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

Optimal. Leaf size=134 \[ \frac{c^2 \left (c+\frac{d}{x^2}\right )^{7/2} (4 b c-3 a d)}{7 d^5}-\frac{c^3 \left (c+\frac{d}{x^2}\right )^{5/2} (b c-a d)}{5 d^5}+\frac{\left (c+\frac{d}{x^2}\right )^{11/2} (4 b c-a d)}{11 d^5}-\frac{c \left (c+\frac{d}{x^2}\right )^{9/2} (2 b c-a d)}{3 d^5}-\frac{b \left (c+\frac{d}{x^2}\right )^{13/2}}{13 d^5} \]

[Out]

-(c^3*(b*c - a*d)*(c + d/x^2)^(5/2))/(5*d^5) + (c^2*(4*b*c - 3*a*d)*(c + d/x^2)^(7/2))/(7*d^5) - (c*(2*b*c - a
*d)*(c + d/x^2)^(9/2))/(3*d^5) + ((4*b*c - a*d)*(c + d/x^2)^(11/2))/(11*d^5) - (b*(c + d/x^2)^(13/2))/(13*d^5)

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Rubi [A]  time = 0.0960258, antiderivative size = 134, 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} \[ \frac{c^2 \left (c+\frac{d}{x^2}\right )^{7/2} (4 b c-3 a d)}{7 d^5}-\frac{c^3 \left (c+\frac{d}{x^2}\right )^{5/2} (b c-a d)}{5 d^5}+\frac{\left (c+\frac{d}{x^2}\right )^{11/2} (4 b c-a d)}{11 d^5}-\frac{c \left (c+\frac{d}{x^2}\right )^{9/2} (2 b c-a d)}{3 d^5}-\frac{b \left (c+\frac{d}{x^2}\right )^{13/2}}{13 d^5} \]

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)*(c + d/x^2)^(5/2))/(5*d^5) + (c^2*(4*b*c - 3*a*d)*(c + d/x^2)^(7/2))/(7*d^5) - (c*(2*b*c - a
*d)*(c + d/x^2)^(9/2))/(3*d^5) + ((4*b*c - a*d)*(c + d/x^2)^(11/2))/(11*d^5) - (b*(c + d/x^2)^(13/2))/(13*d^5)

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 ) \left (c+\frac{d}{x^2}\right )^{3/2}}{x^9} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int 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) (c+d x)^{3/2}}{d^4}-\frac{c^2 (4 b c-3 a d) (c+d x)^{5/2}}{d^4}+\frac{3 c (2 b c-a d) (c+d x)^{7/2}}{d^4}+\frac{(-4 b c+a d) (c+d x)^{9/2}}{d^4}+\frac{b (c+d x)^{11/2}}{d^4}\right ) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{c^3 (b c-a d) \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d^5}+\frac{c^2 (4 b c-3 a d) \left (c+\frac{d}{x^2}\right )^{7/2}}{7 d^5}-\frac{c (2 b c-a d) \left (c+\frac{d}{x^2}\right )^{9/2}}{3 d^5}+\frac{(4 b c-a d) \left (c+\frac{d}{x^2}\right )^{11/2}}{11 d^5}-\frac{b \left (c+\frac{d}{x^2}\right )^{13/2}}{13 d^5}\\ \end{align*}

Mathematica [A]  time = 0.0339789, size = 115, normalized size = 0.86 \[ \frac{\sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right )^2 \left (13 a d x^2 \left (-40 c^2 d x^4+16 c^3 x^6+70 c d^2 x^2-105 d^3\right )+b \left (-560 c^2 d^2 x^4+320 c^3 d x^6-128 c^4 x^8+840 c d^3 x^2-1155 d^4\right )\right )}{15015 d^5 x^{12}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d/x^2]*(d + c*x^2)^2*(13*a*d*x^2*(-105*d^3 + 70*c*d^2*x^2 - 40*c^2*d*x^4 + 16*c^3*x^6) + b*(-1155*d^
4 + 840*c*d^3*x^2 - 560*c^2*d^2*x^4 + 320*c^3*d*x^6 - 128*c^4*x^8)))/(15015*d^5*x^12)

