3.788 \(\int \frac{(a+c x^4)^{3/2}}{x^9} \, dx\)

Optimal. Leaf size=68 \[ -\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{16 \sqrt{a}}-\frac{3 c \sqrt{a+c x^4}}{16 x^4}-\frac{\left (a+c x^4\right )^{3/2}}{8 x^8} \]

[Out]

(-3*c*Sqrt[a + c*x^4])/(16*x^4) - (a + c*x^4)^(3/2)/(8*x^8) - (3*c^2*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]])/(16*Sqr
t[a])

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Rubi [A]  time = 0.0385098, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 47, 63, 208} \[ -\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{16 \sqrt{a}}-\frac{3 c \sqrt{a+c x^4}}{16 x^4}-\frac{\left (a+c x^4\right )^{3/2}}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^(3/2)/x^9,x]

[Out]

(-3*c*Sqrt[a + c*x^4])/(16*x^4) - (a + c*x^4)^(3/2)/(8*x^8) - (3*c^2*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]])/(16*Sqr
t[a])

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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

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{\left (a+c x^4\right )^{3/2}}{x^9} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{(a+c x)^{3/2}}{x^3} \, dx,x,x^4\right )\\ &=-\frac{\left (a+c x^4\right )^{3/2}}{8 x^8}+\frac{1}{16} (3 c) \operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x^2} \, dx,x,x^4\right )\\ &=-\frac{3 c \sqrt{a+c x^4}}{16 x^4}-\frac{\left (a+c x^4\right )^{3/2}}{8 x^8}+\frac{1}{32} \left (3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^4\right )\\ &=-\frac{3 c \sqrt{a+c x^4}}{16 x^4}-\frac{\left (a+c x^4\right )^{3/2}}{8 x^8}+\frac{1}{16} (3 c) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^4}\right )\\ &=-\frac{3 c \sqrt{a+c x^4}}{16 x^4}-\frac{\left (a+c x^4\right )^{3/2}}{8 x^8}-\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{16 \sqrt{a}}\\ \end{align*}

Mathematica [A]  time = 0.0389661, size = 76, normalized size = 1.12 \[ -\frac{2 a^2+3 c^2 x^8 \sqrt{\frac{c x^4}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{c x^4}{a}+1}\right )+7 a c x^4+5 c^2 x^8}{16 x^8 \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^(3/2)/x^9,x]

[Out]

-(2*a^2 + 7*a*c*x^4 + 5*c^2*x^8 + 3*c^2*x^8*Sqrt[1 + (c*x^4)/a]*ArcTanh[Sqrt[1 + (c*x^4)/a]])/(16*x^8*Sqrt[a +
c*x^4])

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Maple [A]  time = 0.014, size = 63, normalized size = 0.9 \begin{align*} -{\frac{3\,{c}^{2}}{16}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{4}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}-{\frac{a}{8\,{x}^{8}}\sqrt{c{x}^{4}+a}}-{\frac{5\,c}{16\,{x}^{4}}\sqrt{c{x}^{4}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^(3/2)/x^9,x)

[Out]

-3/16*c^2/a^(1/2)*ln((2*a+2*a^(1/2)*(c*x^4+a)^(1/2))/x^2)-1/8*a/x^8*(c*x^4+a)^(1/2)-5/16*c*(c*x^4+a)^(1/2)/x^4

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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^4+a)^(3/2)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.48978, size = 320, normalized size = 4.71 \begin{align*} \left [\frac{3 \, \sqrt{a} c^{2} x^{8} \log \left (\frac{c x^{4} - 2 \, \sqrt{c x^{4} + a} \sqrt{a} + 2 \, a}{x^{4}}\right ) - 2 \,{\left (5 \, a c x^{4} + 2 \, a^{2}\right )} \sqrt{c x^{4} + a}}{32 \, a x^{8}}, \frac{3 \, \sqrt{-a} c^{2} x^{8} \arctan \left (\frac{\sqrt{c x^{4} + a} \sqrt{-a}}{a}\right ) -{\left (5 \, a c x^{4} + 2 \, a^{2}\right )} \sqrt{c x^{4} + a}}{16 \, a x^{8}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^9,x, algorithm="fricas")

[Out]

[1/32*(3*sqrt(a)*c^2*x^8*log((c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(a) + 2*a)/x^4) - 2*(5*a*c*x^4 + 2*a^2)*sqrt(c*x^4
+ a))/(a*x^8), 1/16*(3*sqrt(-a)*c^2*x^8*arctan(sqrt(c*x^4 + a)*sqrt(-a)/a) - (5*a*c*x^4 + 2*a^2)*sqrt(c*x^4 +
a))/(a*x^8)]

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Sympy [A]  time = 3.69391, size = 75, normalized size = 1.1 \begin{align*} - \frac{a \sqrt{c} \sqrt{\frac{a}{c x^{4}} + 1}}{8 x^{6}} - \frac{5 c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{4}} + 1}}{16 x^{2}} - \frac{3 c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x^{2}} \right )}}{16 \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**(3/2)/x**9,x)

[Out]

-a*sqrt(c)*sqrt(a/(c*x**4) + 1)/(8*x**6) - 5*c**(3/2)*sqrt(a/(c*x**4) + 1)/(16*x**2) - 3*c**2*asinh(sqrt(a)/(s
qrt(c)*x**2))/(16*sqrt(a))

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Giac [A]  time = 1.11595, size = 82, normalized size = 1.21 \begin{align*} \frac{1}{16} \, c^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{5 \,{\left (c x^{4} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{c x^{4} + a} a}{c^{2} x^{8}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^9,x, algorithm="giac")

[Out]

1/16*c^2*(3*arctan(sqrt(c*x^4 + a)/sqrt(-a))/sqrt(-a) - (5*(c*x^4 + a)^(3/2) - 3*sqrt(c*x^4 + a)*a)/(c^2*x^8))