3.2331 \(\int \frac{1}{x^{3/2} (a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=458 \[ \frac{36 a^2 c^2+b c x \left (5 b^2-32 a c\right )-35 a b^2 c+5 b^4}{4 a^2 \sqrt{x} \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 \sqrt{c} \left (\sqrt{b^2-4 a c} \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )+124 a^2 b c^2-47 a b^3 c+5 b^5\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{c} \left (-60 a^2 c^2 \sqrt{b^2-4 a c}+124 a^2 b c^2-5 b^4 \sqrt{b^2-4 a c}-47 a b^3 c+37 a b^2 c \sqrt{b^2-4 a c}+5 b^5\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{4 a^3 \sqrt{x} \left (b^2-4 a c\right )^2}+\frac{-2 a c+b^2+b c x}{2 a \sqrt{x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[Out]

(-3*(5*b^2 - 12*a*c)*(b^2 - 5*a*c))/(4*a^3*(b^2 - 4*a*c)^2*Sqrt[x]) + (b^2 - 2*a*c + b*c*x)/(2*a*(b^2 - 4*a*c)
*Sqrt[x]*(a + b*x + c*x^2)^2) + (5*b^4 - 35*a*b^2*c + 36*a^2*c^2 + b*c*(5*b^2 - 32*a*c)*x)/(4*a^2*(b^2 - 4*a*c
)^2*Sqrt[x]*(a + b*x + c*x^2)) - (3*Sqrt[c]*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 + Sqrt[b^2 - 4*a*c]*(5*b^4 - 3
7*a*b^2*c + 60*a^2*c^2))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*a^3*(b^2 -
4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[c]*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 - 5*b^4*Sqrt[b^2 -
4*a*c] + 37*a*b^2*c*Sqrt[b^2 - 4*a*c] - 60*a^2*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b
+ Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

________________________________________________________________________________________

Rubi [A]  time = 9.72157, antiderivative size = 458, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {740, 822, 828, 826, 1166, 205} \[ \frac{36 a^2 c^2+b c x \left (5 b^2-32 a c\right )-35 a b^2 c+5 b^4}{4 a^2 \sqrt{x} \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 \sqrt{c} \left (\sqrt{b^2-4 a c} \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )+124 a^2 b c^2-47 a b^3 c+5 b^5\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{c} \left (-60 a^2 c^2 \sqrt{b^2-4 a c}+124 a^2 b c^2-5 b^4 \sqrt{b^2-4 a c}-47 a b^3 c+37 a b^2 c \sqrt{b^2-4 a c}+5 b^5\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{4 a^3 \sqrt{x} \left (b^2-4 a c\right )^2}+\frac{-2 a c+b^2+b c x}{2 a \sqrt{x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*(a + b*x + c*x^2)^3),x]

[Out]

(-3*(5*b^2 - 12*a*c)*(b^2 - 5*a*c))/(4*a^3*(b^2 - 4*a*c)^2*Sqrt[x]) + (b^2 - 2*a*c + b*c*x)/(2*a*(b^2 - 4*a*c)
*Sqrt[x]*(a + b*x + c*x^2)^2) + (5*b^4 - 35*a*b^2*c + 36*a^2*c^2 + b*c*(5*b^2 - 32*a*c)*x)/(4*a^2*(b^2 - 4*a*c
)^2*Sqrt[x]*(a + b*x + c*x^2)) - (3*Sqrt[c]*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 + Sqrt[b^2 - 4*a*c]*(5*b^4 - 3
7*a*b^2*c + 60*a^2*c^2))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*a^3*(b^2 -
4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[c]*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 - 5*b^4*Sqrt[b^2 -
4*a*c] + 37*a*b^2*c*Sqrt[b^2 - 4*a*c] - 60*a^2*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b
+ Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

Rule 740

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 822

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 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), 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] && FractionQ[m] && LtQ[m, -1]

