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

Optimal. Leaf size=110 \[ -\frac {\left (c+\frac {d}{x^2}\right )^{3/2} (3 a d+2 b c)}{6 c}-\frac {1}{2} \sqrt {c+\frac {d}{x^2}} (3 a d+2 b c)+\frac {1}{2} \sqrt {c} (3 a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )+\frac {a x^2 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c} \]

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Rubi [A]  time = 0.07, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {446, 78, 50, 63, 208} \begin {gather*} -\frac {\left (c+\frac {d}{x^2}\right )^{3/2} (3 a d+2 b c)}{6 c}-\frac {1}{2} \sqrt {c+\frac {d}{x^2}} (3 a d+2 b c)+\frac {1}{2} \sqrt {c} (3 a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )+\frac {a x^2 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((2*b*c + 3*a*d)*Sqrt[c + d/x^2])/2 - ((2*b*c + 3*a*d)*(c + d/x^2)^(3/2))/(6*c) + (a*(c + d/x^2)^(5/2)*x^2)/(
2*c) + (Sqrt[c]*(2*b*c + 3*a*d)*ArcTanh[Sqrt[c + d/x^2]/Sqrt[c]])/2

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

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 \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x) (c+d x)^{3/2}}{x^2} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^2}{2 c}-\frac {\left (b c+\frac {3 a d}{2}\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{x} \, dx,x,\frac {1}{x^2}\right )}{2 c}\\ &=-\frac {(2 b c+3 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{6 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^2}{2 c}-\frac {1}{4} (2 b c+3 a d) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {1}{2} (2 b c+3 a d) \sqrt {c+\frac {d}{x^2}}-\frac {(2 b c+3 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{6 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^2}{2 c}-\frac {1}{4} (c (2 b c+3 a d)) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {1}{2} (2 b c+3 a d) \sqrt {c+\frac {d}{x^2}}-\frac {(2 b c+3 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{6 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^2}{2 c}-\frac {(c (2 b c+3 a d)) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+\frac {d}{x^2}}\right )}{2 d}\\ &=-\frac {1}{2} (2 b c+3 a d) \sqrt {c+\frac {d}{x^2}}-\frac {(2 b c+3 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{6 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^2}{2 c}+\frac {1}{2} \sqrt {c} (2 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.10, size = 78, normalized size = 0.71 \begin {gather*} \frac {1}{3} \sqrt {c+\frac {d}{x^2}} \left (-\frac {(3 a d+2 b c) \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {c x^2}{d}\right )}{\sqrt {\frac {c x^2}{d}+1}}-\frac {b \left (c x^2+d\right )^2}{d x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d/x^2]*(-((b*(d + c*x^2)^2)/(d*x^2)) - ((2*b*c + 3*a*d)*Hypergeometric2F1[-3/2, -1/2, 1/2, -((c*x^2)
/d)])/Sqrt[1 + (c*x^2)/d]))/3

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IntegrateAlgebraic [A]  time = 0.17, size = 93, normalized size = 0.85 \begin {gather*} \frac {1}{2} \left (3 a \sqrt {c} d+2 b c^{3/2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {c x^2+d}{x^2}}}{\sqrt {c}}\right )+\frac {\sqrt {\frac {c x^2+d}{x^2}} \left (3 a c x^4-6 a d x^2-8 b c x^2-2 b d\right )}{6 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 195, normalized size = 1.77 \begin {gather*} \left [\frac {3 \, {\left (2 \, b c + 3 \, a d\right )} \sqrt {c} x^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) + 2 \, {\left (3 \, a c x^{4} - 2 \, {\left (4 \, b c + 3 \, a d\right )} x^{2} - 2 \, b d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{12 \, x^{2}}, -\frac {3 \, {\left (2 \, b c + 3 \, a d\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - {\left (3 \, a c x^{4} - 2 \, {\left (4 \, b c + 3 \, a d\right )} x^{2} - 2 \, b d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{6 \, x^{2}}\right ] \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,x, algorithm="fricas")

[Out]

