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

Optimal. Leaf size=122 \[ \frac{2 a^{7/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{c} \sqrt{a+c x^4}}+\frac{2}{7} a x \sqrt{a+c x^4}+\frac{1}{7} x \left (a+c x^4\right )^{3/2} \]

[Out]

(2*a*x*Sqrt[a + c*x^4])/7 + (x*(a + c*x^4)^(3/2))/7 + (2*a^(7/4)*(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])/(7*c^(1/4)*Sqrt[a + c*x^4])

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Rubi [A]  time = 0.0779991, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 a^{7/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{c} \sqrt{a+c x^4}}+\frac{2}{7} a x \sqrt{a+c x^4}+\frac{1}{7} x \left (a+c x^4\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)^(3/2),x]

[Out]

(2*a*x*Sqrt[a + c*x^4])/7 + (x*(a + c*x^4)^(3/2))/7 + (2*a^(7/4)*(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])/(7*c^(1/4)*Sqrt[a + c*x^4])

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Rubi in Sympy [A]  time = 7.26833, size = 110, normalized size = 0.9 \[ \frac{2 a^{\frac{7}{4}} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{7 \sqrt [4]{c} \sqrt{a + c x^{4}}} + \frac{2 a x \sqrt{a + c x^{4}}}{7} + \frac{x \left (a + c x^{4}\right )^{\frac{3}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**(7/4)*sqrt((a + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**
2)*elliptic_f(2*atan(c**(1/4)*x/a**(1/4)), 1/2)/(7*c**(1/4)*sqrt(a + c*x**4)) +
2*a*x*sqrt(a + c*x**4)/7 + x*(a + c*x**4)**(3/2)/7

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Mathematica [C]  time = 0.246741, size = 102, normalized size = 0.84 \[ \frac{-\frac{4 i a^2 \sqrt{\frac{c x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}+3 a^2 x+4 a c x^5+c^2 x^9}{7 \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^4)^(3/2),x]

[Out]

(3*a^2*x + 4*a*c*x^5 + c^2*x^9 - ((4*I)*a^2*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcS
inh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[c])/Sqrt[a]])/(7*Sqrt[a + c*
x^4])

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Maple [C]  time = 0.008, size = 103, normalized size = 0.8 \[{\frac{c{x}^{5}}{7}\sqrt{c{x}^{4}+a}}+{\frac{3\,ax}{7}\sqrt{c{x}^{4}+a}}+{\frac{4\,{a}^{2}}{7}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^(3/2),x)

[Out]

1/7*c*x^5*(c*x^4+a)^(1/2)+3/7*a*x*(c*x^4+a)^(1/2)+4/7*a^2/(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. \[ \int{\left (c x^{4} + a\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^4 + a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^4 + a)^(3/2), x)

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Sympy [A]  time = 2.77626, size = 37, normalized size = 0.3 \[ \frac{a^{\frac{3}{2}} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+a)**(3/2),x)

[Out]

a**(3/2)*x*gamma(1/4)*hyper((-3/2, 1/4), (5/4,), c*x**4*exp_polar(I*pi)/a)/(4*ga
mma(5/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^4 + a)^(3/2), x)