Rule 826

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

Rule 1166

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

Rule 205

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

Rubi steps

\begin{align*} \int \frac{1}{x^{3/2} \left (a+b x+c x^2\right )^3} \, dx &=\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \sqrt{x} \left (a+b x+c x^2\right )^2}-\frac{\int \frac{\frac{1}{2} \left (-5 b^2+18 a c\right )-\frac{7 b c x}{2}}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \sqrt{x} \left (a+b x+c x^2\right )^2}+\frac{5 b^4-35 a b^2 c+36 a^2 c^2+b c \left (5 b^2-32 a c\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt{x} \left (a+b x+c x^2\right )}+\frac{\int \frac{\frac{3}{4} \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )+\frac{3}{4} b c \left (5 b^2-32 a c\right ) x}{x^{3/2} \left (a+b x+c x^2\right )} \, dx}{2 a^2 \left (b^2-4 a c\right )^2}\\ &=-\frac{3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{4 a^3 \left (b^2-4 a c\right )^2 \sqrt{x}}+\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \sqrt{x} \left (a+b x+c x^2\right )^2}+\frac{5 b^4-35 a b^2 c+36 a^2 c^2+b c \left (5 b^2-32 a c\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt{x} \left (a+b x+c x^2\right )}+\frac{\int \frac{-\frac{3}{4} b \left (5 b^4-42 a b^2 c+92 a^2 c^2\right )-\frac{3}{4} c \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right ) x}{\sqrt{x} \left (a+b x+c x^2\right )} \, dx}{2 a^3 \left (b^2-4 a c\right )^2}\\ &=-\frac{3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{4 a^3 \left (b^2-4 a c\right )^2 \sqrt{x}}+\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \sqrt{x} \left (a+b x+c x^2\right )^2}+\frac{5 b^4-35 a b^2 c+36 a^2 c^2+b c \left (5 b^2-32 a c\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt{x} \left (a+b x+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{4} b \left (5 b^4-42 a b^2 c+92 a^2 c^2\right )-\frac{3}{4} c \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right ) x^2}{a+b x^2+c x^4} \, dx,x,\sqrt{x}\right )}{a^3 \left (b^2-4 a c\right )^2}\\ &=-\frac{3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{4 a^3 \left (b^2-4 a c\right )^2 \sqrt{x}}+\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \sqrt{x} \left (a+b x+c x^2\right )^2}+\frac{5 b^4-35 a b^2 c+36 a^2 c^2+b c \left (5 b^2-32 a c\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt{x} \left (a+b x+c x^2\right )}+\frac{\left (3 c \left (5 b^5-47 a b^3 c+124 a^2 b c^2-\sqrt{b^2-4 a c} \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx,x,\sqrt{x}\right )}{8 a^3 \left (b^2-4 a c\right )^{5/2}}-\frac{\left (3 c \left (5 b^5-47 a b^3 c+124 a^2 b c^2+\sqrt{b^2-4 a c} \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx,x,\sqrt{x}\right )}{8 a^3 \left (b^2-4 a c\right )^{5/2}}\\ &=-\frac{3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{4 a^3 \left (b^2-4 a c\right )^2 \sqrt{x}}+\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \sqrt{x} \left (a+b x+c x^2\right )^2}+\frac{5 b^4-35 a b^2 c+36 a^2 c^2+b c \left (5 b^2-32 a c\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt{x} \left (a+b x+c x^2\right )}-\frac{3 \sqrt{c} \left (5 b^5-47 a b^3 c+124 a^2 b c^2+\sqrt{b^2-4 a c} \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{c} \left (5 b^5-47 a b^3 c+124 a^2 b c^2-\sqrt{b^2-4 a c} \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt{b+\sqrt{b^2-4 a c}}}\\ \end{align*}