[1/12*(3*(2*b*c + 3*a*d)*sqrt(c)*x^2*log(-2*c*x^2 - 2*sqrt(c)*x^2*sqrt((c*x^2 + d)/x^2) - d) + 2*(3*a*c*x^4 -
2*(4*b*c + 3*a*d)*x^2 - 2*b*d)*sqrt((c*x^2 + d)/x^2))/x^2, -1/6*(3*(2*b*c + 3*a*d)*sqrt(-c)*x^2*arctan(sqrt(-c
)*x^2*sqrt((c*x^2 + d)/x^2)/(c*x^2 + d)) - (3*a*c*x^4 - 2*(4*b*c + 3*a*d)*x^2 - 2*b*d)*sqrt((c*x^2 + d)/x^2))/
x^2]

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giac [B]  time = 0.54, size = 225, normalized size = 2.05 \begin {gather*} \frac {1}{2} \, \sqrt {c x^{2} + d} a c x \mathrm {sgn}\relax (x) - \frac {1}{4} \, {\left (2 \, b c^{\frac {3}{2}} \mathrm {sgn}\relax (x) + 3 \, a \sqrt {c} d \mathrm {sgn}\relax (x)\right )} \log \left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2}\right ) + \frac {2 \, {\left (6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {3}{2}} d \mathrm {sgn}\relax (x) + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a \sqrt {c} d^{2} \mathrm {sgn}\relax (x) - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {3}{2}} d^{2} \mathrm {sgn}\relax (x) - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a \sqrt {c} d^{3} \mathrm {sgn}\relax (x) + 4 \, b c^{\frac {3}{2}} d^{3} \mathrm {sgn}\relax (x) + 3 \, a \sqrt {c} d^{4} \mathrm {sgn}\relax (x)\right )}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{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,x, algorithm="giac")

[Out]

1/2*sqrt(c*x^2 + d)*a*c*x*sgn(x) - 1/4*(2*b*c^(3/2)*sgn(x) + 3*a*sqrt(c)*d*sgn(x))*log((sqrt(c)*x - sqrt(c*x^2
+ d))^2) + 2/3*(6*(sqrt(c)*x - sqrt(c*x^2 + d))^4*b*c^(3/2)*d*sgn(x) + 3*(sqrt(c)*x - sqrt(c*x^2 + d))^4*a*sq
rt(c)*d^2*sgn(x) - 6*(sqrt(c)*x - sqrt(c*x^2 + d))^2*b*c^(3/2)*d^2*sgn(x) - 6*(sqrt(c)*x - sqrt(c*x^2 + d))^2*
a*sqrt(c)*d^3*sgn(x) + 4*b*c^(3/2)*d^3*sgn(x) + 3*a*sqrt(c)*d^4*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d)^
3

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maple [B]  time = 0.06, size = 216, normalized size = 1.96 \begin {gather*} \frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (9 a c \,d^{3} x^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )+6 b \,c^{2} d^{2} x^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )+9 \sqrt {c \,x^{2}+d}\, a \,c^{\frac {3}{2}} d^{2} x^{4}+6 \sqrt {c \,x^{2}+d}\, b \,c^{\frac {5}{2}} d \,x^{4}+6 \left (c \,x^{2}+d \right )^{\frac {3}{2}} a \,c^{\frac {3}{2}} d \,x^{4}+4 \left (c \,x^{2}+d \right )^{\frac {3}{2}} b \,c^{\frac {5}{2}} x^{4}-6 \left (c \,x^{2}+d \right )^{\frac {5}{2}} a \sqrt {c}\, d \,x^{2}-4 \left (c \,x^{2}+d \right )^{\frac {5}{2}} b \,c^{\frac {3}{2}} x^{2}-2 \left (c \,x^{2}+d \right )^{\frac {5}{2}} b \sqrt {c}\, d \right )}{6 \left (c \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {c}\, d^{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,x)

[Out]

