3.7.4 \(\int \frac {(a+\frac {b}{x^2}) (c+\frac {d}{x^2})^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=74 \[ \frac {\left (c+\frac {d}{x^2}\right )^{7/2} (2 b c-a d)}{7 d^3}-\frac {c \left (c+\frac {d}{x^2}\right )^{5/2} (b c-a d)}{5 d^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{9/2}}{9 d^3} \]

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Rubi [A]  time = 0.06, antiderivative size = 74, 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 {\left (c+\frac {d}{x^2}\right )^{7/2} (2 b c-a d)}{7 d^3}-\frac {c \left (c+\frac {d}{x^2}\right )^{5/2} (b c-a d)}{5 d^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{9/2}}{9 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b/x^2)*(c + d/x^2)^(3/2))/x^5,x]

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 71, normalized size = 0.96 \begin {gather*} \frac {\sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right )^2 \left (9 a d x^2 \left (2 c x^2-5 d\right )+b \left (-8 c^2 x^4+20 c d x^2-35 d^2\right )\right )}{315 d^3 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b/x^2)*(c + d/x^2)^(3/2))/x^5,x]

[Out]

(Sqrt[c + d/x^2]*(d + c*x^2)^2*(9*a*d*x^2*(-5*d + 2*c*x^2) + b*(-35*d^2 + 20*c*d*x^2 - 8*c^2*x^4)))/(315*d^3*x
^8)

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IntegrateAlgebraic [A]  time = 0.07, size = 114, normalized size = 1.54 \begin {gather*} \frac {\sqrt {\frac {c x^2+d}{x^2}} \left (18 a c^3 d x^8-9 a c^2 d^2 x^6-72 a c d^3 x^4-45 a d^4 x^2-8 b c^4 x^8+4 b c^3 d x^6-3 b c^2 d^2 x^4-50 b c d^3 x^2-35 b d^4\right )}{315 d^3 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b/x^2)*(c + d/x^2)^(3/2))/x^5,x]

[Out]

(Sqrt[(d + c*x^2)/x^2]*(-35*b*d^4 - 50*b*c*d^3*x^2 - 45*a*d^4*x^2 - 3*b*c^2*d^2*x^4 - 72*a*c*d^3*x^4 + 4*b*c^3
*d*x^6 - 9*a*c^2*d^2*x^6 - 8*b*c^4*x^8 + 18*a*c^3*d*x^8))/(315*d^3*x^8)

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fricas [A]  time = 0.48, size = 109, normalized size = 1.47 \begin {gather*} -\frac {{\left (2 \, {\left (4 \, b c^{4} - 9 \, a c^{3} d\right )} x^{8} - {\left (4 \, b c^{3} d - 9 \, a c^{2} d^{2}\right )} x^{6} + 35 \, b d^{4} + 3 \, {\left (b c^{2} d^{2} + 24 \, a c d^{3}\right )} x^{4} + 5 \, {\left (10 \, b c d^{3} + 9 \, a d^{4}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{315 \, d^{3} x^{8}} \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^5,x, algorithm="fricas")

[Out]

-1/315*(2*(4*b*c^4 - 9*a*c^3*d)*x^8 - (4*b*c^3*d - 9*a*c^2*d^2)*x^6 + 35*b*d^4 + 3*(b*c^2*d^2 + 24*a*c*d^3)*x^
4 + 5*(10*b*c*d^3 + 9*a*d^4)*x^2)*sqrt((c*x^2 + d)/x^2)/(d^3*x^8)

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giac [B]  time = 2.69, size = 430, normalized size = 5.81 \begin {gather*} \frac {4 \, {\left (315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{14} a c^{\frac {7}{2}} \mathrm {sgn}\relax (x) + 840 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} b c^{\frac {9}{2}} \mathrm {sgn}\relax (x) - 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} a c^{\frac {7}{2}} d \mathrm {sgn}\relax (x) + 1260 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} b c^{\frac {9}{2}} d \mathrm {sgn}\relax (x) + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} a c^{\frac {7}{2}} d^{2} \mathrm {sgn}\relax (x) + 1764 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} b c^{\frac {9}{2}} d^{2} \mathrm {sgn}\relax (x) - 819 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} a c^{\frac {7}{2}} d^{3} \mathrm {sgn}\relax (x) + 504 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} b c^{\frac {9}{2}} d^{3} \mathrm {sgn}\relax (x) + 441 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} a c^{\frac {7}{2}} d^{4} \mathrm {sgn}\relax (x) + 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {9}{2}} d^{4} \mathrm {sgn}\relax (x) - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {7}{2}} d^{5} \mathrm {sgn}\relax (x) - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {9}{2}} d^{5} \mathrm {sgn}\relax (x) + 81 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {7}{2}} d^{6} \mathrm {sgn}\relax (x) + 4 \, b c^{\frac {9}{2}} d^{6} \mathrm {sgn}\relax (x) - 9 \, a c^{\frac {7}{2}} d^{7} \mathrm {sgn}\relax (x)\right )}}{315 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{9}} \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^5,x, algorithm="giac")