Mathematica [A]  time = 1.26227, size = 440, normalized size = 0.96 \[ \frac{\frac{-36 a^2 c^2+35 a b^2 c+32 a b c^2 x-5 b^3 c x-5 b^4}{2 a \sqrt{x} \left (4 a c-b^2\right ) (a+x (b+c x))}-\frac{\frac{3 \sqrt{c} \left (\frac{b \left (124 a^2 c^2-47 a b^2 c+5 b^4\right )}{\sqrt{b^2-4 a c}}+\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{c} \left (60 a^2 c^2 \sqrt{b^2-4 a c}-124 a^2 b c^2+5 b^4 \sqrt{b^2-4 a c}+47 a b^3 c-37 a b^2 c \sqrt{b^2-4 a c}-5 b^5\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{\sqrt{x}}}{2 a^2 \left (b^2-4 a c\right )}+\frac{-2 a c+b^2+b c x}{\sqrt{x} (a+x (b+c x))^2}}{2 a \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(3/2)*(a + b*x + c*x^2)^3),x]

[Out]

((b^2 - 2*a*c + b*c*x)/(Sqrt[x]*(a + x*(b + c*x))^2) + (-5*b^4 + 35*a*b^2*c - 36*a^2*c^2 - 5*b^3*c*x + 32*a*b*
c^2*x)/(2*a*(-b^2 + 4*a*c)*Sqrt[x]*(a + x*(b + c*x))) - ((3*(5*b^2 - 12*a*c)*(b^2 - 5*a*c))/Sqrt[x] + (3*Sqrt[
c]*((5*b^2 - 12*a*c)*(b^2 - 5*a*c) + (b*(5*b^4 - 47*a*b^2*c + 124*a^2*c^2))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]
*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[c]*(-5*b^5 + 4
7*a*b^3*c - 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a
*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b
^2 - 4*a*c]]))/(2*a^2*(b^2 - 4*a*c)))/(2*a*(b^2 - 4*a*c))

________________________________________________________________________________________

Maple [B]  time = 0.201, size = 1571, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/a/(c*x^2+b*x+a)^2*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)-27/(c*x^2+b*x+a)^2*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x
^(1/2)*c^2+93/2/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*
arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b+15/8/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(-4*a*c+b
^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*
b^5+15/8/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh
(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^5-141/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(-4*a*c+b^2
)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))
*b^3-141/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arct
an(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^3+93/2/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/(-4*a*c+b^2)^
(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b-9/
4/a^2/(c*x^2+b*x+a)^2*b^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2)-7/4/a^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4
)*x^(3/2)*b^6-45/2/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*
2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))+47/4/a^2/(c*x^2+b*x+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)*b^2-
34/a/(c*x^2+b*x+a)^2*c^3*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)+45/2/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*2^(1/2)/((
-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))+99/4/a^2/(c*x^2+b
*x+a)^2*c^2*b^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)-25/4/a/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*b
^2*c^2+43/4/a^2/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*b^4*c+33/2/a/(c*x^2+b*x+a)^2*b^3/(16*a^2*c^
2-8*a*b^2*c+b^4)*x^(1/2)*c-7/4/a^3/(c*x^2+b*x+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)*b^4-7/2/a^3/(c*x^2+b
*x+a)^2*c*b^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)-2/a^3/x^(1/2)+111/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*2^(1/2
)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^2+15/8/a^3/(16
*a^2*c^2-8*a*b^2*c+b^4)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2
)^(1/2))*c)^(1/2))*b^4-15/8/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x
^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^4-17/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*c
^3-111/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2
)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^2

