/(sqrt(c + d/x^2)*d^3)) - a*(sqrt(c + d/x^2)/d^
2 + c/(sqrt(c + d/x^2)*d^2))

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mupad [B]  time = 4.64, size = 66, normalized size = 0.97 \begin {gather*} -\frac {\sqrt {c+\frac {d}{x^2}}\,\left (-8\,b\,c^2\,x^4+6\,a\,c\,d\,x^4-4\,b\,c\,d\,x^2+3\,a\,d^2\,x^2+b\,d^2\right )}{3\,d^3\,x^2\,\left (c\,x^2+d\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^2)/(x^5*(c + d/x^2)^(3/2)),x)

[Out]

-((c + d/x^2)^(1/2)*(b*d^2 + 3*a*d^2*x^2 - 8*b*c^2*x^4 + 6*a*c*d*x^4 - 4*b*c*d*x^2))/(3*d^3*x^2*(d + c*x^2))

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sympy [A]  time = 11.07, size = 61, normalized size = 0.90 \begin {gather*} - \frac {b \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3 d^{3}} - \frac {c \left (a d - b c\right )}{d^{3} \sqrt {c + \frac {d}{x^{2}}}} - \frac {\sqrt {c + \frac {d}{x^{2}}} \left (a d - 2 b c\right )}{d^{3}} \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**5,x)

[Out]

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

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