3.57 \(\int \frac{A+B x+C x^2}{x^3 (a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=219 \[ -\frac{35 (4 A b-a C)+93 b B x}{35 a^5 \sqrt{a+b x^2}}-\frac{35 (3 A b-a C)+87 b B x}{105 a^4 \left (a+b x^2\right )^{3/2}}-\frac{7 (2 A b-a C)+13 b B x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac{a \left (\frac{A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}+\frac{(9 A b-2 a C) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{11/2}}-\frac{A \sqrt{a+b x^2}}{2 a^5 x^2}-\frac{B \sqrt{a+b x^2}}{a^5 x} \]

[Out]

-(a*((A*b)/a - C) + b*B*x)/(7*a^2*(a + b*x^2)^(7/2)) - (7*(2*A*b - a*C) + 13*b*B*x)/(35*a^3*(a + b*x^2)^(5/2))
- (35*(3*A*b - a*C) + 87*b*B*x)/(105*a^4*(a + b*x^2)^(3/2)) - (35*(4*A*b - a*C) + 93*b*B*x)/(35*a^5*Sqrt[a +
b*x^2]) - (A*Sqrt[a + b*x^2])/(2*a^5*x^2) - (B*Sqrt[a + b*x^2])/(a^5*x) + ((9*A*b - 2*a*C)*ArcTanh[Sqrt[a + b*
x^2]/Sqrt[a]])/(2*a^(11/2))

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Rubi [A]  time = 0.48031, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {1805, 1807, 807, 266, 63, 208} \[ -\frac{35 (4 A b-a C)+93 b B x}{35 a^5 \sqrt{a+b x^2}}-\frac{35 (3 A b-a C)+87 b B x}{105 a^4 \left (a+b x^2\right )^{3/2}}-\frac{7 (2 A b-a C)+13 b B x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac{a \left (\frac{A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}+\frac{(9 A b-2 a C) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{11/2}}-\frac{A \sqrt{a+b x^2}}{2 a^5 x^2}-\frac{B \sqrt{a+b x^2}}{a^5 x} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2)/(x^3*(a + b*x^2)^(9/2)),x]

[Out]

-(a*((A*b)/a - C) + b*B*x)/(7*a^2*(a + b*x^2)^(7/2)) - (7*(2*A*b - a*C) + 13*b*B*x)/(35*a^3*(a + b*x^2)^(5/2))
- (35*(3*A*b - a*C) + 87*b*B*x)/(105*a^4*(a + b*x^2)^(3/2)) - (35*(4*A*b - a*C) + 93*b*B*x)/(35*a^5*Sqrt[a +
b*x^2]) - (A*Sqrt[a + b*x^2])/(2*a^5*x^2) - (B*Sqrt[a + b*x^2])/(a^5*x) + ((9*A*b - 2*a*C)*ArcTanh[Sqrt[a + b*
x^2]/Sqrt[a]])/(2*a^(11/2))

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1807

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[(R*(c*x)^(m + 1)*(a + b*x^2)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rule 807

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 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]