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [B]  time = 17.034, size = 11857, normalized size = 25.89 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(3*sqrt(1/2)*((a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*x^5 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^
3)*x^4 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*x^3 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^2 + (a^5*b^4 -
8*a^6*b^2*c + 16*a^7*c^2)*x)*sqrt(-(25*b^11 - 495*a*b^9*c + 3894*a^2*b^7*c^2 - 15015*a^3*b^5*c^3 + 27720*a^4*b
^3*c^4 - 18480*a^5*b*c^5 + (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 -
1024*a^12*c^5)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4
- 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*
a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*
c^4 - 1024*a^12*c^5))*log(27/2*sqrt(1/2)*(125*b^17 - 3775*a*b^15*c + 49360*a^2*b^13*c^2 - 362733*a^3*b^11*c^3
+ 1623534*a^4*b^9*c^4 - 4463140*a^5*b^7*c^5 + 7146736*a^6*b^5*c^6 - 5684672*a^7*b^3*c^7 + 1324800*a^8*b*c^8 -
(5*a^7*b^16 - 152*a^8*b^14*c + 2006*a^9*b^12*c^2 - 14960*a^10*b^10*c^3 + 68640*a^11*b^8*c^4 - 197120*a^12*b^6*
c^5 + 342528*a^13*b^4*c^6 - 323584*a^14*b^2*c^7 + 122880*a^15*c^8)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a^2
*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*
b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))*sqrt(-(25*b^11 - 495*a*b^9*
c + 3894*a^2*b^7*c^2 - 15015*a^3*b^5*c^3 + 27720*a^4*b^3*c^4 - 18480*a^5*b*c^5 + (a^7*b^10 - 20*a^8*b^8*c + 16
0*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*
a^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^
15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c
+ 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)) - 27*(4125*b^10*c^4 - 77825*a*b^8*
c^5 + 571030*a^2*b^6*c^6 - 1957349*a^3*b^4*c^7 + 2835000*a^4*b^2*c^8 - 810000*a^5*c^9)*sqrt(x)) - 3*sqrt(1/2)*
((a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*x^5 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^4 + (a^3*b^6
- 6*a^4*b^4*c + 32*a^6*c^3)*x^3 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^2 + (a^5*b^4 - 8*a^6*b^2*c + 16*a
^7*c^2)*x)*sqrt(-(25*b^11 - 495*a*b^9*c + 3894*a^2*b^7*c^2 - 15015*a^3*b^5*c^3 + 27720*a^4*b^3*c^4 - 18480*a^5
*b*c^5 + (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sq
rt((625*b^12 - 12250*a*b^10*c + 94725*a^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 - 312300*a^5*b^2*c
^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 102
4*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^
5))*log(-27/2*sqrt(1/2)*(125*b^17 - 3775*a*b^15*c + 49360*a^2*b^13*c^2 - 362733*a^3*b^11*c^3 + 1623534*a^4*b^9
*c^4 - 4463140*a^5*b^7*c^5 + 7146736*a^6*b^5*c^6 - 5684672*a^7*b^3*c^7 + 1324800*a^8*b*c^8 - (5*a^7*b^16 - 152
*a^8*b^14*c + 2006*a^9*b^12*c^2 - 14960*a^10*b^10*c^3 + 68640*a^11*b^8*c^4 - 197120*a^12*b^6*c^5 + 342528*a^13
*b^4*c^6 - 323584*a^14*b^2*c^7 + 122880*a^15*c^8)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a^2*b^8*c^2 - 351310
*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*
b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))*sqrt(-(25*b^11 - 495*a*b^9*c + 3894*a^2*b^7*
c^2 - 15015*a^3*b^5*c^3 + 27720*a^4*b^3*c^4 - 18480*a^5*b*c^5 + (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 6
40*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a^2*b^8*c^2 - 351
310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^
16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^
2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)) - 27*(4125*b^10*c^4 - 77825*a*b^8*c^5 + 571030*a^2*
b^6*c^6 - 1957349*a^3*b^4*c^7 + 2835000*a^4*b^2*c^8 - 810000*a^5*c^9)*sqrt(x)) + 3*sqrt(1/2)*((a^3*b^4*c^2 - 8
*a^4*b^2*c^3 + 16*a^5*c^4)*x^5 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^4 + (a^3*b^6 - 6*a^4*b^4*c + 3
