3.935 \(\int \frac{(a+\frac{b}{x^2}) \sqrt{c+\frac{d}{x^2}}}{x^7} \, dx\)

Optimal. Leaf size=104 \[ \frac{c^2 \left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^4}+\frac{\left (c+\frac{d}{x^2}\right )^{7/2} (3 b c-a d)}{7 d^4}-\frac{c \left (c+\frac{d}{x^2}\right )^{5/2} (3 b c-2 a d)}{5 d^4}-\frac{b \left (c+\frac{d}{x^2}\right )^{9/2}}{9 d^4} \]

[Out]

(c^2*(b*c - a*d)*(c + d/x^2)^(3/2))/(3*d^4) - (c*(3*b*c - 2*a*d)*(c + d/x^2)^(5/2))/(5*d^4) + ((3*b*c - a*d)*(
c + d/x^2)^(7/2))/(7*d^4) - (b*(c + d/x^2)^(9/2))/(9*d^4)

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Rubi [A]  time = 0.0790056, antiderivative size = 104, 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} \[ \frac{c^2 \left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^4}+\frac{\left (c+\frac{d}{x^2}\right )^{7/2} (3 b c-a d)}{7 d^4}-\frac{c \left (c+\frac{d}{x^2}\right )^{5/2} (3 b c-2 a d)}{5 d^4}-\frac{b \left (c+\frac{d}{x^2}\right )^{9/2}}{9 d^4} \]

Antiderivative was successfully verified.

[In]

Int[((a + b/x^2)*Sqrt[c + d/x^2])/x^7,x]

[Out]

(c^2*(b*c - a*d)*(c + d/x^2)^(3/2))/(3*d^4) - (c*(3*b*c - 2*a*d)*(c + d/x^2)^(5/2))/(5*d^4) + ((3*b*c - a*d)*(
c + d/x^2)^(7/2))/(7*d^4) - (b*(c + d/x^2)^(9/2))/(9*d^4)

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

Mathematica [A]  time = 0.0494806, size = 79, normalized size = 0.76 \[ \frac{\sqrt{c+\frac{d}{x^2}} \left (x^2 \left (\frac{c x^2}{d}+1\right ) \left (8 c^2 x^4-12 c d x^2+15 d^2\right ) (6 b c-9 a d)-105 b d^2 \left (c x^2+d\right )\right )}{945 d^3 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b/x^2)*Sqrt[c + d/x^2])/x^7,x]

[Out]

(Sqrt[c + d/x^2]*(-105*b*d^2*(d + c*x^2) + (6*b*c - 9*a*d)*x^2*(1 + (c*x^2)/d)*(15*d^2 - 12*c*d*x^2 + 8*c^2*x^
4)))/(945*d^3*x^8)

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Maple [A]  time = 0.007, size = 94, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 24\,a{c}^{2}d{x}^{6}-16\,b{c}^{3}{x}^{6}-36\,ac{d}^{2}{x}^{4}+24\,b{c}^{2}d{x}^{4}+45\,a{d}^{3}{x}^{2}-30\,bc{d}^{2}{x}^{2}+35\,b{d}^{3} \right ) \left ( c{x}^{2}+d \right ) }{315\,{d}^{4}{x}^{8}}\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^7,x)

[Out]

-1/315*((c*x^2+d)/x^2)^(1/2)*(24*a*c^2*d*x^6-16*b*c^3*x^6-36*a*c*d^2*x^4+24*b*c^2*d*x^4+45*a*d^3*x^2-30*b*c*d^
2*x^2+35*b*d^3)*(c*x^2+d)/d^4/x^8

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Maxima [A]  time = 0.931659, size = 159, normalized size = 1.53 \begin{align*} -\frac{1}{315} \, b{\left (\frac{35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}}}{d^{4}} - \frac{135 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} c}{d^{4}} + \frac{189 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c^{2}}{d^{4}} - \frac{105 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c^{3}}{d^{4}}\right )} - \frac{1}{105} \, a{\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 )} \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^7,x, algorithm="maxima")

[Out]

-1/315*b*(35*(c + d/x^2)^(9/2)/d^4 - 135*(c + d/x^2)^(7/2)*c/d^4 + 189*(c + d/x^2)^(5/2)*c^2/d^4 - 105*(c + d/
x^2)^(3/2)*c^3/d^4) - 1/105*a*(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)

