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

Optimal. Leaf size=124 \[ \frac{2 a^{3/4} c^{3/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{3 \sqrt{a+c x^4}}-\frac{\left (a+c x^4\right )^{3/2}}{3 x^3}+\frac{2}{3} c x \sqrt{a+c x^4} \]

[Out]

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

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Rubi [A]  time = 0.0300392, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {277, 195, 220} \[ \frac{2 a^{3/4} c^{3/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 \sqrt{a+c x^4}}-\frac{\left (a+c x^4\right )^{3/2}}{3 x^3}+\frac{2}{3} c x \sqrt{a+c x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^(3/2)/x^4,x]

[Out]

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

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 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rubi steps

\begin{align*} \int \frac{\left (a+c x^4\right )^{3/2}}{x^4} \, dx &=-\frac{\left (a+c x^4\right )^{3/2}}{3 x^3}+(2 c) \int \sqrt{a+c x^4} \, dx\\ &=\frac{2}{3} c x \sqrt{a+c x^4}-\frac{\left (a+c x^4\right )^{3/2}}{3 x^3}+\frac{1}{3} (4 a c) \int \frac{1}{\sqrt{a+c x^4}} \, dx\\ &=\frac{2}{3} c x \sqrt{a+c x^4}-\frac{\left (a+c x^4\right )^{3/2}}{3 x^3}+\frac{2 a^{3/4} c^{3/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 \sqrt{a+c x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0085894, size = 52, normalized size = 0.42 \[ -\frac{a \sqrt{a+c x^4} \, _2F_1\left (-\frac{3}{2},-\frac{3}{4};\frac{1}{4};-\frac{c x^4}{a}\right )}{3 x^3 \sqrt{\frac{c x^4}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^(3/2)/x^4,x]

[Out]

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

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Maple [C]  time = 0.01, size = 102, normalized size = 0.8 \begin{align*} -{\frac{a}{3\,{x}^{3}}\sqrt{c{x}^{4}+a}}+{\frac{cx}{3}\sqrt{c{x}^{4}+a}}+{\frac{4\,ac}{3}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\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^4,x)

[Out]

-1/3*a*(c*x^4+a)^(1/2)/x^3+1/3*c*x*(c*x^4+a)^(1/2)+4/3*a*c/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)
^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x^4 + a)^(3/2)/x^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c*x^4 + a)^(3/2)/x^4, x)

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Sympy [C]  time = 1.03264, size = 42, normalized size = 0.34 \begin{align*} \frac{a^{\frac{3}{2}} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{3}{4} \\ \frac{1}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac{1}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**(3/2)*gamma(-3/4)*hyper((-3/2, -3/4), (1/4,), c*x**4*exp_polar(I*pi)/a)/(4*x**3*gamma(1/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x^4 + a)^(3/2)/x^4, x)