3.138 \(\int \frac{A+B x^2}{x^9 \sqrt{b x^2+c x^4}} \, dx\)

Optimal. Leaf size=170 \[ \frac{16 c^3 \sqrt{b x^2+c x^4} (9 b B-8 A c)}{315 b^5 x^2}-\frac{8 c^2 \sqrt{b x^2+c x^4} (9 b B-8 A c)}{315 b^4 x^4}+\frac{2 c \sqrt{b x^2+c x^4} (9 b B-8 A c)}{105 b^3 x^6}-\frac{\sqrt{b x^2+c x^4} (9 b B-8 A c)}{63 b^2 x^8}-\frac{A \sqrt{b x^2+c x^4}}{9 b x^{10}} \]

[Out]

-(A*Sqrt[b*x^2 + c*x^4])/(9*b*x^10) - ((9*b*B - 8*A*c)*Sqrt[b*x^2 + c*x^4])/(63*b^2*x^8) + (2*c*(9*b*B - 8*A*c
)*Sqrt[b*x^2 + c*x^4])/(105*b^3*x^6) - (8*c^2*(9*b*B - 8*A*c)*Sqrt[b*x^2 + c*x^4])/(315*b^4*x^4) + (16*c^3*(9*
b*B - 8*A*c)*Sqrt[b*x^2 + c*x^4])/(315*b^5*x^2)

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Rubi [A]  time = 0.30266, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2034, 792, 658, 650} \[ \frac{16 c^3 \sqrt{b x^2+c x^4} (9 b B-8 A c)}{315 b^5 x^2}-\frac{8 c^2 \sqrt{b x^2+c x^4} (9 b B-8 A c)}{315 b^4 x^4}+\frac{2 c \sqrt{b x^2+c x^4} (9 b B-8 A c)}{105 b^3 x^6}-\frac{\sqrt{b x^2+c x^4} (9 b B-8 A c)}{63 b^2 x^8}-\frac{A \sqrt{b x^2+c x^4}}{9 b x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^9*Sqrt[b*x^2 + c*x^4]),x]

[Out]

-(A*Sqrt[b*x^2 + c*x^4])/(9*b*x^10) - ((9*b*B - 8*A*c)*Sqrt[b*x^2 + c*x^4])/(63*b^2*x^8) + (2*c*(9*b*B - 8*A*c
)*Sqrt[b*x^2 + c*x^4])/(105*b^3*x^6) - (8*c^2*(9*b*B - 8*A*c)*Sqrt[b*x^2 + c*x^4])/(315*b^4*x^4) + (16*c^3*(9*
b*B - 8*A*c)*Sqrt[b*x^2 + c*x^4])/(315*b^5*x^2)

Rule 2034

Int[(x_)^(m_.)*((b_.)*(x_)^(k_.) + (a_.)*(x_)^(j_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n
, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a*x^Simplify[j/n] + b*x^Simplify[k/n])^p*(c + d*x)^q, x], x, x^n], x]
/; FreeQ[{a, b, c, d, j, k, m, n, p, q}, x] &&  !IntegerQ[p] && NeQ[k, j] && IntegerQ[Simplify[j/n]] && Integ
erQ[Simplify[k/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]

Rule 792

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d*g - e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((2*c*d - b*e)*(m + p + 1)), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), 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] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]

Rule 658

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

Rule 650

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

Rubi steps

\begin{align*} \int \frac{A+B x^2}{x^9 \sqrt{b x^2+c x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{x^5 \sqrt{b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac{A \sqrt{b x^2+c x^4}}{9 b x^{10}}+\frac{\left (-5 (-b B+A c)+\frac{1}{2} (-b B+2 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{b x+c x^2}} \, dx,x,x^2\right )}{9 b}\\ &=-\frac{A \sqrt{b x^2+c x^4}}{9 b x^{10}}-\frac{(9 b B-8 A c) \sqrt{b x^2+c x^4}}{63 b^2 x^8}-\frac{(c (9 b B-8 A c)) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{b x+c x^2}} \, dx,x,x^2\right )}{21 b^2}\\ &=-\frac{A \sqrt{b x^2+c x^4}}{9 b x^{10}}-\frac{(9 b B-8 A c) \sqrt{b x^2+c x^4}}{63 b^2 x^8}+\frac{2 c (9 b B-8 A c) \sqrt{b x^2+c x^4}}{105 b^3 x^6}+\frac{\left (4 c^2 (9 b B-8 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{b x+c x^2}} \, dx,x,x^2\right )}{105 b^3}\\ &=-\frac{A \sqrt{b x^2+c x^4}}{9 b x^{10}}-\frac{(9 b B-8 A c) \sqrt{b x^2+c x^4}}{63 b^2 x^8}+\frac{2 c (9 b B-8 A c) \sqrt{b x^2+c x^4}}{105 b^3 x^6}-\frac{8 c^2 (9 b B-8 A c) \sqrt{b x^2+c x^4}}{315 b^4 x^4}-\frac{\left (8 c^3 (9 b B-8 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{b x+c x^2}} \, dx,x,x^2\right )}{315 b^4}\\ &=-\frac{A \sqrt{b x^2+c x^4}}{9 b x^{10}}-\frac{(9 b B-8 A c) \sqrt{b x^2+c x^4}}{63 b^2 x^8}+\frac{2 c (9 b B-8 A c) \sqrt{b x^2+c x^4}}{105 b^3 x^6}-\frac{8 c^2 (9 b B-8 A c) \sqrt{b x^2+c x^4}}{315 b^4 x^4}+\frac{16 c^3 (9 b B-8 A c) \sqrt{b x^2+c x^4}}{315 b^5 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0576179, size = 94, normalized size = 0.55 \[ \frac{x^2 \left (\frac{c x^2}{b}+1\right ) \left (-6 b^2 c x^2+5 b^3+8 b c^2 x^4-16 c^3 x^6\right ) (8 A c-9 b B)-35 A b^3 \left (b+c x^2\right )}{315 b^4 x^8 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^9*Sqrt[b*x^2 + c*x^4]),x]

