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

Optimal. Leaf size=296 \[ -\frac{\sqrt [4]{b} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{2 \sqrt{2} c^{3/2}}+\frac{\sqrt [4]{b} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{2 \sqrt{2} c^{3/2}}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{\sqrt{2} c^{3/2}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}+1\right )}{\sqrt{2} c^{3/2}}-\frac{2 \sqrt [4]{a-b x^2}}{c \sqrt{c x}} \]

[Out]

(-2*(a - b*x^2)^(1/4))/(c*Sqrt[c*x]) + (b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt
[c*x])/(Sqrt[c]*(a - b*x^2)^(1/4))])/(Sqrt[2]*c^(3/2)) - (b^(1/4)*ArcTan[1 + (Sq
rt[2]*b^(1/4)*Sqrt[c*x])/(Sqrt[c]*(a - b*x^2)^(1/4))])/(Sqrt[2]*c^(3/2)) - (b^(1
/4)*Log[Sqrt[c] + (Sqrt[b]*Sqrt[c]*x)/Sqrt[a - b*x^2] - (Sqrt[2]*b^(1/4)*Sqrt[c*
x])/(a - b*x^2)^(1/4)])/(2*Sqrt[2]*c^(3/2)) + (b^(1/4)*Log[Sqrt[c] + (Sqrt[b]*Sq
rt[c]*x)/Sqrt[a - b*x^2] + (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(a - b*x^2)^(1/4)])/(2*Sq
rt[2]*c^(3/2))

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Rubi [A]  time = 0.614862, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ -\frac{\sqrt [4]{b} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{2 \sqrt{2} c^{3/2}}+\frac{\sqrt [4]{b} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{2 \sqrt{2} c^{3/2}}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{\sqrt{2} c^{3/2}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}+1\right )}{\sqrt{2} c^{3/2}}-\frac{2 \sqrt [4]{a-b x^2}}{c \sqrt{c x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(a - b*x^2)^(1/4))/(c*Sqrt[c*x]) + (b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt
[c*x])/(Sqrt[c]*(a - b*x^2)^(1/4))])/(Sqrt[2]*c^(3/2)) - (b^(1/4)*ArcTan[1 + (Sq
rt[2]*b^(1/4)*Sqrt[c*x])/(Sqrt[c]*(a - b*x^2)^(1/4))])/(Sqrt[2]*c^(3/2)) - (b^(1
/4)*Log[Sqrt[c] + (Sqrt[b]*Sqrt[c]*x)/Sqrt[a - b*x^2] - (Sqrt[2]*b^(1/4)*Sqrt[c*
x])/(a - b*x^2)^(1/4)])/(2*Sqrt[2]*c^(3/2)) + (b^(1/4)*Log[Sqrt[c] + (Sqrt[b]*Sq
rt[c]*x)/Sqrt[a - b*x^2] + (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(a - b*x^2)^(1/4)])/(2*Sq
rt[2]*c^(3/2))

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Rubi in Sympy [A]  time = 66.3904, size = 267, normalized size = 0.9 \[ - \frac{\sqrt{2} \sqrt [4]{b} \log{\left (- \frac{\sqrt{2} \sqrt [4]{b} \sqrt{c} \sqrt{c x}}{\sqrt [4]{a - b x^{2}}} + \frac{\sqrt{b} c x}{\sqrt{a - b x^{2}}} + c \right )}}{4 c^{\frac{3}{2}}} + \frac{\sqrt{2} \sqrt [4]{b} \log{\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c} \sqrt{c x}}{\sqrt [4]{a - b x^{2}}} + \frac{\sqrt{b} c x}{\sqrt{a - b x^{2}}} + c \right )}}{4 c^{\frac{3}{2}}} - \frac{\sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a - b x^{2}}} - 1 \right )}}{2 c^{\frac{3}{2}}} - \frac{\sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a - b x^{2}}} + 1 \right )}}{2 c^{\frac{3}{2}}} - \frac{2 \sqrt [4]{a - b x^{2}}}{c \sqrt{c x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0646625, size = 72, normalized size = 0.24 \[ -\frac{2 x \left (b x^2 \left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b x^2}{a}\right )+3 a-3 b x^2\right )}{3 (c x)^{3/2} \left (a-b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{-b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((-b*x^2+a)^(1/4)/(c*x)^(3/2),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)/(c*x)^(3/2),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)/(c*x)^(3/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 14.9041, size = 51, normalized size = 0.17 \[ \frac{\sqrt [4]{a} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{2 c^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.253768, size = 429, normalized size = 1.45 \[ \frac{2 \, \sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}} + \frac{2 \,{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \sqrt{{\left | c \right |}}}{\sqrt{c x}}\right )}}{2 \, b^{\frac{1}{4}} \sqrt{{\left | c \right |}}}\right ) + 2 \, \sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}} - \frac{2 \,{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \sqrt{{\left | c \right |}}}{\sqrt{c x}}\right )}}{2 \, b^{\frac{1}{4}} \sqrt{{\left | c \right |}}}\right ) + \sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}}{\rm ln}\left (\frac{\sqrt{2}{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} b^{\frac{1}{4}}{\left | c \right |}}{\sqrt{c x}} + \sqrt{b}{\left | c \right |} + \frac{\sqrt{-b c^{2} x^{2} + a c^{2}}{\left | c \right |}}{c x}\right ) - \sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}}{\rm ln}\left (-\frac{\sqrt{2}{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} b^{\frac{1}{4}}{\left | c \right |}}{\sqrt{c x}} + \sqrt{b}{\left | c \right |} + \frac{\sqrt{-b c^{2} x^{2} + a c^{2}}{\left | c \right |}}{c x}\right ) - \frac{8 \,{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \sqrt{{\left | c \right |}}}{\sqrt{c x}}}{4 \, c^{2}} \]

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]

1/4*(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)
))) + 2*sqrt(2)*b^(1/4)*sqrt(abs(c))*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4)*sqrt(a
bs(c)) - 2*(-b*c^2*x^2 + a*c^2)^(1/4)*sqrt(abs(c))/sqrt(c*x))/(b^(1/4)*sqrt(abs(
c)))) + 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)) -
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)) - 8*(-b*c
^2*x^2 + a*c^2)^(1/4)*sqrt(abs(c))/sqrt(c*x))/c^2