3.5.54 \(\int \frac {1}{(a+\frac {c}{x^2}+\frac {b}{x})^{5/2}} \, dx\) [454]

Optimal. Leaf size=220 \[ \frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x}{3 a^3 \left (b^2-4 a c\right )^2}-\frac {2 \left (b^2-2 a c+\frac {b c}{x}\right ) x}{3 a \left (b^2-4 a c\right ) \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2}}-\frac {2 \left (5 b^4-32 a b^2 c+32 a^2 c^2+\frac {b c \left (5 b^2-28 a c\right )}{x}\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}-\frac {5 b \tanh ^{-1}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{2 a^{7/2}} \]

[Out]

-2/3*(b^2-2*a*c+b*c/x)*x/a/(-4*a*c+b^2)/(a+c/x^2+b/x)^(3/2)-5/2*b*arctanh(1/2*(2*a+b/x)/a^(1/2)/(a+c/x^2+b/x)^
(1/2))/a^(7/2)-2/3*(5*b^4-32*a*b^2*c+32*a^2*c^2+b*c*(-28*a*c+5*b^2)/x)*x/a^2/(-4*a*c+b^2)^2/(a+c/x^2+b/x)^(1/2
)+1/3*(128*a^2*c^2-100*a*b^2*c+15*b^4)*x*(a+c/x^2+b/x)^(1/2)/a^3/(-4*a*c+b^2)^2

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Rubi [A]
time = 0.13, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1356, 754, 836, 820, 738, 212} \begin {gather*} -\frac {5 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^{7/2}}-\frac {2 x \left (32 a^2 c^2+\frac {b c \left (5 b^2-28 a c\right )}{x}-32 a b^2 c+5 b^4\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}+\frac {x \left (128 a^2 c^2-100 a b^2 c+15 b^4\right ) \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}{3 a^3 \left (b^2-4 a c\right )^2}-\frac {2 x \left (-2 a c+b^2+\frac {b c}{x}\right )}{3 a \left (b^2-4 a c\right ) \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c/x^2 + b/x)^(-5/2),x]

[Out]

((15*b^4 - 100*a*b^2*c + 128*a^2*c^2)*Sqrt[a + c/x^2 + b/x]*x)/(3*a^3*(b^2 - 4*a*c)^2) - (2*(b^2 - 2*a*c + (b*
c)/x)*x)/(3*a*(b^2 - 4*a*c)*(a + c/x^2 + b/x)^(3/2)) - (2*(5*b^4 - 32*a*b^2*c + 32*a^2*c^2 + (b*c*(5*b^2 - 28*
a*c))/x)*x)/(3*a^2*(b^2 - 4*a*c)^2*Sqrt[a + c/x^2 + b/x]) - (5*b*ArcTanh[(2*a + b/x)/(2*Sqrt[a]*Sqrt[a + c/x^2
+ b/x])])/(2*a^(7/2))

