3.5.53 \(\int \frac {1}{(a+\frac {c}{x^2}+\frac {b}{x})^{3/2}} \, dx\) [453]

Optimal. Leaf size=133 \[ \frac {\left (3 b^2-8 a c\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x}{a^2 \left (b^2-4 a c\right )}-\frac {2 \left (b^2-2 a c+\frac {b c}{x}\right ) x}{a \left (b^2-4 a c\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}-\frac {3 b \tanh ^{-1}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{2 a^{5/2}} \]

[Out]

-3/2*b*arctanh(1/2*(2*a+b/x)/a^(1/2)/(a+c/x^2+b/x)^(1/2))/a^(5/2)-2*(b^2-2*a*c+b*c/x)*x/a/(-4*a*c+b^2)/(a+c/x^
2+b/x)^(1/2)+(-8*a*c+3*b^2)*x*(a+c/x^2+b/x)^(1/2)/a^2/(-4*a*c+b^2)

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Rubi [A]
time = 0.07, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1356, 754, 820, 738, 212} \begin {gather*} -\frac {3 b \tanh ^{-1}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}\right )}{2 a^{5/2}}+\frac {x \left (3 b^2-8 a c\right ) \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}{a^2 \left (b^2-4 a c\right )}-\frac {2 x \left (-2 a c+b^2+\frac {b c}{x}\right )}{a \left (b^2-4 a c\right ) \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c/x^2 + b/x)^(-3/2),x]

[Out]

((3*b^2 - 8*a*c)*Sqrt[a + c/x^2 + b/x]*x)/(a^2*(b^2 - 4*a*c)) - (2*(b^2 - 2*a*c + (b*c)/x)*x)/(a*(b^2 - 4*a*c)
*Sqrt[a + c/x^2 + b/x]) - (3*b*ArcTanh[(2*a + b/x)/(2*Sqrt[a]*Sqrt[a + c/x^2 + b/x])])/(2*a^(5/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[S
implify[m + 2*p + 3], 0]

Rule 1356

Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n + c/x^(2*n))^p/x^2,
x], x, 1/x] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && ILtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2}} \, dx &=-\text {Subst}\left (\int \frac {1}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 \left (b^2-2 a c+\frac {b c}{x}\right ) x}{a \left (b^2-4 a c\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}+\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} \left (-3 b^2+8 a c\right )-b c x}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,\frac {1}{x}\right )}{a \left (b^2-4 a c\right )}\\ &=\frac {\left (3 b^2-8 a c\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x}{a^2 \left (b^2-4 a c\right )}-\frac {2 \left (b^2-2 a c+\frac {b c}{x}\right ) x}{a \left (b^2-4 a c\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\frac {1}{x}\right )}{2 a^2}\\ &=\frac {\left (3 b^2-8 a c\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x}{a^2 \left (b^2-4 a c\right )}-\frac {2 \left (b^2-2 a c+\frac {b c}{x}\right ) x}{a \left (b^2-4 a c\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+\frac {b}{x}}{\sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{a^2}\\ &=\frac {\left (3 b^2-8 a c\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x}{a^2 \left (b^2-4 a c\right )}-\frac {2 \left (b^2-2 a c+\frac {b c}{x}\right ) x}{a \left (b^2-4 a c\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}-\frac {3 b \tanh ^{-1}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{2 a^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.44, size = 140, normalized size = 1.05 \begin {gather*} -\frac {2 \sqrt {a} \left (-3 b^3 x+10 a b c x+4 a c \left (2 c+a x^2\right )-b^2 \left (3 c+a x^2\right )\right )-3 b \left (b^2-4 a c\right ) \sqrt {c+x (b+a x)} \log \left (a^2 \left (b+2 a x-2 \sqrt {a} \sqrt {c+x (b+a x)}\right )\right )}{2 a^{5/2} \left (b^2-4 a c\right ) x \sqrt {a+\frac {c+b x}{x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c/x^2 + b/x)^(-3/2),x]

[Out]

-1/2*(2*Sqrt[a]*(-3*b^3*x + 10*a*b*c*x + 4*a*c*(2*c + a*x^2) - b^2*(3*c + a*x^2)) - 3*b*(b^2 - 4*a*c)*Sqrt[c +
x*(b + a*x)]*Log[a^2*(b + 2*a*x - 2*Sqrt[a]*Sqrt[c + x*(b + a*x)])])/(a^(5/2)*(b^2 - 4*a*c)*x*Sqrt[a + (c + b
*x)/x^2])

