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

Optimal. Leaf size=116 \[ -\frac{a \sqrt{c} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{3/4}}+\frac{a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{3/4}}+\frac{(c x)^{3/2} \sqrt [4]{a+b x^2}}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.194926, antiderivative size = 116, 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 \[ -\frac{a \sqrt{c} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{3/4}}+\frac{a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{3/4}}+\frac{(c x)^{3/2} \sqrt [4]{a+b x^2}}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.2935, size = 104, normalized size = 0.9 \[ - \frac{a \sqrt{c} \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a + b x^{2}}} \right )}}{4 b^{\frac{3}{4}}} + \frac{a \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a + b x^{2}}} \right )}}{4 b^{\frac{3}{4}}} + \frac{\left (c x\right )^{\frac{3}{2}} \sqrt [4]{a + b x^{2}}}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0489449, size = 68, normalized size = 0.59 \[ \frac{x \sqrt{c x} \left (a \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )+3 \left (a+b x^2\right )\right )}{6 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[c*x]*(3*(a + b*x^2) + a*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[3/4, 3/4
, 7/4, -((b*x^2)/a)]))/(6*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int \sqrt{cx}\sqrt [4]{b{x}^{2}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((c*x)^(1/2)*(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((b*x^2 + a)^(1/4)*sqrt(c*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [A]  time = 9.16064, size = 46, normalized size = 0.4 \[ \frac{\sqrt [4]{a} \sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/4)*sqrt(c)*x**(3/2)*gamma(3/4)*hyper((-1/4, 3/4), (7/4,), b*x**2*exp_polar
(I*pi)/a)/(2*gamma(7/4))

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GIAC/XCAS [A]  time = 0.261059, size = 498, normalized size = 4.29 \[ \frac{1}{16} \, a c^{2}{\left (\frac{8 \,{\left (b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} x^{2} \sqrt{{\left | c \right |}}}{\sqrt{c x} a c^{2}} + \frac{2 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \sqrt{{\left | c \right |}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} \sqrt{{\left | c \right |}} + \frac{2 \,{\left (b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \sqrt{{\left | c \right |}}}{\sqrt{c x}}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}} \sqrt{{\left | c \right |}}}\right )}{b c^{2}} + \frac{2 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \sqrt{{\left | c \right |}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} \sqrt{{\left | c \right |}} - \frac{2 \,{\left (b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \sqrt{{\left | c \right |}}}{\sqrt{c x}}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}} \sqrt{{\left | c \right |}}}\right )}{b c^{2}} + \frac{\sqrt{2} \left (-b\right )^{\frac{1}{4}} \sqrt{{\left | c \right |}}{\rm ln}\left (\frac{\sqrt{2}{\left (b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}{\left | c \right |}}{\sqrt{c x}} + \sqrt{-b}{\left | c \right |} + \frac{\sqrt{b c^{2} x^{2} + a c^{2}}{\left | c \right |}}{c x}\right )}{b c^{2}} - \frac{\sqrt{2} \left (-b\right )^{\frac{1}{4}} \sqrt{{\left | c \right |}}{\rm ln}\left (-\frac{\sqrt{2}{\left (b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}{\left | c \right |}}{\sqrt{c x}} + \sqrt{-b}{\left | c \right |} + \frac{\sqrt{b c^{2} x^{2} + a c^{2}}{\left | c \right |}}{c x}\right )}{b c^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*a*c^2*(8*(b*c^2*x^2 + a*c^2)^(1/4)*x^2*sqrt(abs(c))/(sqrt(c*x)*a*c^2) + 2*s
qrt(2)*(-b)^(1/4)*sqrt(abs(c))*arctan(1/2*sqrt(2)*(sqrt(2)*(-b)^(1/4)*sqrt(abs(c
)) + 2*(b*c^2*x^2 + a*c^2)^(1/4)*sqrt(abs(c))/sqrt(c*x))/((-b)^(1/4)*sqrt(abs(c)
)))/(b*c^2) + 2*sqrt(2)*(-b)^(1/4)*sqrt(abs(c))*arctan(-1/2*sqrt(2)*(sqrt(2)*(-b
)^(1/4)*sqrt(abs(c)) - 2*(b*c^2*x^2 + a*c^2)^(1/4)*sqrt(abs(c))/sqrt(c*x))/((-b)
^(1/4)*sqrt(abs(c))))/(b*c^2) + sqrt(2)*(-b)^(1/4)*sqrt(abs(c))*ln(sqrt(2)*(b*c^
2*x^2 + a*c^2)^(1/4)*(-b)^(1/4)*abs(c)/sqrt(c*x) + sqrt(-b)*abs(c) + sqrt(b*c^2*
x^2 + a*c^2)*abs(c)/(c*x))/(b*c^2) - sqrt(2)*(-b)^(1/4)*sqrt(abs(c))*ln(-sqrt(2)
*(b*c^2*x^2 + a*c^2)^(1/4)*(-b)^(1/4)*abs(c)/sqrt(c*x) + sqrt(-b)*abs(c) + sqrt(
b*c^2*x^2 + a*c^2)*abs(c)/(c*x))/(b*c^2))