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 836

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

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 )^{5/2}} \, dx &=-\text {Subst}\left (\int \frac {1}{x^2 \left (a+b x+c x^2\right )^{5/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 \left (b^2-2 a c+\frac {b c}{x}\right ) x}{3 a \left (b^2-4 a c\right ) \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2}}+\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} \left (-5 b^2+16 a c\right )-3 b c x}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 a \left (b^2-4 a c\right )}\\ &=-\frac {2 \left (b^2-2 a c+\frac {b c}{x}\right ) x}{3 a \left (b^2-4 a c\right ) \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2}}-\frac {2 \left (5 b^4-32 a b^2 c+32 a^2 c^2+\frac {b c \left (5 b^2-28 a c\right )}{x}\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}-\frac {4 \text {Subst}\left (\int \frac {\frac {1}{4} \left (15 b^4-100 a b^2 c+128 a^2 c^2\right )+\frac {1}{2} b c \left (5 b^2-28 a c\right ) x}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,\frac {1}{x}\right )}{3 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x}{3 a^3 \left (b^2-4 a c\right )^2}-\frac {2 \left (b^2-2 a c+\frac {b c}{x}\right ) x}{3 a \left (b^2-4 a c\right ) \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2}}-\frac {2 \left (5 b^4-32 a b^2 c+32 a^2 c^2+\frac {b c \left (5 b^2-28 a c\right )}{x}\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}+\frac {(5 b) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\frac {1}{x}\right )}{2 a^3}\\ &=\frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x}{3 a^3 \left (b^2-4 a c\right )^2}-\frac {2 \left (b^2-2 a c+\frac {b c}{x}\right ) x}{3 a \left (b^2-4 a c\right ) \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2}}-\frac {2 \left (5 b^4-32 a b^2 c+32 a^2 c^2+\frac {b c \left (5 b^2-28 a c\right )}{x}\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}-\frac {(5 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^3}\\ &=\frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x}{3 a^3 \left (b^2-4 a c\right )^2}-\frac {2 \left (b^2-2 a c+\frac {b c}{x}\right ) x}{3 a \left (b^2-4 a c\right ) \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2}}-\frac {2 \left (5 b^4-32 a b^2 c+32 a^2 c^2+\frac {b c \left (5 b^2-28 a c\right )}{x}\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}-\frac {5 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^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 1.10, size = 246, normalized size = 1.12 \begin {gather*} \frac {\frac {2 \sqrt {a} (c+x (b+a x)) \left (15 b^6 x^2+8 a^2 b c^2 x \left (39 c+32 a x^2\right )-2 a b^3 c x \left (105 c+74 a x^2\right )+10 b^5 \left (3 c x+2 a x^3\right )+3 b^4 \left (5 c^2-30 a c x^2+a^2 x^4\right )+16 a^2 c^2 \left (8 c^2+12 a c x^2+3 a^2 x^4\right )-4 a b^2 c \left (25 c^2-12 a c x^2+6 a^2 x^4\right )\right )}{\left (b^2-4 a c\right )^2}+15 b (c+x (b+a x))^{5/2} \log \left (a^3 \left (b+2 a x-2 \sqrt {a} \sqrt {c+x (b+a x)}\right )\right )}{6 a^{7/2} x^5 \left (a+\frac {c+b x}{x^2}\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c/x^2 + b/x)^(-5/2),x]

[Out]

((2*Sqrt[a]*(c + x*(b + a*x))*(15*b^6*x^2 + 8*a^2*b*c^2*x*(39*c + 32*a*x^2) - 2*a*b^3*c*x*(105*c + 74*a*x^2) +
10*b^5*(3*c*x + 2*a*x^3) + 3*b^4*(5*c^2 - 30*a*c*x^2 + a^2*x^4) + 16*a^2*c^2*(8*c^2 + 12*a*c*x^2 + 3*a^2*x^4)
- 4*a*b^2*c*(25*c^2 - 12*a*c*x^2 + 6*a^2*x^4)))/(b^2 - 4*a*c)^2 + 15*b*(c + x*(b + a*x))^(5/2)*Log[a^3*(b + 2
*a*x - 2*Sqrt[a]*Sqrt[c + x*(b + a*x)])])/(6*a^(7/2)*x^5*(a + (c + b*x)/x^2)^(5/2))

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Maple [A]
time = 0.10, size = 376, normalized size = 1.71