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Maple [A]
time = 0.06, size = 197, normalized size = 1.48

method	result	size
default	\(\frac {\left (a \,x^{2}+b x +c \right ) \left (8 a^{\frac {7}{2}} c \,x^{2}-2 a^{\frac {5}{2}} b^{2} x^{2}+20 a^{\frac {5}{2}} b c x -6 a^{\frac {3}{2}} b^{3} x +16 a^{\frac {5}{2}} c^{2}-6 a^{\frac {3}{2}} b^{2} c -12 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {a \,x^{2}+b x +c}\, a^{2} b c +3 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {a \,x^{2}+b x +c}\, a \,b^{3}\right )}{2 a^{\frac {7}{2}} \left (\frac {a \,x^{2}+b x +c}{x^{2}}\right )^{\frac {3}{2}} x^{3} \left (4 a c -b^{2}\right )}\)	\(197\)
risch	\(\frac {a \,x^{2}+b x +c}{a^{2} \sqrt {\frac {a \,x^{2}+b x +c}{x^{2}}}\, x}+\frac {\left (\frac {3 b x}{2 a^{2} \sqrt {a \,x^{2}+b x +c}}-\frac {b^{2}}{4 a^{3} \sqrt {a \,x^{2}+b x +c}}-\frac {b^{3} x}{2 a^{2} \left (4 a c -b^{2}\right ) \sqrt {a \,x^{2}+b x +c}}-\frac {b^{4}}{4 a^{3} \left (4 a c -b^{2}\right ) \sqrt {a \,x^{2}+b x +c}}-\frac {3 b \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{2 a^{\frac {5}{2}}}+\frac {c}{a^{2} \sqrt {a \,x^{2}+b x +c}}\right ) \sqrt {a \,x^{2}+b x +c}}{\sqrt {\frac {a \,x^{2}+b x +c}{x^{2}}}\, x}\)	\(220\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+c/x^2+b/x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2*(a*x^2+b*x+c)/a^(7/2)*(8*a^(7/2)*c*x^2-2*a^(5/2)*b^2*x^2+20*a^(5/2)*b*c*x-6*a^(3/2)*b^3*x+16*a^(5/2)*c^2-6
*a^(3/2)*b^2*c-12*ln(1/2*(2*(a*x^2+b*x+c)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*(a*x^2+b*x+c)^(1/2)*a^2*b*c+3*ln(1/2
*(2*(a*x^2+b*x+c)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*(a*x^2+b*x+c)^(1/2)*a*b^3)/((a*x^2+b*x+c)/x^2)^(3/2)/x^3/(4*
a*c-b^2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+c/x^2+b/x)^(3/2),x, algorithm="maxima")

[Out]

integrate((a + b/x + c/x^2)^(-3/2), x)

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Fricas [A]
time = 0.40, size = 465, normalized size = 3.50 \begin {gather*} \left [\frac {3 \, {\left (b^{3} c - 4 \, a b c^{2} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2} + {\left (b^{4} - 4 \, a b^{2} c\right )} x\right )} \sqrt {a} \log \left (-8 \, a^{2} x^{2} - 8 \, a b x - b^{2} - 4 \, a c + 4 \, {\left (2 \, a x^{2} + b x\right )} \sqrt {a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right ) + 4 \, {\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{3} + {\left (3 \, a b^{3} - 10 \, a^{2} b c\right )} x^{2} + {\left (3 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{4 \, {\left (a^{3} b^{2} c - 4 \, a^{4} c^{2} + {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{2} + {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x\right )}}, \frac {3 \, {\left (b^{3} c - 4 \, a b c^{2} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2} + {\left (b^{4} - 4 \, a b^{2} c\right )} x\right )} \sqrt {-a} \arctan \left (\frac {{\left (2 \, a x^{2} + b x\right )} \sqrt {-a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a^{2} x^{2} + a b x + a c\right )}}\right ) + 2 \, {\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{3} + {\left (3 \, a b^{3} - 10 \, a^{2} b c\right )} x^{2} + {\left (3 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a^{3} b^{2} c - 4 \, a^{4} c^{2} + {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{2} + {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+c/x^2+b/x)^(3/2),x, algorithm="fricas")

[Out]

