3.934 \(\int \frac{(a+\frac{b}{x^2}) \sqrt{c+\frac{d}{x^2}}}{x^5} \, dx\)

Optimal. Leaf size=74 \[ \frac{\left (c+\frac{d}{x^2}\right )^{5/2} (2 b c-a d)}{5 d^3}-\frac{c \left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{7/2}}{7 d^3} \]

[Out]

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

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Rubi [A]  time = 0.0587087, antiderivative size = 74, normalized size of antiderivative = 1., 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} \[ \frac{\left (c+\frac{d}{x^2}\right )^{5/2} (2 b c-a d)}{5 d^3}-\frac{c \left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{7/2}}{7 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]]

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]))))

Rubi steps

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

Mathematica [A]  time = 0.0221702, size = 69, normalized size = 0.93 \[ \frac{\sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right ) \left (7 a d x^2 \left (2 c x^2-3 d\right )+b \left (-8 c^2 x^4+12 c d x^2-15 d^2\right )\right )}{105 d^3 x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d/x^2]*(d + c*x^2)*(7*a*d*x^2*(-3*d + 2*c*x^2) + b*(-15*d^2 + 12*c*d*x^2 - 8*c^2*x^4)))/(105*d^3*x^6
)

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Maple [A]  time = 0.005, size = 70, normalized size = 1. \begin{align*}{\frac{ \left ( 14\,acd{x}^{4}-8\,b{c}^{2}{x}^{4}-21\,a{d}^{2}{x}^{2}+12\,bcd{x}^{2}-15\,b{d}^{2} \right ) \left ( c{x}^{2}+d \right ) }{105\,{d}^{3}{x}^{6}}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^2)*(c+d/x^2)^(1/2)/x^5,x)

[Out]

1/105*((c*x^2+d)/x^2)^(1/2)*(14*a*c*d*x^4-8*b*c^2*x^4-21*a*d^2*x^2+12*b*c*d*x^2-15*b*d^2)*(c*x^2+d)/d^3/x^6

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Maxima [A]  time = 0.944198, size = 113, normalized size = 1.53 \begin{align*} -\frac{1}{105} \, b{\left (\frac{15 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}}}{d^{3}} - \frac{42 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c}{d^{3}} + \frac{35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c^{2}}{d^{3}}\right )} - \frac{1}{15} \, a{\left (\frac{3 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}}}{d^{2}} - \frac{5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c}{d^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*(c+d/x^2)^(1/2)/x^5,x, algorithm="maxima")

[Out]

-1/105*b*(15*(c + d/x^2)^(7/2)/d^3 - 42*(c + d/x^2)^(5/2)*c/d^3 + 35*(c + d/x^2)^(3/2)*c^2/d^3) - 1/15*a*(3*(c
+ d/x^2)^(5/2)/d^2 - 5*(c + d/x^2)^(3/2)*c/d^2)

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Fricas [A]  time = 1.34685, size = 188, normalized size = 2.54 \begin{align*} -\frac{{\left (2 \,{\left (4 \, b c^{3} - 7 \, a c^{2} d\right )} x^{6} -{\left (4 \, b c^{2} d - 7 \, a c d^{2}\right )} x^{4} + 15 \, b d^{3} + 3 \,{\left (b c d^{2} + 7 \, a d^{3}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{105 \, d^{3} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*(c+d/x^2)^(1/2)/x^5,x, algorithm="fricas")

[Out]

-1/105*(2*(4*b*c^3 - 7*a*c^2*d)*x^6 - (4*b*c^2*d - 7*a*c*d^2)*x^4 + 15*b*d^3 + 3*(b*c*d^2 + 7*a*d^3)*x^2)*sqrt
((c*x^2 + d)/x^2)/(d^3*x^6)

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Sympy [A]  time = 3.83459, size = 78, normalized size = 1.05 \begin{align*} - \frac{a \left (- \frac{c \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5}\right )}{d^{2}} - \frac{b \left (\frac{c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} - \frac{2 c \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7}\right )}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)*(c+d/x**2)**(1/2)/x**5,x)

[Out]

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

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Giac [B]  time = 3.89088, size = 419, normalized size = 5.66 \begin{align*} \frac{4 \,{\left (105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} a c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) + 280 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} b c^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) - 175 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} a c^{\frac{5}{2}} d \mathrm{sgn}\left (x\right ) + 140 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} b c^{\frac{7}{2}} d \mathrm{sgn}\left (x\right ) + 70 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} a c^{\frac{5}{2}} d^{2} \mathrm{sgn}\left (x\right ) + 84 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{7}{2}} d^{2} \mathrm{sgn}\left (x\right ) - 42 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a c^{\frac{5}{2}} d^{3} \mathrm{sgn}\left (x\right ) - 28 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} b c^{\frac{7}{2}} d^{3} \mathrm{sgn}\left (x\right ) + 49 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a c^{\frac{5}{2}} d^{4} \mathrm{sgn}\left (x\right ) + 4 \, b c^{\frac{7}{2}} d^{4} \mathrm{sgn}\left (x\right ) - 7 \, a c^{\frac{5}{2}} d^{5} \mathrm{sgn}\left (x\right )\right )}}{105 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*(c+d/x^2)^(1/2)/x^5,x, algorithm="giac")

[Out]

4/105*(105*(sqrt(c)*x - sqrt(c*x^2 + d))^10*a*c^(5/2)*sgn(x) + 280*(sqrt(c)*x - sqrt(c*x^2 + d))^8*b*c^(7/2)*s
gn(x) - 175*(sqrt(c)*x - sqrt(c*x^2 + d))^8*a*c^(5/2)*d*sgn(x) + 140*(sqrt(c)*x - sqrt(c*x^2 + d))^6*b*c^(7/2)
*d*sgn(x) + 70*(sqrt(c)*x - sqrt(c*x^2 + d))^6*a*c^(5/2)*d^2*sgn(x) + 84*(sqrt(c)*x - sqrt(c*x^2 + d))^4*b*c^(
7/2)*d^2*sgn(x) - 42*(sqrt(c)*x - sqrt(c*x^2 + d))^4*a*c^(5/2)*d^3*sgn(x) - 28*(sqrt(c)*x - sqrt(c*x^2 + d))^2
*b*c^(7/2)*d^3*sgn(x) + 49*(sqrt(c)*x - sqrt(c*x^2 + d))^2*a*c^(5/2)*d^4*sgn(x) + 4*b*c^(7/2)*d^4*sgn(x) - 7*a
*c^(5/2)*d^5*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d)^7