3.920 \(\int (c x)^{3/2} \sqrt [4]{a+b x^2} \, dx\)

Optimal. Leaf size=118 \[ \frac{a^{3/2} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{6 \sqrt{b} \left (a+b x^2\right )^{3/4}}+\frac{(c x)^{5/2} \sqrt [4]{a+b x^2}}{3 c}+\frac{a c \sqrt{c x} \sqrt [4]{a+b x^2}}{6 b} \]

[Out]

(a*c*Sqrt[c*x]*(a + b*x^2)^(1/4))/(6*b) + ((c*x)^(5/2)*(a + b*x^2)^(1/4))/(3*c)
+ (a^(3/2)*(1 + a/(b*x^2))^(3/4)*(c*x)^(3/2)*EllipticF[ArcCot[(Sqrt[b]*x)/Sqrt[a
]]/2, 2])/(6*Sqrt[b]*(a + b*x^2)^(3/4))

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Rubi [A]  time = 0.242395, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ \frac{a^{3/2} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{6 \sqrt{b} \left (a+b x^2\right )^{3/4}}+\frac{(c x)^{5/2} \sqrt [4]{a+b x^2}}{3 c}+\frac{a c \sqrt{c x} \sqrt [4]{a+b x^2}}{6 b} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(3/2)*(a + b*x^2)^(1/4),x]

[Out]

(a*c*Sqrt[c*x]*(a + b*x^2)^(1/4))/(6*b) + ((c*x)^(5/2)*(a + b*x^2)^(1/4))/(3*c)
+ (a^(3/2)*(1 + a/(b*x^2))^(3/4)*(c*x)^(3/2)*EllipticF[ArcCot[(Sqrt[b]*x)/Sqrt[a
]]/2, 2])/(6*Sqrt[b]*(a + b*x^2)^(3/4))

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Rubi in Sympy [A]  time = 26.628, size = 100, normalized size = 0.85 \[ \frac{a^{\frac{3}{2}} \left (c x\right )^{\frac{3}{2}} \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{6 \sqrt{b} \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{a c \sqrt{c x} \sqrt [4]{a + b x^{2}}}{6 b} + \frac{\left (c x\right )^{\frac{5}{2}} \sqrt [4]{a + b x^{2}}}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(3/2)*(b*x**2+a)**(1/4),x)

[Out]

a**(3/2)*(c*x)**(3/2)*(a/(b*x**2) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x
))/2, 2)/(6*sqrt(b)*(a + b*x**2)**(3/4)) + a*c*sqrt(c*x)*(a + b*x**2)**(1/4)/(6*
b) + (c*x)**(5/2)*(a + b*x**2)**(1/4)/(3*c)

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Mathematica [C]  time = 0.0535818, size = 83, normalized size = 0.7 \[ \frac{c \sqrt{c x} \left (-a^2 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+a^2+3 a b x^2+2 b^2 x^4\right )}{6 b \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(3/2)*(a + b*x^2)^(1/4),x]

[Out]

(c*Sqrt[c*x]*(a^2 + 3*a*b*x^2 + 2*b^2*x^4 - a^2*(1 + (b*x^2)/a)^(3/4)*Hypergeome
tric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)]))/(6*b*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{{\frac{3}{2}}}\sqrt [4]{b{x}^{2}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(3/2)*(b*x^2+a)^(1/4),x)

[Out]

int((c*x)^(3/2)*(b*x^2+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/4)*(c*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(1/4)*(c*x)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x} c x, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/4)*(c*x)^(3/2),x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(1/4)*sqrt(c*x)*c*x, x)

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Sympy [A]  time = 22.9869, size = 46, normalized size = 0.39 \[ \frac{\sqrt [4]{a} c^{\frac{3}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)**(3/2)*(b*x**2+a)**(1/4),x)

[Out]

a**(1/4)*c**(3/2)*x**(5/2)*gamma(5/4)*hyper((-1/4, 5/4), (9/4,), b*x**2*exp_pola
r(I*pi)/a)/(2*gamma(9/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/4)*(c*x)^(3/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(1/4)*(c*x)^(3/2), x)