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Fricas [A]  time = 1.55066, size = 238, normalized size = 2.29 \begin{align*} \frac{{\left (8 \,{\left (2 \, b c^{4} - 3 \, a c^{3} d\right )} x^{8} - 4 \,{\left (2 \, b c^{3} d - 3 \, a c^{2} d^{2}\right )} x^{6} - 35 \, b d^{4} + 3 \,{\left (2 \, b c^{2} d^{2} - 3 \, a c d^{3}\right )} x^{4} - 5 \,{\left (b c d^{3} + 9 \, a d^{4}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{315 \, d^{4} x^{8}} \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^7,x, algorithm="fricas")

[Out]

1/315*(8*(2*b*c^4 - 3*a*c^3*d)*x^8 - 4*(2*b*c^3*d - 3*a*c^2*d^2)*x^6 - 35*b*d^4 + 3*(2*b*c^2*d^2 - 3*a*c*d^3)*
x^4 - 5*(b*c*d^3 + 9*a*d^4)*x^2)*sqrt((c*x^2 + d)/x^2)/(d^4*x^8)

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Sympy [A]  time = 4.62581, size = 112, normalized size = 1.08 \begin{align*} - \frac{a \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}} - \frac{b \left (- \frac{c^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{3 c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} - \frac{3 c \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9}\right )}{d^{4}} \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**7,x)

[Out]

-a*(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 - b*(-c**3*(c + d/x**
2)**(3/2)/3 + 3*c**2*(c + d/x**2)**(5/2)/5 - 3*c*(c + d/x**2)**(7/2)/7 + (c + d/x**2)**(9/2)/9)/d**4

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Giac [B]  time = 3.46804, size = 500, normalized size = 4.81 \begin{align*} \frac{16 \,{\left (210 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{12} a c^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) + 630 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} b c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) - 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} a c^{\frac{7}{2}} d \mathrm{sgn}\left (x\right ) + 378 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} b c^{\frac{9}{2}} d \mathrm{sgn}\left (x\right ) + 63 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} a c^{\frac{7}{2}} d^{2} \mathrm{sgn}\left (x\right ) + 168 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} b c^{\frac{9}{2}} d^{2} \mathrm{sgn}\left (x\right ) - 42 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} a c^{\frac{7}{2}} d^{3} \mathrm{sgn}\left (x\right ) - 72 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{9}{2}} d^{3} \mathrm{sgn}\left (x\right ) + 108 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a c^{\frac{7}{2}} d^{4} \mathrm{sgn}\left (x\right ) + 18 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} b c^{\frac{9}{2}} d^{4} \mathrm{sgn}\left (x\right ) - 27 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a c^{\frac{7}{2}} d^{5} \mathrm{sgn}\left (x\right ) - 2 \, b c^{\frac{9}{2}} d^{5} \mathrm{sgn}\left (x\right ) + 3 \, a c^{\frac{7}{2}} d^{6} \mathrm{sgn}\left (x\right )\right )}}{315 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{9}} \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^7,x, algorithm="giac")

[Out]

16/315*(210*(sqrt(c)*x - sqrt(c*x^2 + d))^12*a*c^(7/2)*sgn(x) + 630*(sqrt(c)*x - sqrt(c*x^2 + d))^10*b*c^(9/2)
*sgn(x) - 315*(sqrt(c)*x - sqrt(c*x^2 + d))^10*a*c^(7/2)*d*sgn(x) + 378*(sqrt(c)*x - sqrt(c*x^2 + d))^8*b*c^(9
/2)*d*sgn(x) + 63*(sqrt(c)*x - sqrt(c*x^2 + d))^8*a*c^(7/2)*d^2*sgn(x) + 168*(sqrt(c)*x - sqrt(c*x^2 + d))^6*b
*c^(9/2)*d^2*sgn(x) - 42*(sqrt(c)*x - sqrt(c*x^2 + d))^6*a*c^(7/2)*d^3*sgn(x) - 72*(sqrt(c)*x - sqrt(c*x^2 + d
))^4*b*c^(9/2)*d^3*sgn(x) + 108*(sqrt(c)*x - sqrt(c*x^2 + d))^4*a*c^(7/2)*d^4*sgn(x) + 18*(sqrt(c)*x - sqrt(c*
x^2 + d))^2*b*c^(9/2)*d^4*sgn(x) - 27*(sqrt(c)*x - sqrt(c*x^2 + d))^2*a*c^(7/2)*d^5*sgn(x) - 2*b*c^(9/2)*d^5*s
gn(x) + 3*a*c^(7/2)*d^6*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d)^9