Rubi steps

\begin{align*} \int \frac{A+B x+C x^2}{x^3 \left (a+b x^2\right )^{9/2}} \, dx &=-\frac{a \left (\frac{A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{-7 A-7 B x+7 \left (\frac{A b}{a}-C\right ) x^2+\frac{6 b B x^3}{a}}{x^3 \left (a+b x^2\right )^{7/2}} \, dx}{7 a}\\ &=-\frac{a \left (\frac{A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac{7 (2 A b-a C)+13 b B x}{35 a^3 \left (a+b x^2\right )^{5/2}}+\frac{\int \frac{35 A+35 B x-35 \left (\frac{2 A b}{a}-C\right ) x^2-\frac{52 b B x^3}{a}}{x^3 \left (a+b x^2\right )^{5/2}} \, dx}{35 a^2}\\ &=-\frac{a \left (\frac{A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac{7 (2 A b-a C)+13 b B x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac{35 (3 A b-a C)+87 b B x}{105 a^4 \left (a+b x^2\right )^{3/2}}-\frac{\int \frac{-105 A-105 B x+105 \left (\frac{3 A b}{a}-C\right ) x^2+\frac{174 b B x^3}{a}}{x^3 \left (a+b x^2\right )^{3/2}} \, dx}{105 a^3}\\ &=-\frac{a \left (\frac{A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac{7 (2 A b-a C)+13 b B x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac{35 (3 A b-a C)+87 b B x}{105 a^4 \left (a+b x^2\right )^{3/2}}-\frac{35 (4 A b-a C)+93 b B x}{35 a^5 \sqrt{a+b x^2}}+\frac{\int \frac{105 A+105 B x-105 \left (\frac{4 A b}{a}-C\right ) x^2}{x^3 \sqrt{a+b x^2}} \, dx}{105 a^4}\\ &=-\frac{a \left (\frac{A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac{7 (2 A b-a C)+13 b B x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac{35 (3 A b-a C)+87 b B x}{105 a^4 \left (a+b x^2\right )^{3/2}}-\frac{35 (4 A b-a C)+93 b B x}{35 a^5 \sqrt{a+b x^2}}-\frac{A \sqrt{a+b x^2}}{2 a^5 x^2}-\frac{\int \frac{-210 a B+105 (9 A b-2 a C) x}{x^2 \sqrt{a+b x^2}} \, dx}{210 a^5}\\ &=-\frac{a \left (\frac{A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac{7 (2 A b-a C)+13 b B x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac{35 (3 A b-a C)+87 b B x}{105 a^4 \left (a+b x^2\right )^{3/2}}-\frac{35 (4 A b-a C)+93 b B x}{35 a^5 \sqrt{a+b x^2}}-\frac{A \sqrt{a+b x^2}}{2 a^5 x^2}-\frac{B \sqrt{a+b x^2}}{a^5 x}-\frac{(9 A b-2 a C) \int \frac{1}{x \sqrt{a+b x^2}} \, dx}{2 a^5}\\ &=-\frac{a \left (\frac{A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac{7 (2 A b-a C)+13 b B x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac{35 (3 A b-a C)+87 b B x}{105 a^4 \left (a+b x^2\right )^{3/2}}-\frac{35 (4 A b-a C)+93 b B x}{35 a^5 \sqrt{a+b x^2}}-\frac{A \sqrt{a+b x^2}}{2 a^5 x^2}-\frac{B \sqrt{a+b x^2}}{a^5 x}-\frac{(9 A b-2 a C) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{4 a^5}\\ &=-\frac{a \left (\frac{A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac{7 (2 A b-a C)+13 b B x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac{35 (3 A b-a C)+87 b B x}{105 a^4 \left (a+b x^2\right )^{3/2}}-\frac{35 (4 A b-a C)+93 b B x}{35 a^5 \sqrt{a+b x^2}}-\frac{A \sqrt{a+b x^2}}{2 a^5 x^2}-\frac{B \sqrt{a+b x^2}}{a^5 x}-\frac{(9 A b-2 a C) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{2 a^5 b}\\ &=-\frac{a \left (\frac{A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac{7 (2 A b-a C)+13 b B x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac{35 (3 A b-a C)+87 b B x}{105 a^4 \left (a+b x^2\right )^{3/2}}-\frac{35 (4 A b-a C)+93 b B x}{35 a^5 \sqrt{a+b x^2}}-\frac{A \sqrt{a+b x^2}}{2 a^5 x^2}-\frac{B \sqrt{a+b x^2}}{a^5 x}+\frac{(9 A b-2 a C) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.377404, size = 178, normalized size = 0.81 \[ \frac{42 a^2 b^3 x^4 (x (5 C x-64 B)-75 A)+14 a^3 b^2 x^2 (10 x (5 C x-24 B)-261 A)-4 a^4 b (396 A+7 x (60 B-29 C x))+\frac{a^5 \left (-105 A-210 B x+352 C x^2\right )}{x^2}-3 a b^4 x^6 (315 A+256 B x)+\frac{105 \left (a+b x^2\right )^4 (9 A b-2 a C) \tanh ^{-1}\left (\sqrt{\frac{b x^2}{a}+1}\right )}{\sqrt{\frac{b x^2}{a}+1}}}{210 a^6 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2)/(x^3*(a + b*x^2)^(9/2)),x]

[Out]

(-3*a*b^4*x^6*(315*A + 256*B*x) + (a^5*(-105*A - 210*B*x + 352*C*x^2))/x^2 - 4*a^4*b*(396*A + 7*x*(60*B - 29*C
*x)) + 42*a^2*b^3*x^4*(-75*A + x*(-64*B + 5*C*x)) + 14*a^3*b^2*x^2*(-261*A + 10*x*(-24*B + 5*C*x)) + (105*(9*A
*b - 2*a*C)*(a + b*x^2)^4*ArcTanh[Sqrt[1 + (b*x^2)/a]])/Sqrt[1 + (b*x^2)/a])/(210*a^6*(a + b*x^2)^(7/2))

