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

Optimal. Leaf size=136 \[ -\frac{128 c^4 \left (b x^2+c x^4\right )^{3/2}}{3465 b^5 x^6}+\frac{64 c^3 \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}-\frac{16 c^2 \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}+\frac{8 c \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}-\frac{\left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}} \]

[Out]

-(b*x^2 + c*x^4)^(3/2)/(11*b*x^14) + (8*c*(b*x^2 + c*x^4)^(3/2))/(99*b^2*x^12) - (16*c^2*(b*x^2 + c*x^4)^(3/2)
)/(231*b^3*x^10) + (64*c^3*(b*x^2 + c*x^4)^(3/2))/(1155*b^4*x^8) - (128*c^4*(b*x^2 + c*x^4)^(3/2))/(3465*b^5*x
^6)

________________________________________________________________________________________

Rubi [A]  time = 0.21401, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2016, 2014} \[ -\frac{128 c^4 \left (b x^2+c x^4\right )^{3/2}}{3465 b^5 x^6}+\frac{64 c^3 \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}-\frac{16 c^2 \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}+\frac{8 c \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}-\frac{\left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x^2 + c*x^4]/x^13,x]

[Out]

-(b*x^2 + c*x^4)^(3/2)/(11*b*x^14) + (8*c*(b*x^2 + c*x^4)^(3/2))/(99*b^2*x^12) - (16*c^2*(b*x^2 + c*x^4)^(3/2)
)/(231*b^3*x^10) + (64*c^3*(b*x^2 + c*x^4)^(3/2))/(1155*b^4*x^8) - (128*c^4*(b*x^2 + c*x^4)^(3/2))/(3465*b^5*x
^6)

Rule 2016

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{b x^2+c x^4}}{x^{13}} \, dx &=-\frac{\left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}-\frac{(8 c) \int \frac{\sqrt{b x^2+c x^4}}{x^{11}} \, dx}{11 b}\\ &=-\frac{\left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}+\frac{8 c \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}+\frac{\left (16 c^2\right ) \int \frac{\sqrt{b x^2+c x^4}}{x^9} \, dx}{33 b^2}\\ &=-\frac{\left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}+\frac{8 c \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}-\frac{16 c^2 \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}-\frac{\left (64 c^3\right ) \int \frac{\sqrt{b x^2+c x^4}}{x^7} \, dx}{231 b^3}\\ &=-\frac{\left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}+\frac{8 c \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}-\frac{16 c^2 \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}+\frac{64 c^3 \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}+\frac{\left (128 c^4\right ) \int \frac{\sqrt{b x^2+c x^4}}{x^5} \, dx}{1155 b^4}\\ &=-\frac{\left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}+\frac{8 c \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}-\frac{16 c^2 \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}+\frac{64 c^3 \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}-\frac{128 c^4 \left (b x^2+c x^4\right )^{3/2}}{3465 b^5 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0152996, size = 68, normalized size = 0.5 \[ -\frac{\left (x^2 \left (b+c x^2\right )\right )^{3/2} \left (240 b^2 c^2 x^4-280 b^3 c x^2+315 b^4-192 b c^3 x^6+128 c^4 x^8\right )}{3465 b^5 x^{14}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x^2 + c*x^4]/x^13,x]

[Out]

-((x^2*(b + c*x^2))^(3/2)*(315*b^4 - 280*b^3*c*x^2 + 240*b^2*c^2*x^4 - 192*b*c^3*x^6 + 128*c^4*x^8))/(3465*b^5
*x^14)

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 72, normalized size = 0.5 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( 128\,{c}^{4}{x}^{8}-192\,{c}^{3}{x}^{6}b+240\,{c}^{2}{x}^{4}{b}^{2}-280\,c{x}^{2}{b}^{3}+315\,{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((c*x^4+b*x^2)^(1/2)/x^13,x)

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.67381, size = 174, normalized size = 1.28 \begin{align*} -\frac{{\left (128 \, c^{5} x^{10} - 64 \, b c^{4} x^{8} + 48 \, b^{2} c^{3} x^{6} - 40 \, b^{3} c^{2} x^{4} + 35 \, b^{4} c x^{2} + 315 \, b^{5}\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((c*x^4+b*x^2)^(1/2)/x^13,x, algorithm="fricas")

[Out]

-1/3465*(128*c^5*x^10 - 64*b*c^4*x^8 + 48*b^2*c^3*x^6 - 40*b^3*c^2*x^4 + 35*b^4*c*x^2 + 315*b^5)*sqrt(c*x^4 +
b*x^2)/(b^5*x^12)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.35179, size = 278, normalized size = 2.04 \begin{align*} \frac{256 \,{\left (1386 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{12} c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) + 924 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{10} b c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) + 330 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} b^{2} c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) - 165 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} b^{3} c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) + 55 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} b^{4} c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) - 11 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} b^{5} c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) + 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((c*x^4+b*x^2)^(1/2)/x^13,x, algorithm="giac")

[Out]

256/3465*(1386*(sqrt(c)*x - sqrt(c*x^2 + b))^12*c^(11/2)*sgn(x) + 924*(sqrt(c)*x - sqrt(c*x^2 + b))^10*b*c^(11
/2)*sgn(x) + 330*(sqrt(c)*x - sqrt(c*x^2 + b))^8*b^2*c^(11/2)*sgn(x) - 165*(sqrt(c)*x - sqrt(c*x^2 + b))^6*b^3
*c^(11/2)*sgn(x) + 55*(sqrt(c)*x - sqrt(c*x^2 + b))^4*b^4*c^(11/2)*sgn(x) - 11*(sqrt(c)*x - sqrt(c*x^2 + b))^2
*b^5*c^(11/2)*sgn(x) + b^6*c^(11/2)*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + b))^2 - b)^11