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

Optimal. Leaf size=46 \[ \frac {\left (c+\frac {d}{x^2}\right )^{5/2} (b c-a d)}{5 d^2}-\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^2} \]

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Rubi [A]  time = 0.04, antiderivative size = 46, 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 = {444, 43} \begin {gather*} \frac {\left (c+\frac {d}{x^2}\right )^{5/2} (b c-a d)}{5 d^2}-\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 49, normalized size = 1.07 \begin {gather*} -\frac {\sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right )^2 \left (7 a d x^2-2 b c x^2+5 b d\right )}{35 d^2 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/35*(Sqrt[c + d/x^2]*(d + c*x^2)^2*(5*b*d - 2*b*c*x^2 + 7*a*d*x^2))/(d^2*x^6)

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IntegrateAlgebraic [A]  time = 0.06, size = 90, normalized size = 1.96 \begin {gather*} \frac {\sqrt {\frac {c x^2+d}{x^2}} \left (-7 a c^2 d x^6-14 a c d^2 x^4-7 a d^3 x^2+2 b c^3 x^6-b c^2 d x^4-8 b c d^2 x^2-5 b d^3\right )}{35 d^2 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(d + c*x^2)/x^2]*(-5*b*d^3 - 8*b*c*d^2*x^2 - 7*a*d^3*x^2 - b*c^2*d*x^4 - 14*a*c*d^2*x^4 + 2*b*c^3*x^6 -
7*a*c^2*d*x^6))/(35*d^2*x^6)

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fricas [B]  time = 0.42, size = 84, normalized size = 1.83 \begin {gather*} \frac {{\left ({\left (2 \, b c^{3} - 7 \, a c^{2} d\right )} x^{6} - {\left (b c^{2} d + 14 \, a c d^{2}\right )} x^{4} - 5 \, b d^{3} - {\left (8 \, b c d^{2} + 7 \, a d^{3}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{35 \, d^{2} x^{6}} \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^3,x, algorithm="fricas")

[Out]

1/35*((2*b*c^3 - 7*a*c^2*d)*x^6 - (b*c^2*d + 14*a*c*d^2)*x^4 - 5*b*d^3 - (8*b*c*d^2 + 7*a*d^3)*x^2)*sqrt((c*x^
2 + d)/x^2)/(d^2*x^6)

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giac [B]  time = 1.87, size = 370, normalized size = 8.04 \begin {gather*} \frac {2 \, {\left (35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} a c^{\frac {5}{2}} \mathrm {sgn}\relax (x) + 70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} b c^{\frac {7}{2}} \mathrm {sgn}\relax (x) - 70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} a c^{\frac {5}{2}} d \mathrm {sgn}\relax (x) + 70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} b c^{\frac {7}{2}} d \mathrm {sgn}\relax (x) + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} a c^{\frac {5}{2}} d^{2} \mathrm {sgn}\relax (x) + 140 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} b c^{\frac {7}{2}} d^{2} \mathrm {sgn}\relax (x) - 140 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} a c^{\frac {5}{2}} d^{3} \mathrm {sgn}\relax (x) + 28 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {7}{2}} d^{3} \mathrm {sgn}\relax (x) + 77 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {5}{2}} d^{4} \mathrm {sgn}\relax (x) + 14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {7}{2}} d^{4} \mathrm {sgn}\relax (x) - 14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {5}{2}} d^{5} \mathrm {sgn}\relax (x) - 2 \, b c^{\frac {7}{2}} d^{5} \mathrm {sgn}\relax (x) + 7 \, a c^{\frac {5}{2}} d^{6} \mathrm {sgn}\relax (x)\right )}}{35 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{7}} \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^3,x, algorithm="giac")

[Out]

