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

Optimal. Leaf size=117 \[ -\frac{a c^{3/2} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{5/4}}-\frac{a c^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{5/4}}+\frac{c \sqrt{c x} \left (a+b x^2\right )^{3/4}}{2 b} \]

[Out]

(c*Sqrt[c*x]*(a + b*x^2)^(3/4))/(2*b) - (a*c^(3/2)*ArcTan[(b^(1/4)*Sqrt[c*x])/(S
qrt[c]*(a + b*x^2)^(1/4))])/(4*b^(5/4)) - (a*c^(3/2)*ArcTanh[(b^(1/4)*Sqrt[c*x])
/(Sqrt[c]*(a + b*x^2)^(1/4))])/(4*b^(5/4))

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Rubi [A]  time = 0.160331, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{a c^{3/2} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{5/4}}-\frac{a c^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{5/4}}+\frac{c \sqrt{c x} \left (a+b x^2\right )^{3/4}}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

(c*Sqrt[c*x]*(a + b*x^2)^(3/4))/(2*b) - (a*c^(3/2)*ArcTan[(b^(1/4)*Sqrt[c*x])/(S
qrt[c]*(a + b*x^2)^(1/4))])/(4*b^(5/4)) - (a*c^(3/2)*ArcTanh[(b^(1/4)*Sqrt[c*x])
/(Sqrt[c]*(a + b*x^2)^(1/4))])/(4*b^(5/4))

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Rubi in Sympy [A]  time = 20.8795, size = 105, normalized size = 0.9 \[ - \frac{a c^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a + b x^{2}}} \right )}}{4 b^{\frac{5}{4}}} - \frac{a c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a + b x^{2}}} \right )}}{4 b^{\frac{5}{4}}} + \frac{c \sqrt{c x} \left (a + b x^{2}\right )^{\frac{3}{4}}}{2 b} \]

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*c**(3/2)*atan(b**(1/4)*sqrt(c*x)/(sqrt(c)*(a + b*x**2)**(1/4)))/(4*b**(5/4))
- a*c**(3/2)*atanh(b**(1/4)*sqrt(c*x)/(sqrt(c)*(a + b*x**2)**(1/4)))/(4*b**(5/4)
) + c*sqrt(c*x)*(a + b*x**2)**(3/4)/(2*b)

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Mathematica [C]  time = 0.0612729, size = 69, normalized size = 0.59 \[ \frac{c \sqrt{c x} \left (-a \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+a+b x^2\right )}{2 b \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(c*Sqrt[c*x]*(a + b*x^2 - a*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/
4, -((b*x^2)/a)]))/(2*b*(a + b*x^2)^(1/4))

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}}\, dx \]

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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240898, size = 382, normalized size = 3.26 \[ \frac{4 \,{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x} c + 4 \, \left (\frac{a^{4} c^{6}}{b^{5}}\right )^{\frac{1}{4}} b \arctan \left (\frac{\left (\frac{a^{4} c^{6}}{b^{5}}\right )^{\frac{1}{4}}{\left (b^{2} x^{2} + a b\right )}}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x} a c +{\left (b x^{2} + a\right )} \sqrt{\frac{\sqrt{b x^{2} + a} a^{2} c^{3} x + \sqrt{\frac{a^{4} c^{6}}{b^{5}}}{\left (b^{3} x^{2} + a b^{2}\right )}}{b x^{2} + a}}}\right ) - \left (\frac{a^{4} c^{6}}{b^{5}}\right )^{\frac{1}{4}} b \log \left (\frac{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x} a c + \left (\frac{a^{4} c^{6}}{b^{5}}\right )^{\frac{1}{4}}{\left (b^{2} x^{2} + a b\right )}}{b x^{2} + a}\right ) + \left (\frac{a^{4} c^{6}}{b^{5}}\right )^{\frac{1}{4}} b \log \left (\frac{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x} a c - \left (\frac{a^{4} c^{6}}{b^{5}}\right )^{\frac{1}{4}}{\left (b^{2} x^{2} + a b\right )}}{b x^{2} + a}\right )}{8 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(4*(b*x^2 + a)^(3/4)*sqrt(c*x)*c + 4*(a^4*c^6/b^5)^(1/4)*b*arctan((a^4*c^6/b
^5)^(1/4)*(b^2*x^2 + a*b)/((b*x^2 + a)^(3/4)*sqrt(c*x)*a*c + (b*x^2 + a)*sqrt((s
qrt(b*x^2 + a)*a^2*c^3*x + sqrt(a^4*c^6/b^5)*(b^3*x^2 + a*b^2))/(b*x^2 + a)))) -
(a^4*c^6/b^5)^(1/4)*b*log(((b*x^2 + a)^(3/4)*sqrt(c*x)*a*c + (a^4*c^6/b^5)^(1/4
)*(b^2*x^2 + a*b))/(b*x^2 + a)) + (a^4*c^6/b^5)^(1/4)*b*log(((b*x^2 + a)^(3/4)*s
qrt(c*x)*a*c - (a^4*c^6/b^5)^(1/4)*(b^2*x^2 + a*b))/(b*x^2 + a)))/b

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Sympy [A]  time = 13.9146, size = 44, normalized size = 0.38 \[ \frac{c^{\frac{3}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt [4]{a} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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