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Maple [A]  time = 0.009, size = 118, normalized size = 0.9 \begin{align*}{\frac{ \left ( 208\,a{c}^{3}d{x}^{8}-128\,b{c}^{4}{x}^{8}-520\,a{c}^{2}{d}^{2}{x}^{6}+320\,b{c}^{3}d{x}^{6}+910\,ac{d}^{3}{x}^{4}-560\,b{c}^{2}{d}^{2}{x}^{4}-1365\,a{d}^{4}{x}^{2}+840\,bc{d}^{3}{x}^{2}-1155\,b{d}^{4} \right ) \left ( c{x}^{2}+d \right ) }{15015\,{d}^{5}{x}^{10}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}}} \end{align*}

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]

1/15015*((c*x^2+d)/x^2)^(3/2)*(208*a*c^3*d*x^8-128*b*c^4*x^8-520*a*c^2*d^2*x^6+320*b*c^3*d*x^6+910*a*c*d^3*x^4
-560*b*c^2*d^2*x^4-1365*a*d^4*x^2+840*b*c*d^3*x^2-1155*b*d^4)*(c*x^2+d)/d^5/x^10

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Maxima [A]  time = 0.944169, size = 205, normalized size = 1.53 \begin{align*} -\frac{1}{1155} \,{\left (\frac{105 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{11}{2}}}{d^{4}} - \frac{385 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}} c}{d^{4}} + \frac{495 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} c^{2}}{d^{4}} - \frac{231 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c^{3}}{d^{4}}\right )} a - \frac{1}{15015} \,{\left (\frac{1155 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{13}{2}}}{d^{5}} - \frac{5460 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{11}{2}} c}{d^{5}} + \frac{10010 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}} c^{2}}{d^{5}} - \frac{8580 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} c^{3}}{d^{5}} + \frac{3003 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c^{4}}{d^{5}}\right )} b \end{align*}

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]

-1/1155*(105*(c + d/x^2)^(11/2)/d^4 - 385*(c + d/x^2)^(9/2)*c/d^4 + 495*(c + d/x^2)^(7/2)*c^2/d^4 - 231*(c + d
/x^2)^(5/2)*c^3/d^4)*a - 1/15015*(1155*(c + d/x^2)^(13/2)/d^5 - 5460*(c + d/x^2)^(11/2)*c/d^5 + 10010*(c + d/x
^2)^(9/2)*c^2/d^5 - 8580*(c + d/x^2)^(7/2)*c^3/d^5 + 3003*(c + d/x^2)^(5/2)*c^4/d^5)*b

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Fricas [A]  time = 2.15573, size = 360, normalized size = 2.69 \begin{align*} -\frac{{\left (16 \,{\left (8 \, b c^{6} - 13 \, a c^{5} d\right )} x^{12} - 8 \,{\left (8 \, b c^{5} d - 13 \, a c^{4} d^{2}\right )} x^{10} + 6 \,{\left (8 \, b c^{4} d^{2} - 13 \, a c^{3} d^{3}\right )} x^{8} + 1155 \, b d^{6} - 5 \,{\left (8 \, b c^{3} d^{3} - 13 \, a c^{2} d^{4}\right )} x^{6} + 35 \,{\left (b c^{2} d^{4} + 52 \, a c d^{5}\right )} x^{4} + 105 \,{\left (14 \, b c d^{5} + 13 \, a d^{6}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{15015 \, d^{5} x^{12}} \end{align*}

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]

-1/15015*(16*(8*b*c^6 - 13*a*c^5*d)*x^12 - 8*(8*b*c^5*d - 13*a*c^4*d^2)*x^10 + 6*(8*b*c^4*d^2 - 13*a*c^3*d^3)*
x^8 + 1155*b*d^6 - 5*(8*b*c^3*d^3 - 13*a*c^2*d^4)*x^6 + 35*(b*c^2*d^4 + 52*a*c*d^5)*x^4 + 105*(14*b*c*d^5 + 13
*a*d^6)*x^2)*sqrt((c*x^2 + d)/x^2)/(d^5*x^12)

