3.380 \(\int \frac{x^{9/2}}{\sqrt{b x^2+c x^4}} \, dx\)

Optimal. Leaf size=149 \[ \frac{5 b^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{10 b \sqrt{b x^2+c x^4}}{21 c^2 \sqrt{x}}+\frac{2 x^{3/2} \sqrt{b x^2+c x^4}}{7 c} \]

[Out]

(-10*b*Sqrt[b*x^2 + c*x^4])/(21*c^2*Sqrt[x]) + (2*x^(3/2)*Sqrt[b*x^2 + c*x^4])/(
7*c) + (5*b^(7/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)
^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(21*c^(9/4)*Sqrt[b*x^2
+ c*x^4])

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Rubi [A]  time = 0.363642, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{5 b^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{10 b \sqrt{b x^2+c x^4}}{21 c^2 \sqrt{x}}+\frac{2 x^{3/2} \sqrt{b x^2+c x^4}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(-10*b*Sqrt[b*x^2 + c*x^4])/(21*c^2*Sqrt[x]) + (2*x^(3/2)*Sqrt[b*x^2 + c*x^4])/(
7*c) + (5*b^(7/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)
^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(21*c^(9/4)*Sqrt[b*x^2
+ c*x^4])

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Rubi in Sympy [A]  time = 36.5821, size = 143, normalized size = 0.96 \[ \frac{5 b^{\frac{7}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{21 c^{\frac{9}{4}} x \left (b + c x^{2}\right )} - \frac{10 b \sqrt{b x^{2} + c x^{4}}}{21 c^{2} \sqrt{x}} + \frac{2 x^{\frac{3}{2}} \sqrt{b x^{2} + c x^{4}}}{7 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*b**(7/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*sqr
t(b*x**2 + c*x**4)*elliptic_f(2*atan(c**(1/4)*sqrt(x)/b**(1/4)), 1/2)/(21*c**(9/
4)*x*(b + c*x**2)) - 10*b*sqrt(b*x**2 + c*x**4)/(21*c**2*sqrt(x)) + 2*x**(3/2)*s
qrt(b*x**2 + c*x**4)/(7*c)

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Mathematica [C]  time = 0.161149, size = 144, normalized size = 0.97 \[ \frac{x \left (b+c x^2\right ) \left (\frac{2 x^{5/2}}{7 c}-\frac{10 b \sqrt{x}}{21 c^2}\right )}{\sqrt{x^2 \left (b+c x^2\right )}}+\frac{10 i b^2 x^2 \sqrt{\frac{b}{c x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{21 c^2 \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(b + c*x^2)*((-10*b*Sqrt[x])/(21*c^2) + (2*x^(5/2))/(7*c)))/Sqrt[x^2*(b + c*x
^2)] + (((10*I)/21)*b^2*Sqrt[1 + b/(c*x^2)]*x^2*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt
[b])/Sqrt[c]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[b])/Sqrt[c]]*c^2*Sqrt[x^2*(b + c*x^2)
])

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Maple [A]  time = 0.018, size = 137, normalized size = 0.9 \[{\frac{1}{21\,{c}^{3}}\sqrt{x} \left ( 5\,{b}^{2}\sqrt{-bc}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) +6\,{c}^{3}{x}^{5}-4\,b{c}^{2}{x}^{3}-10\,{b}^{2}cx \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21/(c*x^4+b*x^2)^(1/2)*x^(1/2)*(5*b^2*(-b*c)^(1/2)*((c*x+(-b*c)^(1/2))/(-b*c)^
(1/2))^(1/2)*2^(1/2)*((-c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*(-x*c/(-b*c)^(1/2)
)^(1/2)*EllipticF(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))+6*c^3*x^5
-4*b*c^2*x^3-10*b^2*c*x)/c^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{9}{2}}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{\frac{9}{2}}}{\sqrt{c x^{4} + b x^{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^(9/2)/sqrt(c*x^4 + b*x^2), 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(x**(9/2)/(c*x**4+b*x**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{9}{2}}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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