3.951 \(\int \frac{(a+\frac{b}{x^2}) (c+\frac{d}{x^2})^{3/2}}{x^7} \, dx\)

Optimal. Leaf size=104 \[ \frac{c^2 \left (c+\frac{d}{x^2}\right )^{5/2} (b c-a d)}{5 d^4}+\frac{\left (c+\frac{d}{x^2}\right )^{9/2} (3 b c-a d)}{9 d^4}-\frac{c \left (c+\frac{d}{x^2}\right )^{7/2} (3 b c-2 a d)}{7 d^4}-\frac{b \left (c+\frac{d}{x^2}\right )^{11/2}}{11 d^4} \]

[Out]

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

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Rubi [A]  time = 0.0720556, 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} \[ \frac{c^2 \left (c+\frac{d}{x^2}\right )^{5/2} (b c-a d)}{5 d^4}+\frac{\left (c+\frac{d}{x^2}\right )^{9/2} (3 b c-a d)}{9 d^4}-\frac{c \left (c+\frac{d}{x^2}\right )^{7/2} (3 b c-2 a d)}{7 d^4}-\frac{b \left (c+\frac{d}{x^2}\right )^{11/2}}{11 d^4} \]

Antiderivative was successfully verified.

[In]

Int[((a + b/x^2)*(c + d/x^2)^(3/2))/x^7,x]

[Out]

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

Mathematica [A]  time = 0.0303498, size = 94, normalized size = 0.9 \[ \frac{\sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right )^2 \left (-11 a d x^2 \left (8 c^2 x^4-20 c d x^2+35 d^2\right )-3 b \left (40 c^2 d x^4-16 c^3 x^6-70 c d^2 x^2+105 d^3\right )\right )}{3465 d^4 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b/x^2)*(c + d/x^2)^(3/2))/x^7,x]

[Out]

(Sqrt[c + d/x^2]*(d + c*x^2)^2*(-11*a*d*x^2*(35*d^2 - 20*c*d*x^2 + 8*c^2*x^4) - 3*b*(105*d^3 - 70*c*d^2*x^2 +
40*c^2*d*x^4 - 16*c^3*x^6)))/(3465*d^4*x^10)

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Maple [A]  time = 0.008, size = 94, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 88\,a{c}^{2}d{x}^{6}-48\,b{c}^{3}{x}^{6}-220\,ac{d}^{2}{x}^{4}+120\,b{c}^{2}d{x}^{4}+385\,a{d}^{3}{x}^{2}-210\,bc{d}^{2}{x}^{2}+315\,b{d}^{3} \right ) \left ( c{x}^{2}+d \right ) }{3465\,{d}^{4}{x}^{8}} \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^7,x)

[Out]

-1/3465*((c*x^2+d)/x^2)^(3/2)*(88*a*c^2*d*x^6-48*b*c^3*x^6-220*a*c*d^2*x^4+120*b*c^2*d*x^4+385*a*d^3*x^2-210*b
*c*d^2*x^2+315*b*d^3)*(c*x^2+d)/d^4/x^8

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Maxima [A]  time = 0.93332, size = 159, normalized size = 1.53 \begin{align*} -\frac{1}{315} \,{\left (\frac{35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}}}{d^{3}} - \frac{90 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} c}{d^{3}} + \frac{63 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c^{2}}{d^{3}}\right )} a - \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 )} 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^7,x, algorithm="maxima")

[Out]

-1/315*(35*(c + d/x^2)^(9/2)/d^3 - 90*(c + d/x^2)^(7/2)*c/d^3 + 63*(c + d/x^2)^(5/2)*c^2/d^3)*a - 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)*b

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Fricas [A]  time = 1.73843, size = 305, normalized size = 2.93 \begin{align*} \frac{{\left (8 \,{\left (6 \, b c^{5} - 11 \, a c^{4} d\right )} x^{10} - 4 \,{\left (6 \, b c^{4} d - 11 \, a c^{3} d^{2}\right )} x^{8} + 3 \,{\left (6 \, b c^{3} d^{2} - 11 \, a c^{2} d^{3}\right )} x^{6} - 315 \, b d^{5} - 5 \,{\left (3 \, b c^{2} d^{3} + 110 \, a c d^{4}\right )} x^{4} - 35 \,{\left (12 \, b c d^{4} + 11 \, a d^{5}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3465 \, d^{4} x^{10}} \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^7,x, algorithm="fricas")

[Out]

