3.7.33 \(\int \frac {a+\frac {b}{x^2}}{(c+\frac {d}{x^2})^{3/2} x^9} \, dx\)

Optimal. Leaf size=126 \[ \frac {c^3 (b c-a d)}{d^5 \sqrt {c+\frac {d}{x^2}}}+\frac {c^2 \sqrt {c+\frac {d}{x^2}} (4 b c-3 a d)}{d^5}+\frac {\left (c+\frac {d}{x^2}\right )^{5/2} (4 b c-a d)}{5 d^5}-\frac {c \left (c+\frac {d}{x^2}\right )^{3/2} (2 b c-a d)}{d^5}-\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^5} \]

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Rubi [A]  time = 0.09, antiderivative size = 126, 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 {c^2 \sqrt {c+\frac {d}{x^2}} (4 b c-3 a d)}{d^5}+\frac {c^3 (b c-a d)}{d^5 \sqrt {c+\frac {d}{x^2}}}+\frac {\left (c+\frac {d}{x^2}\right )^{5/2} (4 b c-a d)}{5 d^5}-\frac {c \left (c+\frac {d}{x^2}\right )^{3/2} (2 b c-a d)}{d^5}-\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)/((c + d/x^2)^(3/2)*x^9),x]

[Out]

(c^3*(b*c - a*d))/(d^5*Sqrt[c + d/x^2]) + (c^2*(4*b*c - 3*a*d)*Sqrt[c + d/x^2])/d^5 - (c*(2*b*c - a*d)*(c + d/
x^2)^(3/2))/d^5 + ((4*b*c - a*d)*(c + d/x^2)^(5/2))/(5*d^5) - (b*(c + d/x^2)^(7/2))/(7*d^5)

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^9} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3 (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^3 (b c-a d)}{d^4 (c+d x)^{3/2}}-\frac {c^2 (4 b c-3 a d)}{d^4 \sqrt {c+d x}}+\frac {3 c (2 b c-a d) \sqrt {c+d x}}{d^4}+\frac {(-4 b c+a d) (c+d x)^{3/2}}{d^4}+\frac {b (c+d x)^{5/2}}{d^4}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {c^3 (b c-a d)}{d^5 \sqrt {c+\frac {d}{x^2}}}+\frac {c^2 (4 b c-3 a d) \sqrt {c+\frac {d}{x^2}}}{d^5}-\frac {c (2 b c-a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{d^5}+\frac {(4 b c-a d) \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^5}-\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^5}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 104, normalized size = 0.83 \begin {gather*} \frac {b \left (128 c^4 x^8+64 c^3 d x^6-16 c^2 d^2 x^4+8 c d^3 x^2-5 d^4\right )-7 a d x^2 \left (16 c^3 x^6+8 c^2 d x^4-2 c d^2 x^2+d^3\right )}{35 d^5 x^8 \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^9),x]

[Out]

(-7*a*d*x^2*(d^3 - 2*c*d^2*x^2 + 8*c^2*d*x^4 + 16*c^3*x^6) + b*(-5*d^4 + 8*c*d^3*x^2 - 16*c^2*d^2*x^4 + 64*c^3
*d*x^6 + 128*c^4*x^8))/(35*d^5*Sqrt[c + d/x^2]*x^8)

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IntegrateAlgebraic [A]  time = 0.11, size = 123, normalized size = 0.98 \begin {gather*} \frac {\sqrt {\frac {c x^2+d}{x^2}} \left (-112 a c^3 d x^8-56 a c^2 d^2 x^6+14 a c d^3 x^4-7 a d^4 x^2+128 b c^4 x^8+64 b c^3 d x^6-16 b c^2 d^2 x^4+8 b c d^3 x^2-5 b d^4\right )}{35 d^5 x^6 \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^9),x]

[Out]

(Sqrt[(d + c*x^2)/x^2]*(-5*b*d^4 + 8*b*c*d^3*x^2 - 7*a*d^4*x^2 - 16*b*c^2*d^2*x^4 + 14*a*c*d^3*x^4 + 64*b*c^3*
d*x^6 - 56*a*c^2*d^2*x^6 + 128*b*c^4*x^8 - 112*a*c^3*d*x^8))/(35*d^5*x^6*(d + c*x^2))

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fricas [A]  time = 0.45, size = 121, normalized size = 0.96 \begin {gather*} \frac {{\left (16 \, {\left (8 \, b c^{4} - 7 \, a c^{3} d\right )} x^{8} + 8 \, {\left (8 \, b c^{3} d - 7 \, a c^{2} d^{2}\right )} x^{6} - 5 \, b d^{4} - 2 \, {\left (8 \, b c^{2} d^{2} - 7 \, a c d^{3}\right )} x^{4} + {\left (8 \, b c d^{3} - 7 \, a d^{4}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{35 \, {\left (c d^{5} x^{8} + d^{6} x^{6}\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^9,x, algorithm="fricas")

[Out]

1/35*(16*(8*b*c^4 - 7*a*c^3*d)*x^8 + 8*(8*b*c^3*d - 7*a*c^2*d^2)*x^6 - 5*b*d^4 - 2*(8*b*c^2*d^2 - 7*a*c*d^3)*x
^4 + (8*b*c*d^3 - 7*a*d^4)*x^2)*sqrt((c*x^2 + d)/x^2)/(c*d^5*x^8 + d^6*x^6)

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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^{9}}\,{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^9,x, algorithm="giac")