2/35*(35*(sqrt(c)*x - sqrt(c*x^2 + d))^12*a*c^(5/2)*sgn(x) + 70*(sqrt(c)*x - sqrt(c*x^2 + d))^10*b*c^(7/2)*sgn
(x) - 70*(sqrt(c)*x - sqrt(c*x^2 + d))^10*a*c^(5/2)*d*sgn(x) + 70*(sqrt(c)*x - sqrt(c*x^2 + d))^8*b*c^(7/2)*d*
sgn(x) + 105*(sqrt(c)*x - sqrt(c*x^2 + d))^8*a*c^(5/2)*d^2*sgn(x) + 140*(sqrt(c)*x - sqrt(c*x^2 + d))^6*b*c^(7
/2)*d^2*sgn(x) - 140*(sqrt(c)*x - sqrt(c*x^2 + d))^6*a*c^(5/2)*d^3*sgn(x) + 28*(sqrt(c)*x - sqrt(c*x^2 + d))^4
*b*c^(7/2)*d^3*sgn(x) + 77*(sqrt(c)*x - sqrt(c*x^2 + d))^4*a*c^(5/2)*d^4*sgn(x) + 14*(sqrt(c)*x - sqrt(c*x^2 +
d))^2*b*c^(7/2)*d^4*sgn(x) - 14*(sqrt(c)*x - sqrt(c*x^2 + d))^2*a*c^(5/2)*d^5*sgn(x) - 2*b*c^(7/2)*d^5*sgn(x)
+ 7*a*c^(5/2)*d^6*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d)^7

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maple [A]  time = 0.05, size = 48, normalized size = 1.04 \begin {gather*} -\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (7 a d \,x^{2}-2 b c \,x^{2}+5 b d \right ) \left (c \,x^{2}+d \right )}{35 d^{2} x^{4}} \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^3,x)

[Out]

-1/35*((c*x^2+d)/x^2)^(3/2)*(7*a*d*x^2-2*b*c*x^2+5*b*d)*(c*x^2+d)/d^2/x^4

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maxima [A]  time = 0.59, size = 49, normalized size = 1.07 \begin {gather*} -\frac {a {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}}}{5 \, d} - \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 )} 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^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.33, size = 122, normalized size = 2.65 \begin {gather*} \frac {2\,b\,c^3\,\sqrt {c+\frac {d}{x^2}}}{35\,d^2}-\frac {a\,c^2\,\sqrt {c+\frac {d}{x^2}}}{5\,d}-\frac {2\,a\,c\,\sqrt {c+\frac {d}{x^2}}}{5\,x^2}-\frac {a\,d\,\sqrt {c+\frac {d}{x^2}}}{5\,x^4}-\frac {8\,b\,c\,\sqrt {c+\frac {d}{x^2}}}{35\,x^4}-\frac {b\,d\,\sqrt {c+\frac {d}{x^2}}}{7\,x^6}-\frac {b\,c^2\,\sqrt {c+\frac {d}{x^2}}}{35\,d\,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^3,x)

[Out]

(2*b*c^3*(c + d/x^2)^(1/2))/(35*d^2) - (a*c^2*(c + d/x^2)^(1/2))/(5*d) - (2*a*c*(c + d/x^2)^(1/2))/(5*x^2) - (
a*d*(c + d/x^2)^(1/2))/(5*x^4) - (8*b*c*(c + d/x^2)^(1/2))/(35*x^4) - (b*d*(c + d/x^2)^(1/2))/(7*x^6) - (b*c^2
*(c + d/x^2)^(1/2))/(35*d*x^2)

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sympy [A]  time = 14.16, size = 138, normalized size = 3.00 \begin {gather*} - \frac {a c \left (\begin {cases} \frac {\sqrt {c}}{x^{2}} & \text {for}\: d = 0 \\\frac {2 \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3 d} & \text {otherwise} \end {cases}\right )}{2} - \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} - \frac {b 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 {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^{2}} \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**3,x)

[Out]

-a*c*Piecewise((sqrt(c)/x**2, Eq(d, 0)), (2*(c + d/x**2)**(3/2)/(3*d), True))/2 - a*(-c*(c + d/x**2)**(3/2)/3
+ (c + d/x**2)**(5/2)/5)/d - b*c*(-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**2

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