1/3465*(8*(6*b*c^5 - 11*a*c^4*d)*x^10 - 4*(6*b*c^4*d - 11*a*c^3*d^2)*x^8 + 3*(6*b*c^3*d^2 - 11*a*c^2*d^3)*x^6
- 315*b*d^5 - 5*(3*b*c^2*d^3 + 110*a*c*d^4)*x^4 - 35*(12*b*c*d^4 + 11*a*d^5)*x^2)*sqrt((c*x^2 + d)/x^2)/(d^4*x
^10)

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Sympy [B]  time = 13.995, size = 262, normalized size = 2.52 \begin{align*} - \frac{a c \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{a \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^{3}} - \frac{b 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{b \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}} \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**7,x)

[Out]

-a*c*(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 - a*(-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**3 - b*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 - b*(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

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Giac [B]  time = 7.53456, size = 662, normalized size = 6.37 \begin{align*} \frac{16 \,{\left (2310 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{16} a c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 6930 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{14} b c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) - 1155 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{14} a c^{\frac{9}{2}} d \mathrm{sgn}\left (x\right ) + 12474 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{12} b c^{\frac{11}{2}} d \mathrm{sgn}\left (x\right ) + 231 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{12} a c^{\frac{9}{2}} d^{2} \mathrm{sgn}\left (x\right ) + 15246 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} b c^{\frac{11}{2}} d^{2} \mathrm{sgn}\left (x\right ) - 4851 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} a c^{\frac{9}{2}} d^{3} \mathrm{sgn}\left (x\right ) + 4950 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} b c^{\frac{11}{2}} d^{3} \mathrm{sgn}\left (x\right ) + 2475 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} a c^{\frac{9}{2}} d^{4} \mathrm{sgn}\left (x\right ) + 990 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} b c^{\frac{11}{2}} d^{4} \mathrm{sgn}\left (x\right ) + 495 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} a c^{\frac{9}{2}} d^{5} \mathrm{sgn}\left (x\right ) - 330 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{11}{2}} d^{5} \mathrm{sgn}\left (x\right ) + 605 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a c^{\frac{9}{2}} d^{6} \mathrm{sgn}\left (x\right ) + 66 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} b c^{\frac{11}{2}} d^{6} \mathrm{sgn}\left (x\right ) - 121 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a c^{\frac{9}{2}} d^{7} \mathrm{sgn}\left (x\right ) - 6 \, b c^{\frac{11}{2}} d^{7} \mathrm{sgn}\left (x\right ) + 11 \, a c^{\frac{9}{2}} d^{8} \mathrm{sgn}\left (x\right )\right )}}{3465 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{11}} \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^7,x, algorithm="giac")

[Out]

16/3465*(2310*(sqrt(c)*x - sqrt(c*x^2 + d))^16*a*c^(9/2)*sgn(x) + 6930*(sqrt(c)*x - sqrt(c*x^2 + d))^14*b*c^(1
1/2)*sgn(x) - 1155*(sqrt(c)*x - sqrt(c*x^2 + d))^14*a*c^(9/2)*d*sgn(x) + 12474*(sqrt(c)*x - sqrt(c*x^2 + d))^1
2*b*c^(11/2)*d*sgn(x) + 231*(sqrt(c)*x - sqrt(c*x^2 + d))^12*a*c^(9/2)*d^2*sgn(x) + 15246*(sqrt(c)*x - sqrt(c*
x^2 + d))^10*b*c^(11/2)*d^2*sgn(x) - 4851*(sqrt(c)*x - sqrt(c*x^2 + d))^10*a*c^(9/2)*d^3*sgn(x) + 4950*(sqrt(c
)*x - sqrt(c*x^2 + d))^8*b*c^(11/2)*d^3*sgn(x) + 2475*(sqrt(c)*x - sqrt(c*x^2 + d))^8*a*c^(9/2)*d^4*sgn(x) + 9
90*(sqrt(c)*x - sqrt(c*x^2 + d))^6*b*c^(11/2)*d^4*sgn(x) + 495*(sqrt(c)*x - sqrt(c*x^2 + d))^6*a*c^(9/2)*d^5*s
gn(x) - 330*(sqrt(c)*x - sqrt(c*x^2 + d))^4*b*c^(11/2)*d^5*sgn(x) + 605*(sqrt(c)*x - sqrt(c*x^2 + d))^4*a*c^(9
/2)*d^6*sgn(x) + 66*(sqrt(c)*x - sqrt(c*x^2 + d))^2*b*c^(11/2)*d^6*sgn(x) - 121*(sqrt(c)*x - sqrt(c*x^2 + d))^
2*a*c^(9/2)*d^7*sgn(x) - 6*b*c^(11/2)*d^7*sgn(x) + 11*a*c^(9/2)*d^8*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + d))^2 -
d)^11