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Maple [A]  time = 0.011, size = 288, normalized size = 1.3 \begin{align*}{\frac{C}{7\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{C}{5\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{C}{3\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{C}{{a}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{C\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{9}{2}}}}-{\frac{A}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{9\,Ab}{14\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{9\,Ab}{10\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{3\,Ab}{2\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{9\,Ab}{2\,{a}^{5}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{9\,Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{11}{2}}}}-{\frac{B}{ax} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{8\,bBx}{7\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{48\,bBx}{35\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{64\,bBx}{35\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{128\,bBx}{35\,{a}^{5}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)/x^3/(b*x^2+a)^(9/2),x)

[Out]

1/7*C/a/(b*x^2+a)^(7/2)+1/5*C/a^2/(b*x^2+a)^(5/2)+1/3*C/a^3/(b*x^2+a)^(3/2)+C/a^4/(b*x^2+a)^(1/2)-C/a^(9/2)*ln
((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-1/2*A/a/x^2/(b*x^2+a)^(7/2)-9/14*A*b/a^2/(b*x^2+a)^(7/2)-9/10*A*b/a^3/(b*x
^2+a)^(5/2)-3/2*A*b/a^4/(b*x^2+a)^(3/2)-9/2*A*b/a^5/(b*x^2+a)^(1/2)+9/2*A*b/a^(11/2)*ln((2*a+2*a^(1/2)*(b*x^2+
a)^(1/2))/x)-B/a/x/(b*x^2+a)^(7/2)-8/7*B*b/a^2*x/(b*x^2+a)^(7/2)-48/35*B*b/a^3*x/(b*x^2+a)^(5/2)-64/35*B*b/a^4
*x/(b*x^2+a)^(3/2)-128/35*B*b/a^5*x/(b*x^2+a)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/x^3/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.12306, size = 1527, normalized size = 6.97 \begin{align*} \left [-\frac{105 \,{\left ({\left (2 \, C a b^{4} - 9 \, A b^{5}\right )} x^{10} + 4 \,{\left (2 \, C a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{8} + 6 \,{\left (2 \, C a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{6} + 4 \,{\left (2 \, C a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{4} +{\left (2 \, C a^{5} - 9 \, A a^{4} b\right )} x^{2}\right )} \sqrt{a} \log \left (-\frac{b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (768 \, B a b^{4} x^{9} + 2688 \, B a^{2} b^{3} x^{7} + 3360 \, B a^{3} b^{2} x^{5} + 1680 \, B a^{4} b x^{3} - 105 \,{\left (2 \, C a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{8} + 210 \, B a^{5} x - 350 \,{\left (2 \, C a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{6} + 105 \, A a^{5} - 406 \,{\left (2 \, C a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{4} - 176 \,{\left (2 \, C a^{5} - 9 \, A a^{4} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{420 \,{\left (a^{6} b^{4} x^{10} + 4 \, a^{7} b^{3} x^{8} + 6 \, a^{8} b^{2} x^{6} + 4 \, a^{9} b x^{4} + a^{10} x^{2}\right )}}, \frac{105 \,{\left ({\left (2 \, C a b^{4} - 9 \, A b^{5}\right )} x^{10} + 4 \,{\left (2 \, C a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{8} + 6 \,{\left (2 \, C a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{6} + 4 \,{\left (2 \, C a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{4} +{\left (2 \, C a^{5} - 9 \, A a^{4} b\right )} x^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (768 \, B a b^{4} x^{9} + 2688 \, B a^{2} b^{3} x^{7} + 3360 \, B a^{3} b^{2} x^{5} + 1680 \, B a^{4} b x^{3} - 105 \,{\left (2 \, C a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{8} + 210 \, B a^{5} x - 350 \,{\left (2 \, C a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{6} + 105 \, A a^{5} - 406 \,{\left (2 \, C a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{4} - 176 \,{\left (2 \, C a^{5} - 9 \, A a^{4} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{210 \,{\left (a^{6} b^{4} x^{10} + 4 \, a^{7} b^{3} x^{8} + 6 \, a^{8} b^{2} x^{6} + 4 \, a^{9} b x^{4} + a^{10} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/x^3/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