2*a^6*c^3)*x^3 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^2 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*x)*sqrt(-
(25*b^11 - 495*a*b^9*c + 3894*a^2*b^7*c^2 - 15015*a^3*b^5*c^3 + 27720*a^4*b^3*c^4 - 18480*a^5*b*c^5 - (a^7*b^1
0 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((625*b^12 - 12
250*a*b^10*c + 94725*a^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^
6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^
7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5))*log(27/2*sqrt
(1/2)*(125*b^17 - 3775*a*b^15*c + 49360*a^2*b^13*c^2 - 362733*a^3*b^11*c^3 + 1623534*a^4*b^9*c^4 - 4463140*a^5
*b^7*c^5 + 7146736*a^6*b^5*c^6 - 5684672*a^7*b^3*c^7 + 1324800*a^8*b*c^8 + (5*a^7*b^16 - 152*a^8*b^14*c + 2006
*a^9*b^12*c^2 - 14960*a^10*b^10*c^3 + 68640*a^11*b^8*c^4 - 197120*a^12*b^6*c^5 + 342528*a^13*b^4*c^6 - 323584*
a^14*b^2*c^7 + 122880*a^15*c^8)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591
886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17
*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))*sqrt(-(25*b^11 - 495*a*b^9*c + 3894*a^2*b^7*c^2 - 15015*a^3*b^
5*c^3 + 27720*a^4*b^3*c^4 - 18480*a^5*b*c^5 - (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 +
1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a^2*b^8*c^2 - 351310*a^3*b^6*c^3 +
591886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a
^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c
^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)) - 27*(4125*b^10*c^4 - 77825*a*b^8*c^5 + 571030*a^2*b^6*c^6 - 1957349*
a^3*b^4*c^7 + 2835000*a^4*b^2*c^8 - 810000*a^5*c^9)*sqrt(x)) - 3*sqrt(1/2)*((a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*
a^5*c^4)*x^5 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^4 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*x^3 + 2
*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^2 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*x)*sqrt(-(25*b^11 - 495*a*b
^9*c + 3894*a^2*b^7*c^2 - 15015*a^3*b^5*c^3 + 27720*a^4*b^3*c^4 - 18480*a^5*b*c^5 - (a^7*b^10 - 20*a^8*b^8*c +
160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((625*b^12 - 12250*a*b^10*c + 947
25*a^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20
*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^
8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5))*log(-27/2*sqrt(1/2)*(125*b^17 -
3775*a*b^15*c + 49360*a^2*b^13*c^2 - 362733*a^3*b^11*c^3 + 1623534*a^4*b^9*c^4 - 4463140*a^5*b^7*c^5 + 714673
6*a^6*b^5*c^6 - 5684672*a^7*b^3*c^7 + 1324800*a^8*b*c^8 + (5*a^7*b^16 - 152*a^8*b^14*c + 2006*a^9*b^12*c^2 - 1
4960*a^10*b^10*c^3 + 68640*a^11*b^8*c^4 - 197120*a^12*b^6*c^5 + 342528*a^13*b^4*c^6 - 323584*a^14*b^2*c^7 + 12
2880*a^15*c^8)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 -
312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a
^18*b^2*c^4 - 1024*a^19*c^5)))*sqrt(-(25*b^11 - 495*a*b^9*c + 3894*a^2*b^7*c^2 - 15015*a^3*b^5*c^3 + 27720*a^4
*b^3*c^4 - 18480*a^5*b*c^5 - (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4
- 1024*a^12*c^5)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^
4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 128
0*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^
2*c^4 - 1024*a^12*c^5)) - 27*(4125*b^10*c^4 - 77825*a*b^8*c^5 + 571030*a^2*b^6*c^6 - 1957349*a^3*b^4*c^7 + 283
5000*a^4*b^2*c^8 - 810000*a^5*c^9)*sqrt(x)) + 2*(8*a^2*b^4 - 64*a^3*b^2*c + 128*a^4*c^2 + 3*(5*b^4*c^2 - 37*a*
b^2*c^3 + 60*a^2*c^4)*x^4 + (30*b^5*c - 227*a*b^3*c^2 + 392*a^2*b*c^3)*x^3 + (15*b^6 - 91*a*b^4*c + 25*a^2*b^2
*c^2 + 324*a^3*c^3)*x^2 + (25*a*b^5 - 194*a^2*b^3*c + 364*a^3*b*c^2)*x)*sqrt(x))/((a^3*b^4*c^2 - 8*a^4*b^2*c^3
+ 16*a^5*c^4)*x^5 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^4 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*x
^3 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^2 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out