3.99 \(\int \frac{(A+B x^2) \sqrt{b x^2+c x^4}}{x^{13}} \, dx\)

Optimal. Leaf size=170 \[ \frac{16 c^3 \left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{3465 b^5 x^6}-\frac{8 c^2 \left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{1155 b^4 x^8}+\frac{2 c \left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{231 b^3 x^{10}}-\frac{\left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{99 b^2 x^{12}}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}} \]

[Out]

-(A*(b*x^2 + c*x^4)^(3/2))/(11*b*x^14) - ((11*b*B - 8*A*c)*(b*x^2 + c*x^4)^(3/2))/(99*b^2*x^12) + (2*c*(11*b*B
- 8*A*c)*(b*x^2 + c*x^4)^(3/2))/(231*b^3*x^10) - (8*c^2*(11*b*B - 8*A*c)*(b*x^2 + c*x^4)^(3/2))/(1155*b^4*x^8
) + (16*c^3*(11*b*B - 8*A*c)*(b*x^2 + c*x^4)^(3/2))/(3465*b^5*x^6)

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Rubi [A]  time = 0.300112, 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} \[ \frac{16 c^3 \left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{3465 b^5 x^6}-\frac{8 c^2 \left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{1155 b^4 x^8}+\frac{2 c \left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{231 b^3 x^{10}}-\frac{\left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{99 b^2 x^{12}}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^13,x]

[Out]

-(A*(b*x^2 + c*x^4)^(3/2))/(11*b*x^14) - ((11*b*B - 8*A*c)*(b*x^2 + c*x^4)^(3/2))/(99*b^2*x^12) + (2*c*(11*b*B
- 8*A*c)*(b*x^2 + c*x^4)^(3/2))/(231*b^3*x^10) - (8*c^2*(11*b*B - 8*A*c)*(b*x^2 + c*x^4)^(3/2))/(1155*b^4*x^8
) + (16*c^3*(11*b*B - 8*A*c)*(b*x^2 + c*x^4)^(3/2))/(3465*b^5*x^6)

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{\left (A+B x^2\right ) \sqrt{b x^2+c x^4}}{x^{13}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(A+B x) \sqrt{b x+c x^2}}{x^7} \, dx,x,x^2\right )\\ &=-\frac{A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}+\frac{\left (-7 (-b B+A c)+\frac{3}{2} (-b B+2 A c)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b x+c x^2}}{x^6} \, dx,x,x^2\right )}{11 b}\\ &=-\frac{A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}-\frac{(11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}-\frac{(c (11 b B-8 A c)) \operatorname{Subst}\left (\int \frac{\sqrt{b x+c x^2}}{x^5} \, dx,x,x^2\right )}{33 b^2}\\ &=-\frac{A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}-\frac{(11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}+\frac{2 c (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}+\frac{\left (4 c^2 (11 b B-8 A c)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b x+c x^2}}{x^4} \, dx,x,x^2\right )}{231 b^3}\\ &=-\frac{A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}-\frac{(11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}+\frac{2 c (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}-\frac{8 c^2 (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}-\frac{\left (8 c^3 (11 b B-8 A c)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b x+c x^2}}{x^3} \, dx,x,x^2\right )}{1155 b^4}\\ &=-\frac{A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}-\frac{(11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}+\frac{2 c (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}-\frac{8 c^2 (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}+\frac{16 c^3 (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{3465 b^5 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0684124, size = 94, normalized size = 0.55 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (x^2 \left (\frac{c x^2}{b}+1\right ) \left (-30 b^2 c x^2+35 b^3+24 b c^2 x^4-16 c^3 x^6\right ) (8 A c-11 b B)-315 A b^3 \left (b+c x^2\right )\right )}{3465 b^4 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^13,x]

[Out]

(Sqrt[x^2*(b + c*x^2)]*(-315*A*b^3*(b + c*x^2) + (-11*b*B + 8*A*c)*x^2*(1 + (c*x^2)/b)*(35*b^3 - 30*b^2*c*x^2
+ 24*b*c^2*x^4 - 16*c^3*x^6)))/(3465*b^4*x^12)

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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}-176\,Bb{c}^{3}{x}^{8}-192\,Ab{c}^{3}{x}^{6}+264\,B{b}^{2}{c}^{2}{x}^{6}+240\,A{b}^{2}{c}^{2}{x}^{4}-330\,B{b}^{3}c{x}^{4}-280\,A{b}^{3}c{x}^{2}+385\,B{b}^{4}{x}^{2}+315\,A{b}^{4} \right ) }{3465\,{x}^{12}{b}^{5}}\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)*(c*x^4+b*x^2)^(1/2)/x^13,x)