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Sympy [B]  time = 15.9664, size = 326, normalized size = 2.43 \begin{align*} - \frac{a c \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}} - \frac{a \left (\frac{c^{4} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} - \frac{4 c^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} + \frac{6 c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7} - \frac{4 c \left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{11}{2}}}{11}\right )}{d^{4}} - \frac{b c \left (\frac{c^{4} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} - \frac{4 c^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} + \frac{6 c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7} - \frac{4 c \left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{11}{2}}}{11}\right )}{d^{5}} - \frac{b \left (- \frac{c^{5} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} + c^{4} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}} - \frac{10 c^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7} + \frac{10 c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9} - \frac{5 c \left (c + \frac{d}{x^{2}}\right )^{\frac{11}{2}}}{11} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{13}{2}}}{13}\right )}{d^{5}} \end{align*}

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]

-a*c*(-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 - a*(c**4*(c + d/x**2)**(3/2)/3 - 4*c**3*(c + d/x**2)**(5/2)/5 + 6*c**2*(c + d/x**2)**(7/2)/7 - 4
*c*(c + d/x**2)**(9/2)/9 + (c + d/x**2)**(11/2)/11)/d**4 - b*c*(c**4*(c + d/x**2)**(3/2)/3 - 4*c**3*(c + d/x**
2)**(5/2)/5 + 6*c**2*(c + d/x**2)**(7/2)/7 - 4*c*(c + d/x**2)**(9/2)/9 + (c + d/x**2)**(11/2)/11)/d**5 - b*(-c
**5*(c + d/x**2)**(3/2)/3 + c**4*(c + d/x**2)**(5/2) - 10*c**3*(c + d/x**2)**(7/2)/7 + 10*c**2*(c + d/x**2)**(
9/2)/9 - 5*c*(c + d/x**2)**(11/2)/11 + (c + d/x**2)**(13/2)/13)/d**5

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Giac [B]  time = 12.433, size = 743, normalized size = 5.54 \begin{align*} \text{result too large to display} \end{align*}

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]

32/15015*(15015*(sqrt(c)*x - sqrt(c*x^2 + d))^18*a*c^(11/2)*sgn(x) + 48048*(sqrt(c)*x - sqrt(c*x^2 + d))^16*b*
c^(13/2)*sgn(x) - 3003*(sqrt(c)*x - sqrt(c*x^2 + d))^16*a*c^(11/2)*d*sgn(x) + 96096*(sqrt(c)*x - sqrt(c*x^2 +
d))^14*b*c^(13/2)*d*sgn(x) - 6006*(sqrt(c)*x - sqrt(c*x^2 + d))^14*a*c^(11/2)*d^2*sgn(x) + 109824*(sqrt(c)*x -
sqrt(c*x^2 + d))^12*b*c^(13/2)*d^2*sgn(x) - 28314*(sqrt(c)*x - sqrt(c*x^2 + d))^12*a*c^(11/2)*d^3*sgn(x) + 37
752*(sqrt(c)*x - sqrt(c*x^2 + d))^10*b*c^(13/2)*d^3*sgn(x) + 13728*(sqrt(c)*x - sqrt(c*x^2 + d))^10*a*c^(11/2)
*d^4*sgn(x) + 5720*(sqrt(c)*x - sqrt(c*x^2 + d))^8*b*c^(13/2)*d^4*sgn(x) + 5720*(sqrt(c)*x - sqrt(c*x^2 + d))^
8*a*c^(11/2)*d^5*sgn(x) - 2288*(sqrt(c)*x - sqrt(c*x^2 + d))^6*b*c^(13/2)*d^5*sgn(x) + 3718*(sqrt(c)*x - sqrt(
c*x^2 + d))^6*a*c^(11/2)*d^6*sgn(x) + 624*(sqrt(c)*x - sqrt(c*x^2 + d))^4*b*c^(13/2)*d^6*sgn(x) - 1014*(sqrt(c
)*x - sqrt(c*x^2 + d))^4*a*c^(11/2)*d^7*sgn(x) - 104*(sqrt(c)*x - sqrt(c*x^2 + d))^2*b*c^(13/2)*d^7*sgn(x) + 1
69*(sqrt(c)*x - sqrt(c*x^2 + d))^2*a*c^(11/2)*d^8*sgn(x) + 8*b*c^(13/2)*d^8*sgn(x) - 13*a*c^(11/2)*d^9*sgn(x))
/((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d)^13