3.950 \(\int \frac{(c x)^{5/2}}{\sqrt [4]{a+b x^2}} \, dx\)

Optimal. Leaf size=125 \[ -\frac{a^{3/2} c^2 \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 b^{3/2} \sqrt [4]{a+b x^2}}-\frac{a c^2 x \sqrt{c x}}{2 b \sqrt [4]{a+b x^2}}+\frac{c (c x)^{3/2} \left (a+b x^2\right )^{3/4}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.153705, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{a^{3/2} c^2 \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 b^{3/2} \sqrt [4]{a+b x^2}}-\frac{a c^2 x \sqrt{c x}}{2 b \sqrt [4]{a+b x^2}}+\frac{c (c x)^{3/2} \left (a+b x^2\right )^{3/4}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(5/2)/(a + b*x^2)^(1/4),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} c^{2} \sqrt{c x} \sqrt [4]{\frac{a}{b x^{2}} + 1} \int ^{\frac{1}{x}} \frac{1}{\sqrt [4]{\frac{a x^{2}}{b} + 1}}\, dx}{4 b^{2} \sqrt [4]{a + b x^{2}}} - \frac{a^{2} c^{2} \sqrt{c x}}{2 b^{2} x \sqrt [4]{a + b x^{2}}} - \frac{a c^{2} x \sqrt{c x}}{2 b \sqrt [4]{a + b x^{2}}} + \frac{c \left (c x\right )^{\frac{3}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(5/2)/(b*x**2+a)**(1/4),x)

[Out]

a**2*c**2*sqrt(c*x)*(a/(b*x**2) + 1)**(1/4)*Integral((a*x**2/b + 1)**(-1/4), (x,
1/x))/(4*b**2*(a + b*x**2)**(1/4)) - a**2*c**2*sqrt(c*x)/(2*b**2*x*(a + b*x**2)
**(1/4)) - a*c**2*x*sqrt(c*x)/(2*b*(a + b*x**2)**(1/4)) + c*(c*x)**(3/2)*(a + b*
x**2)**(3/4)/(3*b)

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Mathematica [C]  time = 0.0634539, size = 69, normalized size = 0.55 \[ \frac{c (c x)^{3/2} \left (-a \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )+a+b x^2\right )}{3 b \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(5/2)/(a + b*x^2)^(1/4),x]

[Out]

(c*(c*x)^(3/2)*(a + b*x^2 - a*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4,
7/4, -((b*x^2)/a)]))/(3*b*(a + b*x^2)^(1/4))

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(5/2)/(b*x^2+a)^(1/4),x)

[Out]

int((c*x)^(5/2)/(b*x^2+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(5/2)/(b*x^2 + a)^(1/4),x, algorithm="maxima")

[Out]

integrate((c*x)^(5/2)/(b*x^2 + a)^(1/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(5/2)/(b*x^2 + a)^(1/4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 150.86, size = 44, normalized size = 0.35 \[ \frac{c^{\frac{5}{2}} x^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt [4]{a} \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)**(5/2)/(b*x**2+a)**(1/4),x)

[Out]

c**(5/2)*x**(7/2)*gamma(7/4)*hyper((1/4, 7/4), (11/4,), b*x**2*exp_polar(I*pi)/a
)/(2*a**(1/4)*gamma(11/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(5/2)/(b*x^2 + a)^(1/4),x, algorithm="giac")

[Out]

integrate((c*x)^(5/2)/(b*x^2 + a)^(1/4), x)