3.388 \(\int \frac{1}{x^{7/2} \sqrt{b x^2+c x^4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.366148, 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 \[ \frac{5 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 b^{9/4} \sqrt{b x^2+c x^4}}+\frac{10 c \sqrt{b x^2+c x^4}}{21 b^2 x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(7/2)*Sqrt[b*x^2 + c*x^4]),x]

[Out]

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

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Rubi in Sympy [A]  time = 34.905, size = 143, normalized size = 0.96 \[ - \frac{2 \sqrt{b x^{2} + c x^{4}}}{7 b x^{\frac{9}{2}}} + \frac{10 c \sqrt{b x^{2} + c x^{4}}}{21 b^{2} x^{\frac{5}{2}}} + \frac{5 c^{\frac{7}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{21 b^{\frac{9}{4}} x \left (b + c x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(7/2)/(c*x**4+b*x**2)**(1/2),x)

[Out]

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

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Mathematica [C]  time = 0.161682, size = 144, normalized size = 0.97 \[ \frac{x \left (\frac{10 c}{21 b^2 x^{3/2}}-\frac{2}{7 b x^{7/2}}\right ) \left (b+c x^2\right )}{\sqrt{x^2 \left (b+c x^2\right )}}+\frac{10 i c^2 x^2 \sqrt{\frac{b}{c x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{21 b^2 \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(7/2)*Sqrt[b*x^2 + c*x^4]),x]

[Out]

((-2/(7*b*x^(7/2)) + (10*c)/(21*b^2*x^(3/2)))*x*(b + c*x^2))/Sqrt[x^2*(b + c*x^2
)] + (((10*I)/21)*c^2*Sqrt[1 + b/(c*x^2)]*x^2*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b
])/Sqrt[c]]/Sqrt[x]], -1])/(b^2*Sqrt[(I*Sqrt[b])/Sqrt[c]]*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.022, size = 134, normalized size = 0.9 \[{\frac{1}{21\,{b}^{2}} \left ( 5\,\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}\sqrt{2}{x}^{3}c+10\,{c}^{2}{x}^{4}+4\,bc{x}^{2}-6\,{b}^{2} \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(7/2)/(c*x^4+b*x^2)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{4} + b x^{2}} x^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2)*x^(7/2)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*x^4 + b*x^2)*x^(7/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{c x^{4} + b x^{2}} x^{\frac{7}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2)*x^(7/2)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(c*x^4 + b*x^2)*x^(7/2)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{\frac{7}{2}} \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(7/2)/(c*x**4+b*x**2)**(1/2),x)

[Out]

Integral(1/(x**(7/2)*sqrt(x**2*(b + c*x**2))), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{4} + b x^{2}} x^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2)*x^(7/2)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^4 + b*x^2)*x^(7/2)), x)