[Out]

integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^9), x)

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maple [A]  time = 0.05, size = 118, normalized size = 0.94 \begin {gather*} -\frac {\left (112 a \,c^{3} d \,x^{8}-128 b \,c^{4} x^{8}+56 a \,c^{2} d^{2} x^{6}-64 b \,c^{3} d \,x^{6}-14 a c \,d^{3} x^{4}+16 b \,c^{2} d^{2} x^{4}+7 a \,d^{4} x^{2}-8 b c \,d^{3} x^{2}+5 b \,d^{4}\right ) \left (c \,x^{2}+d \right )}{35 \left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} d^{5} x^{10}} \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^9,x)

[Out]

-1/35*(112*a*c^3*d*x^8-128*b*c^4*x^8+56*a*c^2*d^2*x^6-64*b*c^3*d*x^6-14*a*c*d^3*x^4+16*b*c^2*d^2*x^4+7*a*d^4*x
^2-8*b*c*d^3*x^2+5*b*d^4)*(c*x^2+d)/((c*x^2+d)/x^2)^(3/2)/d^5/x^10

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maxima [A]  time = 0.55, size = 151, normalized size = 1.20 \begin {gather*} -\frac {1}{35} \, b {\left (\frac {5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}}}{d^{5}} - \frac {28 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c}{d^{5}} + \frac {70 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{2}}{d^{5}} - \frac {140 \, \sqrt {c + \frac {d}{x^{2}}} c^{3}}{d^{5}} - \frac {35 \, c^{4}}{\sqrt {c + \frac {d}{x^{2}}} d^{5}}\right )} - \frac {1}{5} \, a {\left (\frac {{\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}}}{d^{4}} - \frac {5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c}{d^{4}} + \frac {15 \, \sqrt {c + \frac {d}{x^{2}}} c^{2}}{d^{4}} + \frac {5 \, c^{3}}{\sqrt {c + \frac {d}{x^{2}}} d^{4}}\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^9,x, algorithm="maxima")

[Out]

-1/35*b*(5*(c + d/x^2)^(7/2)/d^5 - 28*(c + d/x^2)^(5/2)*c/d^5 + 70*(c + d/x^2)^(3/2)*c^2/d^5 - 140*sqrt(c + d/
x^2)*c^3/d^5 - 35*c^4/(sqrt(c + d/x^2)*d^5)) - 1/5*a*((c + d/x^2)^(5/2)/d^4 - 5*(c + d/x^2)^(3/2)*c/d^4 + 15*s
qrt(c + d/x^2)*c^2/d^4 + 5*c^3/(sqrt(c + d/x^2)*d^4))

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mupad [B]  time = 4.92, size = 154, normalized size = 1.22 \begin {gather*} \frac {c\,\sqrt {c+\frac {d}{x^2}}\,\left (21\,a\,d-29\,b\,c\right )}{35\,d^4\,x^2}-\frac {b\,\sqrt {c+\frac {d}{x^2}}}{7\,d^2\,x^6}-\frac {\sqrt {c+\frac {d}{x^2}}\,\left (7\,a\,d^2-13\,b\,c\,d\right )}{35\,d^4\,x^4}-\frac {\sqrt {c+\frac {d}{x^2}}\,\left (x^2\,\left (\frac {58\,b\,c^4-42\,a\,c^3\,d}{35\,d^5}+\frac {2\,c^3\,\left (77\,a\,d-93\,b\,c\right )}{35\,d^5}\right )+\frac {c^2\,\left (77\,a\,d-93\,b\,c\right )}{35\,d^4}\right )}{c\,x^2+d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^2)/(x^9*(c + d/x^2)^(3/2)),x)

[Out]

(c*(c + d/x^2)^(1/2)*(21*a*d - 29*b*c))/(35*d^4*x^2) - (b*(c + d/x^2)^(1/2))/(7*d^2*x^6) - ((c + d/x^2)^(1/2)*
(7*a*d^2 - 13*b*c*d))/(35*d^4*x^4) - ((c + d/x^2)^(1/2)*(x^2*((58*b*c^4 - 42*a*c^3*d)/(35*d^5) + (2*c^3*(77*a*
d - 93*b*c))/(35*d^5)) + (c^2*(77*a*d - 93*b*c))/(35*d^4)))/(d + c*x^2)

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sympy [A]  time = 16.14, size = 122, normalized size = 0.97 \begin {gather*} - \frac {b \left (c + \frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7 d^{5}} - \frac {c^{3} \left (a d - b c\right )}{d^{5} \sqrt {c + \frac {d}{x^{2}}}} - \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}} \left (a d - 4 b c\right )}{5 d^{5}} - \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}} \left (- 3 a c d + 6 b c^{2}\right )}{3 d^{5}} - \frac {\sqrt {c + \frac {d}{x^{2}}} \left (3 a c^{2} d - 4 b c^{3}\right )}{d^{5}} \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**9,x)

[Out]

-b*(c + d/x**2)**(7/2)/(7*d**5) - c**3*(a*d - b*c)/(d**5*sqrt(c + d/x**2)) - (c + d/x**2)**(5/2)*(a*d - 4*b*c)
/(5*d**5) - (c + d/x**2)**(3/2)*(-3*a*c*d + 6*b*c**2)/(3*d**5) - sqrt(c + d/x**2)*(3*a*c**2*d - 4*b*c**3)/d**5

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