3.923 \(\int \frac{\sqrt [4]{a+b x^2}}{(c x)^{9/2}} \, dx\)

Optimal. Leaf size=123 \[ \frac{4 b^{5/2} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{21 a^{3/2} c^6 \left (a+b x^2\right )^{3/4}}-\frac{2 b \sqrt [4]{a+b x^2}}{21 a c^3 (c x)^{3/2}}-\frac{2 \sqrt [4]{a+b x^2}}{7 c (c x)^{7/2}} \]

[Out]

(-2*(a + b*x^2)^(1/4))/(7*c*(c*x)^(7/2)) - (2*b*(a + b*x^2)^(1/4))/(21*a*c^3*(c*
x)^(3/2)) + (4*b^(5/2)*(1 + a/(b*x^2))^(3/4)*(c*x)^(3/2)*EllipticF[ArcCot[(Sqrt[
b]*x)/Sqrt[a]]/2, 2])/(21*a^(3/2)*c^6*(a + b*x^2)^(3/4))

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Rubi [A]  time = 0.256944, antiderivative size = 123, 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 \[ \frac{4 b^{5/2} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{21 a^{3/2} c^6 \left (a+b x^2\right )^{3/4}}-\frac{2 b \sqrt [4]{a+b x^2}}{21 a c^3 (c x)^{3/2}}-\frac{2 \sqrt [4]{a+b x^2}}{7 c (c x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(1/4)/(c*x)^(9/2),x]

[Out]

(-2*(a + b*x^2)^(1/4))/(7*c*(c*x)^(7/2)) - (2*b*(a + b*x^2)^(1/4))/(21*a*c^3*(c*
x)^(3/2)) + (4*b^(5/2)*(1 + a/(b*x^2))^(3/4)*(c*x)^(3/2)*EllipticF[ArcCot[(Sqrt[
b]*x)/Sqrt[a]]/2, 2])/(21*a^(3/2)*c^6*(a + b*x^2)^(3/4))

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Rubi in Sympy [A]  time = 28.7322, size = 110, normalized size = 0.89 \[ - \frac{2 \sqrt [4]{a + b x^{2}}}{7 c \left (c x\right )^{\frac{7}{2}}} - \frac{2 b \sqrt [4]{a + b x^{2}}}{21 a c^{3} \left (c x\right )^{\frac{3}{2}}} + \frac{4 b^{\frac{5}{2}} \left (c x\right )^{\frac{3}{2}} \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{21 a^{\frac{3}{2}} c^{6} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(1/4)/(c*x)**(9/2),x)

[Out]

-2*(a + b*x**2)**(1/4)/(7*c*(c*x)**(7/2)) - 2*b*(a + b*x**2)**(1/4)/(21*a*c**3*(
c*x)**(3/2)) + 4*b**(5/2)*(c*x)**(3/2)*(a/(b*x**2) + 1)**(3/4)*elliptic_f(atan(s
qrt(a)/(sqrt(b)*x))/2, 2)/(21*a**(3/2)*c**6*(a + b*x**2)**(3/4))

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Mathematica [C]  time = 0.068166, size = 92, normalized size = 0.75 \[ -\frac{2 \sqrt{c x} \left (3 a^2+2 b^2 x^4 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+4 a b x^2+b^2 x^4\right )}{21 a c^5 x^4 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(1/4)/(c*x)^(9/2),x]

[Out]

(-2*Sqrt[c*x]*(3*a^2 + 4*a*b*x^2 + b^2*x^4 + 2*b^2*x^4*(1 + (b*x^2)/a)^(3/4)*Hyp
ergeometric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)]))/(21*a*c^5*x^4*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{9}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/4)/(c*x)^(9/2),x)

[Out]

int((b*x^2+a)^(1/4)/(c*x)^(9/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}{\left (c x\right )^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/4)/(c*x)^(9/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(1/4)/(c*x)^(9/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/4)/(c*x)^(9/2),x, algorithm="fricas")

[Out]

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

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Sympy [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)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}{\left (c x\right )^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/4)/(c*x)^(9/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(1/4)/(c*x)^(9/2), x)