[-1/420*(105*((2*C*a*b^4 - 9*A*b^5)*x^10 + 4*(2*C*a^2*b^3 - 9*A*a*b^4)*x^8 + 6*(2*C*a^3*b^2 - 9*A*a^2*b^3)*x^6
+ 4*(2*C*a^4*b - 9*A*a^3*b^2)*x^4 + (2*C*a^5 - 9*A*a^4*b)*x^2)*sqrt(a)*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a
) + 2*a)/x^2) + 2*(768*B*a*b^4*x^9 + 2688*B*a^2*b^3*x^7 + 3360*B*a^3*b^2*x^5 + 1680*B*a^4*b*x^3 - 105*(2*C*a^2
*b^3 - 9*A*a*b^4)*x^8 + 210*B*a^5*x - 350*(2*C*a^3*b^2 - 9*A*a^2*b^3)*x^6 + 105*A*a^5 - 406*(2*C*a^4*b - 9*A*a
^3*b^2)*x^4 - 176*(2*C*a^5 - 9*A*a^4*b)*x^2)*sqrt(b*x^2 + a))/(a^6*b^4*x^10 + 4*a^7*b^3*x^8 + 6*a^8*b^2*x^6 +
4*a^9*b*x^4 + a^10*x^2), 1/210*(105*((2*C*a*b^4 - 9*A*b^5)*x^10 + 4*(2*C*a^2*b^3 - 9*A*a*b^4)*x^8 + 6*(2*C*a^3
*b^2 - 9*A*a^2*b^3)*x^6 + 4*(2*C*a^4*b - 9*A*a^3*b^2)*x^4 + (2*C*a^5 - 9*A*a^4*b)*x^2)*sqrt(-a)*arctan(sqrt(-a
)/sqrt(b*x^2 + a)) - (768*B*a*b^4*x^9 + 2688*B*a^2*b^3*x^7 + 3360*B*a^3*b^2*x^5 + 1680*B*a^4*b*x^3 - 105*(2*C*
a^2*b^3 - 9*A*a*b^4)*x^8 + 210*B*a^5*x - 350*(2*C*a^3*b^2 - 9*A*a^2*b^3)*x^6 + 105*A*a^5 - 406*(2*C*a^4*b - 9*
A*a^3*b^2)*x^4 - 176*(2*C*a^5 - 9*A*a^4*b)*x^2)*sqrt(b*x^2 + a))/(a^6*b^4*x^10 + 4*a^7*b^3*x^8 + 6*a^8*b^2*x^6
+ 4*a^9*b*x^4 + a^10*x^2)]

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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((C*x**2+B*x+A)/x**3/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.20897, size = 439, normalized size = 2. \begin{align*} -\frac{{\left ({\left ({\left ({\left (3 \,{\left ({\left (\frac{93 \, B b^{4} x}{a^{5}} - \frac{35 \,{\left (C a^{24} b^{6} - 4 \, A a^{23} b^{7}\right )}}{a^{28} b^{3}}\right )} x + \frac{308 \, B b^{3}}{a^{4}}\right )} x - \frac{35 \,{\left (10 \, C a^{25} b^{5} - 39 \, A a^{24} b^{6}\right )}}{a^{28} b^{3}}\right )} x + \frac{1050 \, B b^{2}}{a^{3}}\right )} x - \frac{14 \,{\left (29 \, C a^{26} b^{4} - 108 \, A a^{25} b^{5}\right )}}{a^{28} b^{3}}\right )} x + \frac{420 \, B b}{a^{2}}\right )} x - \frac{2 \,{\left (88 \, C a^{27} b^{3} - 291 \, A a^{26} b^{4}\right )}}{a^{28} b^{3}}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} + \frac{{\left (2 \, C a - 9 \, A b\right )} \arctan \left (-\frac{\sqrt{b} x - \sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{5}} + \frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{3} A b + 2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a \sqrt{b} +{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )} A a b - 2 \, B a^{2} \sqrt{b}}{{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{2} a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/x^3/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

-1/105*(((((3*((93*B*b^4*x/a^5 - 35*(C*a^24*b^6 - 4*A*a^23*b^7)/(a^28*b^3))*x + 308*B*b^3/a^4)*x - 35*(10*C*a^
25*b^5 - 39*A*a^24*b^6)/(a^28*b^3))*x + 1050*B*b^2/a^3)*x - 14*(29*C*a^26*b^4 - 108*A*a^25*b^5)/(a^28*b^3))*x
+ 420*B*b/a^2)*x - 2*(88*C*a^27*b^3 - 291*A*a^26*b^4)/(a^28*b^3))/(b*x^2 + a)^(7/2) + (2*C*a - 9*A*b)*arctan(-
(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^5) + ((sqrt(b)*x - sqrt(b*x^2 + a))^3*A*b + 2*(sqrt(b)*x -
sqrt(b*x^2 + a))^2*B*a*sqrt(b) + (sqrt(b)*x - sqrt(b*x^2 + a))*A*a*b - 2*B*a^2*sqrt(b))/(((sqrt(b)*x - sqrt(b
*x^2 + a))^2 - a)^2*a^5)