[Out]

-1/3465*(c*x^2+b)*(128*A*c^4*x^8-176*B*b*c^3*x^8-192*A*b*c^3*x^6+264*B*b^2*c^2*x^6+240*A*b^2*c^2*x^4-330*B*b^3
*c*x^4-280*A*b^3*c*x^2+385*B*b^4*x^2+315*A*b^4)*(c*x^4+b*x^2)^(1/2)/x^12/b^5

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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)*(c*x^4+b*x^2)^(1/2)/x^13,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.78683, size = 298, normalized size = 1.75 \begin{align*} \frac{{\left (16 \,{\left (11 \, B b c^{4} - 8 \, A c^{5}\right )} x^{10} - 8 \,{\left (11 \, B b^{2} c^{3} - 8 \, A b c^{4}\right )} x^{8} + 6 \,{\left (11 \, B b^{3} c^{2} - 8 \, A b^{2} c^{3}\right )} x^{6} - 315 \, A b^{5} - 5 \,{\left (11 \, B b^{4} c - 8 \, A b^{3} c^{2}\right )} x^{4} - 35 \,{\left (11 \, B b^{5} + A b^{4} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{3465 \, b^{5} x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2)^(1/2)/x^13,x, algorithm="fricas")

[Out]

1/3465*(16*(11*B*b*c^4 - 8*A*c^5)*x^10 - 8*(11*B*b^2*c^3 - 8*A*b*c^4)*x^8 + 6*(11*B*b^3*c^2 - 8*A*b^2*c^3)*x^6
- 315*A*b^5 - 5*(11*B*b^4*c - 8*A*b^3*c^2)*x^4 - 35*(11*B*b^5 + A*b^4*c)*x^2)*sqrt(c*x^4 + b*x^2)/(b^5*x^12)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )}{x^{13}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)*(c*x**4+b*x**2)**(1/2)/x**13,x)

[Out]

Integral(sqrt(x**2*(b + c*x**2))*(A + B*x**2)/x**13, x)

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Giac [B]  time = 4.42151, size = 581, normalized size = 3.42 \begin{align*} \frac{32 \,{\left (3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{14} B c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) - 4851 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{12} B b c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 11088 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{12} A c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) + 231 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{10} B b^{2} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 7392 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{10} A b c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) - 165 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} B b^{3} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 2640 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} A b^{2} c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) + 1815 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} B b^{4} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) - 1320 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} A b^{3} c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) - 605 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} B b^{5} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 440 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} A b^{4} c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) + 121 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} B b^{6} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) - 88 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} A b^{5} c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) - 11 \, B b^{7} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 8 \, A b^{6} c^{\frac{11}{2}} \mathrm{sgn}\left (x\right )\right )}}{3465 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2)^(1/2)/x^13,x, algorithm="giac")

[Out]

32/3465*(3465*(sqrt(c)*x - sqrt(c*x^2 + b))^14*B*c^(9/2)*sgn(x) - 4851*(sqrt(c)*x - sqrt(c*x^2 + b))^12*B*b*c^
(9/2)*sgn(x) + 11088*(sqrt(c)*x - sqrt(c*x^2 + b))^12*A*c^(11/2)*sgn(x) + 231*(sqrt(c)*x - sqrt(c*x^2 + b))^10
*B*b^2*c^(9/2)*sgn(x) + 7392*(sqrt(c)*x - sqrt(c*x^2 + b))^10*A*b*c^(11/2)*sgn(x) - 165*(sqrt(c)*x - sqrt(c*x^
2 + b))^8*B*b^3*c^(9/2)*sgn(x) + 2640*(sqrt(c)*x - sqrt(c*x^2 + b))^8*A*b^2*c^(11/2)*sgn(x) + 1815*(sqrt(c)*x
- sqrt(c*x^2 + b))^6*B*b^4*c^(9/2)*sgn(x) - 1320*(sqrt(c)*x - sqrt(c*x^2 + b))^6*A*b^3*c^(11/2)*sgn(x) - 605*(
sqrt(c)*x - sqrt(c*x^2 + b))^4*B*b^5*c^(9/2)*sgn(x) + 440*(sqrt(c)*x - sqrt(c*x^2 + b))^4*A*b^4*c^(11/2)*sgn(x
) + 121*(sqrt(c)*x - sqrt(c*x^2 + b))^2*B*b^6*c^(9/2)*sgn(x) - 88*(sqrt(c)*x - sqrt(c*x^2 + b))^2*A*b^5*c^(11/
2)*sgn(x) - 11*B*b^7*c^(9/2)*sgn(x) + 8*A*b^6*c^(11/2)*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + b))^2 - b)^11