method	result	size
default	\(\frac {\left (a \,x^{2}+b x +c \right ) \left (96 a^{\frac {13}{2}} c^{2} x^{4}-48 a^{\frac {11}{2}} b^{2} c \,x^{4}+512 a^{\frac {11}{2}} b \,c^{2} x^{3}+6 a^{\frac {9}{2}} b^{4} x^{4}+384 a^{\frac {11}{2}} c^{3} x^{2}-296 a^{\frac {9}{2}} b^{3} c \,x^{3}+96 a^{\frac {9}{2}} b^{2} c^{2} x^{2}+40 a^{\frac {7}{2}} b^{5} x^{3}+624 a^{\frac {9}{2}} b \,c^{3} x -180 a^{\frac {7}{2}} b^{4} c \,x^{2}+256 a^{\frac {9}{2}} c^{4}-420 a^{\frac {7}{2}} b^{3} c^{2} x +30 a^{\frac {5}{2}} b^{6} x^{2}-200 a^{\frac {7}{2}} b^{2} c^{3}+60 a^{\frac {5}{2}} b^{5} c x +30 a^{\frac {5}{2}} b^{4} c^{2}-240 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} a^{4} b \,c^{2}+120 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} a^{3} b^{3} c -15 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} a^{2} b^{5}\right )}{6 a^{\frac {11}{2}} \left (\frac {a \,x^{2}+b x +c}{x^{2}}\right )^{\frac {5}{2}} x^{5} \left (4 a c -b^{2}\right )^{2}}\)	\(376\)
risch	\(\text {Expression too large to display}\)	\(3225\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+c/x^2+b/x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/6*(a*x^2+b*x+c)*(96*a^(13/2)*c^2*x^4-48*a^(11/2)*b^2*c*x^4+512*a^(11/2)*b*c^2*x^3+6*a^(9/2)*b^4*x^4+384*a^(1
1/2)*c^3*x^2-296*a^(9/2)*b^3*c*x^3+96*a^(9/2)*b^2*c^2*x^2+40*a^(7/2)*b^5*x^3+624*a^(9/2)*b*c^3*x-180*a^(7/2)*b
^4*c*x^2+256*a^(9/2)*c^4-420*a^(7/2)*b^3*c^2*x+30*a^(5/2)*b^6*x^2-200*a^(7/2)*b^2*c^3+60*a^(5/2)*b^5*c*x+30*a^
(5/2)*b^4*c^2-240*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)^(3/2)*a^4*b*c^2+120*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)^(3/2)*a^3*b^3*c-15*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)^(3/2)*a^2*b^5)/a^(11/2)/((a*x^2+b*x+c)/x^2)^(5/2)/x^5/(4*a*c-b^
2)^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)^(5/2),x, algorithm="maxima")

[Out]

integrate((a + b/x + c/x^2)^(-5/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (198) = 396\).
time = 0.54, size = 1081, normalized size = 4.91 \begin {gather*} \left [\frac {15 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4} + {\left (a^{2} b^{5} - 8 \, a^{3} b^{3} c + 16 \, a^{4} b c^{2}\right )} x^{4} + 2 \, {\left (a b^{6} - 8 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2}\right )} x^{3} + {\left (b^{7} - 6 \, a b^{5} c + 32 \, a^{3} b c^{3}\right )} x^{2} + 2 \, {\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\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 (3 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2}\right )} x^{5} + 4 \, {\left (5 \, a^{2} b^{5} - 37 \, a^{3} b^{3} c + 64 \, a^{4} b c^{2}\right )} x^{4} + 3 \, {\left (5 \, a b^{6} - 30 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2} + 64 \, a^{4} c^{3}\right )} x^{3} + 6 \, {\left (5 \, a b^{5} c - 35 \, a^{2} b^{3} c^{2} + 52 \, a^{3} b c^{3}\right )} x^{2} + {\left (15 \, a b^{4} c^{2} - 100 \, a^{2} b^{2} c^{3} + 128 \, a^{3} c^{4}\right )} x\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{12 \, {\left (a^{4} b^{4} c^{2} - 8 \, a^{5} b^{2} c^{3} + 16 \, a^{6} c^{4} + {\left (a^{6} b^{4} - 8 \, a^{7} b^{2} c + 16 \, a^{8} c^{2}\right )} x^{4} + 2 \, {\left (a^{5} b^{5} - 8 \, a^{6} b^{3} c + 16 \, a^{7} b c^{2}\right )} x^{3} + {\left (a^{4} b^{6} - 6 \, a^{5} b^{4} c + 32 \, a^{7} c^{3}\right )} x^{2} + 2 \, {\left (a^{4} b^{5} c - 8 \, a^{5} b^{3} c^{2} + 16 \, a^{6} b c^{3}\right )} x\right )}}, \frac {15 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4} + {\left (a^{2} b^{5} - 8 \, a^{3} b^{3} c + 16 \, a^{4} b c^{2}\right )} x^{4} + 2 \, {\left (a b^{6} - 8 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2}\right )} x^{3} + {\left (b^{7} - 6 \, a b^{5} c + 32 \, a^{3} b c^{3}\right )} x^{2} + 2 \, {\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\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 (3 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2}\right )} x^{5} + 4 \, {\left (5 \, a^{2} b^{5} - 37 \, a^{3} b^{3} c + 64 \, a^{4} b c^{2}\right )} x^{4} + 3 \, {\left (5 \, a b^{6} - 30 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2} + 64 \, a^{4} c^{3}\right )} x^{3} + 6 \, {\left (5 \, a b^{5} c - 35 \, a^{2} b^{3} c^{2} + 52 \, a^{3} b c^{3}\right )} x^{2} + {\left (15 \, a b^{4} c^{2} - 100 \, a^{2} b^{2} c^{3} + 128 \, a^{3} c^{4}\right )} x\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{6 \, {\left (a^{4} b^{4} c^{2} - 8 \, a^{5} b^{2} c^{3} + 16 \, a^{6} c^{4} + {\left (a^{6} b^{4} - 8 \, a^{7} b^{2} c + 16 \, a^{8} c^{2}\right )} x^{4} + 2 \, {\left (a^{5} b^{5} - 8 \, a^{6} b^{3} c + 16 \, a^{7} b c^{2}\right )} x^{3} + {\left (a^{4} b^{6} - 6 \, a^{5} b^{4} c + 32 \, a^{7} c^{3}\right )} x^{2} + 2 \, {\left (a^{4} b^{5} c - 8 \, a^{5} b^{3} c^{2} + 16 \, a^{6} b c^{3}\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)^(5/2),x, algorithm="fricas")