[Out]

(-35*A*b^3*(b + c*x^2) + (-9*b*B + 8*A*c)*x^2*(1 + (c*x^2)/b)*(5*b^3 - 6*b^2*c*x^2 + 8*b*c^2*x^4 - 16*c^3*x^6)
)/(315*b^4*x^8*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.006, size = 118, normalized size = 0.7 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( 128\,A{c}^{4}{x}^{8}-144\,Bb{c}^{3}{x}^{8}-64\,Ab{c}^{3}{x}^{6}+72\,B{b}^{2}{c}^{2}{x}^{6}+48\,A{b}^{2}{c}^{2}{x}^{4}-54\,B{b}^{3}c{x}^{4}-40\,A{b}^{3}c{x}^{2}+45\,B{b}^{4}{x}^{2}+35\,A{b}^{4} \right ) }{315\,{x}^{8}{b}^{5}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/x^9/(c*x^4+b*x^2)^(1/2),x)

[Out]

-1/315*(c*x^2+b)*(128*A*c^4*x^8-144*B*b*c^3*x^8-64*A*b*c^3*x^6+72*B*b^2*c^2*x^6+48*A*b^2*c^2*x^4-54*B*b^3*c*x^
4-40*A*b^3*c*x^2+45*B*b^4*x^2+35*A*b^4)/x^8/b^5/(c*x^4+b*x^2)^(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((B*x^2+A)/x^9/(c*x^4+b*x^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.26573, size = 240, normalized size = 1.41 \begin{align*} \frac{{\left (16 \,{\left (9 \, B b c^{3} - 8 \, A c^{4}\right )} x^{8} - 8 \,{\left (9 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x^{6} - 35 \, A b^{4} + 6 \,{\left (9 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} x^{4} - 5 \,{\left (9 \, B b^{4} - 8 \, A b^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{315 \, b^{5} x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^9/(c*x^4+b*x^2)^(1/2),x, algorithm="fricas")

[Out]

1/315*(16*(9*B*b*c^3 - 8*A*c^4)*x^8 - 8*(9*B*b^2*c^2 - 8*A*b*c^3)*x^6 - 35*A*b^4 + 6*(9*B*b^3*c - 8*A*b^2*c^2)
*x^4 - 5*(9*B*b^4 - 8*A*b^3*c)*x^2)*sqrt(c*x^4 + b*x^2)/(b^5*x^10)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x^{2}}{x^{9} \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**9/(c*x**4+b*x**2)**(1/2),x)

[Out]

Integral((A + B*x**2)/(x**9*sqrt(x**2*(b + c*x**2))), x)

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Giac [A]  time = 1.18785, size = 182, normalized size = 1.07 \begin{align*} -\frac{45 \, B b{\left (c + \frac{b}{x^{2}}\right )}^{\frac{7}{2}} + 35 \, A{\left (c + \frac{b}{x^{2}}\right )}^{\frac{9}{2}} - 189 \, B b{\left (c + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} c - 180 \, A{\left (c + \frac{b}{x^{2}}\right )}^{\frac{7}{2}} c + 315 \, B b{\left (c + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} c^{2} + 378 \, A{\left (c + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} c^{2} - 315 \, B b \sqrt{c + \frac{b}{x^{2}}} c^{3} - 420 \, A{\left (c + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} c^{3} + 315 \, A \sqrt{c + \frac{b}{x^{2}}} c^{4}}{315 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^9/(c*x^4+b*x^2)^(1/2),x, algorithm="giac")

[Out]

-1/315*(45*B*b*(c + b/x^2)^(7/2) + 35*A*(c + b/x^2)^(9/2) - 189*B*b*(c + b/x^2)^(5/2)*c - 180*A*(c + b/x^2)^(7
/2)*c + 315*B*b*(c + b/x^2)^(3/2)*c^2 + 378*A*(c + b/x^2)^(5/2)*c^2 - 315*B*b*sqrt(c + b/x^2)*c^3 - 420*A*(c +
b/x^2)^(3/2)*c^3 + 315*A*sqrt(c + b/x^2)*c^4)/b^5