3.787 \(\int \frac{(a+c x^4)^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=63 \[ -\frac{\left (a+c x^4\right )^{3/2}}{4 x^4}+\frac{3}{4} c \sqrt{a+c x^4}-\frac{3}{4} \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0377259, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {266, 47, 50, 63, 208} \[ -\frac{\left (a+c x^4\right )^{3/2}}{4 x^4}+\frac{3}{4} c \sqrt{a+c x^4}-\frac{3}{4} \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^(3/2)/x^5,x]

[Out]

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

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 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^5} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{(a+c x)^{3/2}}{x^2} \, dx,x,x^4\right )\\ &=-\frac{\left (a+c x^4\right )^{3/2}}{4 x^4}+\frac{1}{8} (3 c) \operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x} \, dx,x,x^4\right )\\ &=\frac{3}{4} c \sqrt{a+c x^4}-\frac{\left (a+c x^4\right )^{3/2}}{4 x^4}+\frac{1}{8} (3 a c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^4\right )\\ &=\frac{3}{4} c \sqrt{a+c x^4}-\frac{\left (a+c x^4\right )^{3/2}}{4 x^4}+\frac{1}{4} (3 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^4}\right )\\ &=\frac{3}{4} c \sqrt{a+c x^4}-\frac{\left (a+c x^4\right )^{3/2}}{4 x^4}-\frac{3}{4} \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )\\ \end{align*}

Mathematica [C]  time = 0.0084739, size = 37, normalized size = 0.59 \[ \frac{c \left (a+c x^4\right )^{5/2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{c x^4}{a}+1\right )}{10 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^(3/2)/x^5,x]

[Out]

(c*(a + c*x^4)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 + (c*x^4)/a])/(10*a^2)

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Maple [A]  time = 0.015, size = 58, normalized size = 0.9 \begin{align*}{\frac{c}{2}\sqrt{c{x}^{4}+a}}-{\frac{3\,c}{4}\sqrt{a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{4}+a} \right ) } \right ) }-{\frac{a}{4\,{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^5,x)

[Out]

1/2*c*(c*x^4+a)^(1/2)-3/4*a^(1/2)*c*ln((2*a+2*a^(1/2)*(c*x^4+a)^(1/2))/x^2)-1/4*a/x^4*(c*x^4+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^4+a)^(3/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.55509, size = 285, normalized size = 4.52 \begin{align*} \left [\frac{3 \, \sqrt{a} c x^{4} \log \left (\frac{c x^{4} - 2 \, \sqrt{c x^{4} + a} \sqrt{a} + 2 \, a}{x^{4}}\right ) + 2 \,{\left (2 \, c x^{4} - a\right )} \sqrt{c x^{4} + a}}{8 \, x^{4}}, \frac{3 \, \sqrt{-a} c x^{4} \arctan \left (\frac{\sqrt{c x^{4} + a} \sqrt{-a}}{a}\right ) +{\left (2 \, c x^{4} - a\right )} \sqrt{c x^{4} + a}}{4 \, x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.75046, size = 95, normalized size = 1.51 \begin{align*} - \frac{3 \sqrt{a} c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x^{2}} \right )}}{4} - \frac{a^{2}}{4 \sqrt{c} x^{6} \sqrt{\frac{a}{c x^{4}} + 1}} + \frac{a \sqrt{c}}{4 x^{2} \sqrt{\frac{a}{c x^{4}} + 1}} + \frac{c^{\frac{3}{2}} x^{2}}{2 \sqrt{\frac{a}{c x^{4}} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**(3/2)/x**5,x)

[Out]

-3*sqrt(a)*c*asinh(sqrt(a)/(sqrt(c)*x**2))/4 - a**2/(4*sqrt(c)*x**6*sqrt(a/(c*x**4) + 1)) + a*sqrt(c)/(4*x**2*
sqrt(a/(c*x**4) + 1)) + c**(3/2)*x**2/(2*sqrt(a/(c*x**4) + 1))

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Giac [A]  time = 1.10319, size = 77, normalized size = 1.22 \begin{align*} \frac{1}{4} \,{\left (\frac{3 \, a \arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \, \sqrt{c x^{4} + a} - \frac{\sqrt{c x^{4} + a} a}{c x^{4}}\right )} c \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^5,x, algorithm="giac")

[Out]

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