[Out]

[1/12*(15*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^4 + 2*(a*b^6 - 8*a^
2*b^4*c + 16*a^3*b^2*c^2)*x^3 + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*x^2 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3
)*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*(3*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*x^5 + 4*(5*a^2*b^5 - 37*a^3*b^3*c + 64*a^4*b*c^2)*x^4 + 3*(5*a*b^6
- 30*a^2*b^4*c + 16*a^3*b^2*c^2 + 64*a^4*c^3)*x^3 + 6*(5*a*b^5*c - 35*a^2*b^3*c^2 + 52*a^3*b*c^3)*x^2 + (15*a*
b^4*c^2 - 100*a^2*b^2*c^3 + 128*a^3*c^4)*x)*sqrt((a*x^2 + b*x + c)/x^2))/(a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6
*c^4 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*x^4 + 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*x^3 + (a^4*b^6 - 6*
a^5*b^4*c + 32*a^7*c^3)*x^2 + 2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*x), 1/6*(15*(b^5*c^2 - 8*a*b^3*c^3
+ 16*a^2*b*c^4 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^4 + 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*x^3 + (
b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*x^2 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*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*(3*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c
^2)*x^5 + 4*(5*a^2*b^5 - 37*a^3*b^3*c + 64*a^4*b*c^2)*x^4 + 3*(5*a*b^6 - 30*a^2*b^4*c + 16*a^3*b^2*c^2 + 64*a^
4*c^3)*x^3 + 6*(5*a*b^5*c - 35*a^2*b^3*c^2 + 52*a^3*b*c^3)*x^2 + (15*a*b^4*c^2 - 100*a^2*b^2*c^3 + 128*a^3*c^4
)*x)*sqrt((a*x^2 + b*x + c)/x^2))/(a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*
c^2)*x^4 + 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*x^3 + (a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*x^2 + 2*(a^4*b^
5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*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 {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+c/x**2+b/x)**(5/2),x)

[Out]