1/6*((c*x^2+d)/x^2)^(3/2)*(4*c^(5/2)*(c*x^2+d)^(3/2)*x^4*b+6*c^(5/2)*(c*x^2+d)^(1/2)*x^4*b*d+6*c^(3/2)*(c*x^2+
d)^(3/2)*x^4*a*d-4*c^(3/2)*(c*x^2+d)^(5/2)*x^2*b+9*c^(3/2)*(c*x^2+d)^(1/2)*x^4*a*d^2-6*c^(1/2)*(c*x^2+d)^(5/2)
*x^2*a*d+9*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*x^3*a*c*d^3+6*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*x^3*b*c^2*d^2-2*c^(1/2)*(
c*x^2+d)^(5/2)*b*d)/(c*x^2+d)^(3/2)/d^2/c^(1/2)

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maxima [A]  time = 1.23, size = 134, normalized size = 1.22 \begin {gather*} \frac {1}{4} \, {\left (2 \, \sqrt {c + \frac {d}{x^{2}}} c x^{2} - 3 \, \sqrt {c} d \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right ) - 4 \, \sqrt {c + \frac {d}{x^{2}}} d\right )} a - \frac {1}{6} \, {\left (3 \, c^{\frac {3}{2}} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right ) + 2 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} + 6 \, \sqrt {c + \frac {d}{x^{2}}} c\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,x, algorithm="maxima")

[Out]

1/4*(2*sqrt(c + d/x^2)*c*x^2 - 3*sqrt(c)*d*log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c))) - 4*sq
rt(c + d/x^2)*d)*a - 1/6*(3*c^(3/2)*log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c))) + 2*(c + d/x^
2)^(3/2) + 6*sqrt(c + d/x^2)*c)*b

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mupad [B]  time = 5.65, size = 95, normalized size = 0.86 \begin {gather*} b\,c^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )-\frac {b\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{3}-a\,d\,\sqrt {c+\frac {d}{x^2}}-b\,c\,\sqrt {c+\frac {d}{x^2}}+\frac {3\,a\,\sqrt {c}\,d\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2}+\frac {a\,c\,x^2\,\sqrt {c+\frac {d}{x^2}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b/x^2)*(c + d/x^2)^(3/2),x)

[Out]

b*c^(3/2)*atanh((c + d/x^2)^(1/2)/c^(1/2)) - (b*(c + d/x^2)^(3/2))/3 - a*d*(c + d/x^2)^(1/2) - b*c*(c + d/x^2)
^(1/2) + (3*a*c^(1/2)*d*atanh((c + d/x^2)^(1/2)/c^(1/2)))/2 + (a*c*x^2*(c + d/x^2)^(1/2))/2

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sympy [A]  time = 56.16, size = 187, normalized size = 1.70 \begin {gather*} \frac {3 a \sqrt {c} d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{2} + \frac {a c \sqrt {d} x \sqrt {\frac {c x^{2}}{d} + 1}}{2} - \frac {a c \sqrt {d} x}{\sqrt {\frac {c x^{2}}{d} + 1}} - \frac {a d^{\frac {3}{2}}}{x \sqrt {\frac {c x^{2}}{d} + 1}} + b c^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )} - \frac {b c^{2} x}{\sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {b c \sqrt {d}}{x \sqrt {\frac {c x^{2}}{d} + 1}} + b d \left (\begin {cases} - \frac {\sqrt {c}}{2 x^{2}} & \text {for}\: d = 0 \\- \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3 d} & \text {otherwise} \end {cases}\right ) \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,x)

[Out]

3*a*sqrt(c)*d*asinh(sqrt(c)*x/sqrt(d))/2 + a*c*sqrt(d)*x*sqrt(c*x**2/d + 1)/2 - a*c*sqrt(d)*x/sqrt(c*x**2/d +
1) - a*d**(3/2)/(x*sqrt(c*x**2/d + 1)) + b*c**(3/2)*asinh(sqrt(c)*x/sqrt(d)) - b*c**2*x/(sqrt(d)*sqrt(c*x**2/d
+ 1)) - b*c*sqrt(d)/(x*sqrt(c*x**2/d + 1)) + b*d*Piecewise((-sqrt(c)/(2*x**2), Eq(d, 0)), (-(c + d/x**2)**(3/
2)/(3*d), True))

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