[1/4*(3*(b^3*c - 4*a*b*c^2 + (a*b^3 - 4*a^2*b*c)*x^2 + (b^4 - 4*a*b^2*c)*x)*sqrt(a)*log(-8*a^2*x^2 - 8*a*b*x -
b^2 - 4*a*c + 4*(2*a*x^2 + b*x)*sqrt(a)*sqrt((a*x^2 + b*x + c)/x^2)) + 4*((a^2*b^2 - 4*a^3*c)*x^3 + (3*a*b^3
- 10*a^2*b*c)*x^2 + (3*a*b^2*c - 8*a^2*c^2)*x)*sqrt((a*x^2 + b*x + c)/x^2))/(a^3*b^2*c - 4*a^4*c^2 + (a^4*b^2
- 4*a^5*c)*x^2 + (a^3*b^3 - 4*a^4*b*c)*x), 1/2*(3*(b^3*c - 4*a*b*c^2 + (a*b^3 - 4*a^2*b*c)*x^2 + (b^4 - 4*a*b^
2*c)*x)*sqrt(-a)*arctan(1/2*(2*a*x^2 + b*x)*sqrt(-a)*sqrt((a*x^2 + b*x + c)/x^2)/(a^2*x^2 + a*b*x + a*c)) + 2*
((a^2*b^2 - 4*a^3*c)*x^3 + (3*a*b^3 - 10*a^2*b*c)*x^2 + (3*a*b^2*c - 8*a^2*c^2)*x)*sqrt((a*x^2 + b*x + c)/x^2)
)/(a^3*b^2*c - 4*a^4*c^2 + (a^4*b^2 - 4*a^5*c)*x^2 + (a^3*b^3 - 4*a^4*b*c)*x)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + \frac {b}{x} + \frac {c}{x^{2}}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+c/x**2+b/x)**(3/2),x)

[Out]

Integral((a + b/x + c/x**2)**(-3/2), x)

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Giac [A]
time = 5.56, size = 237, normalized size = 1.78 \begin {gather*} -\frac {{\left (3 \, b^{3} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 12 \, a b c \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 6 \, \sqrt {a} b^{2} \sqrt {c} - 16 \, a^{\frac {3}{2}} c^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{2 \, {\left (a^{\frac {5}{2}} b^{2} - 4 \, a^{\frac {7}{2}} c\right )}} + \frac {{\left (\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} x}{a^{2} b^{2} \mathrm {sgn}\left (x\right ) - 4 \, a^{3} c \mathrm {sgn}\left (x\right )} + \frac {3 \, b^{3} - 10 \, a b c}{a^{2} b^{2} \mathrm {sgn}\left (x\right ) - 4 \, a^{3} c \mathrm {sgn}\left (x\right )}\right )} x + \frac {3 \, b^{2} c - 8 \, a c^{2}}{a^{2} b^{2} \mathrm {sgn}\left (x\right ) - 4 \, a^{3} c \mathrm {sgn}\left (x\right )}}{\sqrt {a x^{2} + b x + c}} + \frac {3 \, b \log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x + c}\right )} \sqrt {a} - b \right |}\right )}{2 \, a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+c/x^2+b/x)^(3/2),x, algorithm="giac")

[Out]

-1/2*(3*b^3*log(abs(-b + 2*sqrt(a)*sqrt(c))) - 12*a*b*c*log(abs(-b + 2*sqrt(a)*sqrt(c))) + 6*sqrt(a)*b^2*sqrt(
c) - 16*a^(3/2)*c^(3/2))*sgn(x)/(a^(5/2)*b^2 - 4*a^(7/2)*c) + (((a*b^2 - 4*a^2*c)*x/(a^2*b^2*sgn(x) - 4*a^3*c*
sgn(x)) + (3*b^3 - 10*a*b*c)/(a^2*b^2*sgn(x) - 4*a^3*c*sgn(x)))*x + (3*b^2*c - 8*a*c^2)/(a^2*b^2*sgn(x) - 4*a^
3*c*sgn(x)))/sqrt(a*x^2 + b*x + c) + 3/2*b*log(abs(-2*(sqrt(a)*x - sqrt(a*x^2 + b*x + c))*sqrt(a) - b))/(a^(5/
2)*sgn(x))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a+\frac {b}{x}+\frac {c}{x^2}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/x + c/x^2)^(3/2),x)

[Out]

int(1/(a + b/x + c/x^2)^(3/2), x)

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