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

Optimal. Leaf size=69 \[ \frac{3}{4} c x^2 \sqrt{a+c x^4}-\frac{\left (a+c x^4\right )^{3/2}}{2 x^2}+\frac{3}{4} a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0389532, antiderivative size = 69, 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 = {275, 277, 195, 217, 206} \[ \frac{3}{4} c x^2 \sqrt{a+c x^4}-\frac{\left (a+c x^4\right )^{3/2}}{2 x^2}+\frac{3}{4} a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^(3/2)/x^3,x]

[Out]

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

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 277

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [C]  time = 0.0115287, size = 52, normalized size = 0.75 \[ -\frac{a \sqrt{a+c x^4} \, _2F_1\left (-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{c x^4}{a}\right )}{2 x^2 \sqrt{\frac{c x^4}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^(3/2)/x^3,x]

[Out]

-(a*Sqrt[a + c*x^4]*Hypergeometric2F1[-3/2, -1/2, 1/2, -((c*x^4)/a)])/(2*x^2*Sqrt[1 + (c*x^4)/a])

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Maple [A]  time = 0.014, size = 56, normalized size = 0.8 \begin{align*}{\frac{c{x}^{2}}{4}\sqrt{c{x}^{4}+a}}+{\frac{3\,a}{4}\sqrt{c}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ) }-{\frac{a}{2\,{x}^{2}}\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^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.59299, size = 288, normalized size = 4.17 \begin{align*} \left [\frac{3 \, a \sqrt{c} x^{2} \log \left (-2 \, c x^{4} - 2 \, \sqrt{c x^{4} + a} \sqrt{c} x^{2} - a\right ) + 2 \, \sqrt{c x^{4} + a}{\left (c x^{4} - 2 \, a\right )}}{8 \, x^{2}}, -\frac{3 \, a \sqrt{-c} x^{2} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + a}}\right ) - \sqrt{c x^{4} + a}{\left (c x^{4} - 2 \, a\right )}}{4 \, x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.13113, size = 95, normalized size = 1.38 \begin{align*} - \frac{a^{\frac{3}{2}}}{2 x^{2} \sqrt{1 + \frac{c x^{4}}{a}}} - \frac{\sqrt{a} c x^{2}}{4 \sqrt{1 + \frac{c x^{4}}{a}}} + \frac{3 a \sqrt{c} \operatorname{asinh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4} + \frac{c^{2} x^{6}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{4}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19419, size = 72, normalized size = 1.04 \begin{align*} \frac{1}{4} \, \sqrt{c x^{4} + a} c x^{2} - \frac{3 \, a c \arctan \left (\frac{\sqrt{c + \frac{a}{x^{4}}}}{\sqrt{-c}}\right )}{4 \, \sqrt{-c}} - \frac{1}{2} \, a \sqrt{c + \frac{a}{x^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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