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

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

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

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Rubi [A]  time = 0.0849027, 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 \[ \frac{2 a^{3/4} c^{3/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 \sqrt{a+c x^4}}+\frac{2}{3} c x \sqrt{a+c x^4}-\frac{\left (a+c x^4\right )^{3/2}}{3 x^3} \]

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

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Rubi in Sympy [A]  time = 8.85012, size = 112, normalized size = 0.9 \[ \frac{2 a^{\frac{3}{4}} c^{\frac{3}{4}} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt{a + c x^{4}}} + \frac{2 c x \sqrt{a + c x^{4}}}{3} - \frac{\left (a + c x^{4}\right )^{\frac{3}{2}}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+a)**(3/2)/x**4,x)

[Out]

2*a**(3/4)*c**(3/4)*sqrt((a + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sq
rt(c)*x**2)*elliptic_f(2*atan(c**(1/4)*x/a**(1/4)), 1/2)/(3*sqrt(a + c*x**4)) +
2*c*x*sqrt(a + c*x**4)/3 - (a + c*x**4)**(3/2)/(3*x**3)

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Mathematica [C]  time = 0.210883, size = 96, normalized size = 0.77 \[ \frac{-\frac{a^2}{x^3}-\frac{4 i a c \sqrt{\frac{c x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}+c^2 x^5}{3 \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^4)^(3/2)/x^4,x]

[Out]

(-(a^2/x^3) + c^2*x^5 - ((4*I)*a*c*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[
(I*Sqrt[c])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[c])/Sqrt[a]])/(3*Sqrt[a + c*x^4])

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Maple [C]  time = 0.017, size = 102, normalized size = 0.8 \[ -{\frac{a}{3\,{x}^{3}}\sqrt{c{x}^{4}+a}}+{\frac{cx}{3}\sqrt{c{x}^{4}+a}}+{\frac{4\,ac}{3}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \]

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. \[ \int \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{x^{4}}\,{d x} \]

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. \[{\rm integral}\left (\frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{x^{4}}, x\right ) \]

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 [A]  time = 3.15984, size = 42, normalized size = 0.34 \[ \frac{a^{\frac{3}{2}} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{3}{4} \\ \frac{1}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac{1}{4}\right )} \]

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

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)