[Out]

4/315*(315*(sqrt(c)*x - sqrt(c*x^2 + d))^14*a*c^(7/2)*sgn(x) + 840*(sqrt(c)*x - sqrt(c*x^2 + d))^12*b*c^(9/2)*
sgn(x) - 315*(sqrt(c)*x - sqrt(c*x^2 + d))^12*a*c^(7/2)*d*sgn(x) + 1260*(sqrt(c)*x - sqrt(c*x^2 + d))^10*b*c^(
9/2)*d*sgn(x) + 315*(sqrt(c)*x - sqrt(c*x^2 + d))^10*a*c^(7/2)*d^2*sgn(x) + 1764*(sqrt(c)*x - sqrt(c*x^2 + d))
^8*b*c^(9/2)*d^2*sgn(x) - 819*(sqrt(c)*x - sqrt(c*x^2 + d))^8*a*c^(7/2)*d^3*sgn(x) + 504*(sqrt(c)*x - sqrt(c*x
^2 + d))^6*b*c^(9/2)*d^3*sgn(x) + 441*(sqrt(c)*x - sqrt(c*x^2 + d))^6*a*c^(7/2)*d^4*sgn(x) + 144*(sqrt(c)*x -
sqrt(c*x^2 + d))^4*b*c^(9/2)*d^4*sgn(x) - 9*(sqrt(c)*x - sqrt(c*x^2 + d))^4*a*c^(7/2)*d^5*sgn(x) - 36*(sqrt(c)
*x - sqrt(c*x^2 + d))^2*b*c^(9/2)*d^5*sgn(x) + 81*(sqrt(c)*x - sqrt(c*x^2 + d))^2*a*c^(7/2)*d^6*sgn(x) + 4*b*c
^(9/2)*d^6*sgn(x) - 9*a*c^(7/2)*d^7*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d)^9

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maple [A]  time = 0.05, size = 70, normalized size = 0.95 \begin {gather*} \frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (18 a c d \,x^{4}-8 b \,c^{2} x^{4}-45 a \,d^{2} x^{2}+20 b c d \,x^{2}-35 b \,d^{2}\right ) \left (c \,x^{2}+d \right )}{315 d^{3} x^{6}} \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^5,x)

[Out]

1/315*((c*x^2+d)/x^2)^(3/2)*(18*a*c*d*x^4-8*b*c^2*x^4-45*a*d^2*x^2+20*b*c*d*x^2-35*b*d^2)*(c*x^2+d)/d^3/x^6

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maxima [A]  time = 0.64, size = 84, normalized size = 1.14 \begin {gather*} -\frac {1}{35} \, {\left (\frac {5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}}}{d^{2}} - \frac {7 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c}{d^{2}}\right )} a - \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 )} b \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^5,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.76, size = 164, normalized size = 2.22 \begin {gather*} \frac {2\,a\,c^3\,\sqrt {c+\frac {d}{x^2}}}{35\,d^2}-\frac {8\,b\,c^4\,\sqrt {c+\frac {d}{x^2}}}{315\,d^3}-\frac {8\,a\,c\,\sqrt {c+\frac {d}{x^2}}}{35\,x^4}-\frac {a\,d\,\sqrt {c+\frac {d}{x^2}}}{7\,x^6}-\frac {10\,b\,c\,\sqrt {c+\frac {d}{x^2}}}{63\,x^6}-\frac {b\,d\,\sqrt {c+\frac {d}{x^2}}}{9\,x^8}-\frac {a\,c^2\,\sqrt {c+\frac {d}{x^2}}}{35\,d\,x^2}-\frac {b\,c^2\,\sqrt {c+\frac {d}{x^2}}}{105\,d\,x^4}+\frac {4\,b\,c^3\,\sqrt {c+\frac {d}{x^2}}}{315\,d^2\,x^2} \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^5,x)

[Out]

(2*a*c^3*(c + d/x^2)^(1/2))/(35*d^2) - (8*b*c^4*(c + d/x^2)^(1/2))/(315*d^3) - (8*a*c*(c + d/x^2)^(1/2))/(35*x
^4) - (a*d*(c + d/x^2)^(1/2))/(7*x^6) - (10*b*c*(c + d/x^2)^(1/2))/(63*x^6) - (b*d*(c + d/x^2)^(1/2))/(9*x^8)
- (a*c^2*(c + d/x^2)^(1/2))/(35*d*x^2) - (b*c^2*(c + d/x^2)^(1/2))/(105*d*x^4) + (4*b*c^3*(c + d/x^2)^(1/2))/(
315*d^2*x^2)

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sympy [B]  time = 15.61, size = 194, normalized size = 2.62 \begin {gather*} - \frac {a c \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 {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^{2}} - \frac {b 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 {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^{3}} \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**5,x)

[Out]

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

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