Integral((a + b/x + c/x**2)**(-5/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 507 vs. \(2 (198) = 396\).
time = 5.42, size = 507, normalized size = 2.30 \begin {gather*} -\frac {{\left (15 \, b^{5} \sqrt {c} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 120 \, a b^{3} c^{\frac {3}{2}} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 240 \, a^{2} b c^{\frac {5}{2}} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 30 \, \sqrt {a} b^{4} c - 200 \, a^{\frac {3}{2}} b^{2} c^{2} + 256 \, a^{\frac {5}{2}} c^{3}\right )} \mathrm {sgn}\left (x\right )}{6 \, {\left (a^{\frac {7}{2}} b^{4} \sqrt {c} - 8 \, a^{\frac {9}{2}} b^{2} c^{\frac {3}{2}} + 16 \, a^{\frac {11}{2}} c^{\frac {5}{2}}\right )}} + \frac {{\left ({\left ({\left (\frac {3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} x}{a^{3} b^{4} \mathrm {sgn}\left (x\right ) - 8 \, a^{4} b^{2} c \mathrm {sgn}\left (x\right ) + 16 \, a^{5} c^{2} \mathrm {sgn}\left (x\right )} + \frac {4 \, {\left (5 \, a b^{5} - 37 \, a^{2} b^{3} c + 64 \, a^{3} b c^{2}\right )}}{a^{3} b^{4} \mathrm {sgn}\left (x\right ) - 8 \, a^{4} b^{2} c \mathrm {sgn}\left (x\right ) + 16 \, a^{5} c^{2} \mathrm {sgn}\left (x\right )}\right )} x + \frac {3 \, {\left (5 \, b^{6} - 30 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3}\right )}}{a^{3} b^{4} \mathrm {sgn}\left (x\right ) - 8 \, a^{4} b^{2} c \mathrm {sgn}\left (x\right ) + 16 \, a^{5} c^{2} \mathrm {sgn}\left (x\right )}\right )} x + \frac {6 \, {\left (5 \, b^{5} c - 35 \, a b^{3} c^{2} + 52 \, a^{2} b c^{3}\right )}}{a^{3} b^{4} \mathrm {sgn}\left (x\right ) - 8 \, a^{4} b^{2} c \mathrm {sgn}\left (x\right ) + 16 \, a^{5} c^{2} \mathrm {sgn}\left (x\right )}\right )} x + \frac {15 \, b^{4} c^{2} - 100 \, a b^{2} c^{3} + 128 \, a^{2} c^{4}}{a^{3} b^{4} \mathrm {sgn}\left (x\right ) - 8 \, a^{4} b^{2} c \mathrm {sgn}\left (x\right ) + 16 \, a^{5} c^{2} \mathrm {sgn}\left (x\right )}}{3 \, {\left (a x^{2} + b x + c\right )}^{\frac {3}{2}}} + \frac {5 \, 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 {7}{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)^(5/2),x, algorithm="giac")

[Out]

-1/6*(15*b^5*sqrt(c)*log(abs(-b + 2*sqrt(a)*sqrt(c))) - 120*a*b^3*c^(3/2)*log(abs(-b + 2*sqrt(a)*sqrt(c))) + 2
40*a^2*b*c^(5/2)*log(abs(-b + 2*sqrt(a)*sqrt(c))) + 30*sqrt(a)*b^4*c - 200*a^(3/2)*b^2*c^2 + 256*a^(5/2)*c^3)*
sgn(x)/(a^(7/2)*b^4*sqrt(c) - 8*a^(9/2)*b^2*c^(3/2) + 16*a^(11/2)*c^(5/2)) + 1/3*((((3*(a^2*b^4 - 8*a^3*b^2*c
+ 16*a^4*c^2)*x/(a^3*b^4*sgn(x) - 8*a^4*b^2*c*sgn(x) + 16*a^5*c^2*sgn(x)) + 4*(5*a*b^5 - 37*a^2*b^3*c + 64*a^3
*b*c^2)/(a^3*b^4*sgn(x) - 8*a^4*b^2*c*sgn(x) + 16*a^5*c^2*sgn(x)))*x + 3*(5*b^6 - 30*a*b^4*c + 16*a^2*b^2*c^2
+ 64*a^3*c^3)/(a^3*b^4*sgn(x) - 8*a^4*b^2*c*sgn(x) + 16*a^5*c^2*sgn(x)))*x + 6*(5*b^5*c - 35*a*b^3*c^2 + 52*a^
2*b*c^3)/(a^3*b^4*sgn(x) - 8*a^4*b^2*c*sgn(x) + 16*a^5*c^2*sgn(x)))*x + (15*b^4*c^2 - 100*a*b^2*c^3 + 128*a^2*
c^4)/(a^3*b^4*sgn(x) - 8*a^4*b^2*c*sgn(x) + 16*a^5*c^2*sgn(x)))/(a*x^2 + b*x + c)^(3/2) + 5/2*b*log(abs(-2*(sq
rt(a)*x - sqrt(a*x^2 + b*x + c))*sqrt(a) - b))/(a^(7/2)*sgn(x))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (a+\frac {b}{x}+\frac {c}{x^2}\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/x + c/x^2)^(5/2),x)

[Out]

int(1/(a + b/x + c/x^2)^(5/2), x)

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