3.248 \(\int \frac{x^{5/2} \left (A+B x^2\right )}{\sqrt{b x^2+c x^4}} \, dx\)

Optimal. Leaf size=167 \[ \frac{b^{3/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-7 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4} (5 b B-7 A c)}{21 c^2 \sqrt{x}}+\frac{2 B x^{3/2} \sqrt{b x^2+c x^4}}{7 c} \]

[Out]

(-2*(5*b*B - 7*A*c)*Sqrt[b*x^2 + c*x^4])/(21*c^2*Sqrt[x]) + (2*B*x^(3/2)*Sqrt[b*
x^2 + c*x^4])/(7*c) + (b^(3/4)*(5*b*B - 7*A*c)*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 = 0.473889, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{b^{3/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-7 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4} (5 b B-7 A c)}{21 c^2 \sqrt{x}}+\frac{2 B x^{3/2} \sqrt{b x^2+c x^4}}{7 c} \]

Antiderivative was successfully verified.

[In]  Int[(x^(5/2)*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]

[Out]

(-2*(5*b*B - 7*A*c)*Sqrt[b*x^2 + c*x^4])/(21*c^2*Sqrt[x]) + (2*B*x^(3/2)*Sqrt[b*
x^2 + c*x^4])/(7*c) + (b^(3/4)*(5*b*B - 7*A*c)*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 = 35.9155, size = 162, normalized size = 0.97 \[ \frac{2 B x^{\frac{3}{2}} \sqrt{b x^{2} + c x^{4}}}{7 c} - \frac{b^{\frac{3}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (7 A c - 5 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{21 c^{\frac{9}{4}} x \left (b + c x^{2}\right )} + \frac{2 \left (7 A c - 5 B b\right ) \sqrt{b x^{2} + c x^{4}}}{21 c^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(B*x**2+A)/(c*x**4+b*x**2)**(1/2),x)

[Out]

2*B*x**(3/2)*sqrt(b*x**2 + c*x**4)/(7*c) - b**(3/4)*sqrt((b + c*x**2)/(sqrt(b) +
sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*(7*A*c - 5*B*b)*sqrt(b*x**2 + c*x**4)*elli
ptic_f(2*atan(c**(1/4)*sqrt(x)/b**(1/4)), 1/2)/(21*c**(9/4)*x*(b + c*x**2)) + 2*
(7*A*c - 5*B*b)*sqrt(b*x**2 + c*x**4)/(21*c**2*sqrt(x))

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Mathematica [C]  time = 0.281183, size = 151, normalized size = 0.9 \[ \frac{-2 i b x^2 \sqrt{\frac{b}{c x^2}+1} (7 A c-5 b B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-2 x^{3/2} \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right ) \left (-7 A c+5 b B-3 B c x^2\right )}{21 c^2 \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(5/2)*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]

[Out]

(-2*Sqrt[(I*Sqrt[b])/Sqrt[c]]*x^(3/2)*(b + c*x^2)*(5*b*B - 7*A*c - 3*B*c*x^2) -
(2*I)*b*(-5*b*B + 7*A*c)*Sqrt[1 + b/(c*x^2)]*x^2*EllipticF[I*ArcSinh[Sqrt[(I*Sqr
t[b])/Sqrt[c]]/Sqrt[x]], -1])/(21*Sqrt[(I*Sqrt[b])/Sqrt[c]]*c^2*Sqrt[x^2*(b + c*
x^2)])

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Maple [A]  time = 0.021, size = 248, normalized size = 1.5 \[ -{\frac{1}{21\,{c}^{3}}\sqrt{x} \left ( 7\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}bc-5\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{b}^{2}-6\,B{c}^{3}{x}^{5}-14\,A{x}^{3}{c}^{3}+4\,B{x}^{3}b{c}^{2}-14\,Axb{c}^{2}+10\,Bx{b}^{2}c \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x)

[Out]

-1/21/(c*x^4+b*x^2)^(1/2)*x^(1/2)*(7*A*((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)*Ellipt
icF(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*(-b*c)^(1/2)*b*c-5*B*((
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))*(-b*c)^(1/2)*b^2-6*B*c^3*x^5-14*A*x^3*c^3+4*B*x^3*b*c^2-14*A*x*b
*c^2+10*B*x*b^2*c)/c^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} x^{\frac{5}{2}}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(5/2)/sqrt(c*x^4 + b*x^2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*x^(5/2)/sqrt(c*x^4 + b*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(5/2)/sqrt(c*x^4 + b*x^2),x, algorithm="fricas")

[Out]

integral((B*x^4 + A*x^2)*sqrt(x)/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**(5/2)*(B*x**2+A)/(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. \[ \int \frac{{\left (B x^{2} + A\right )} x^{\frac{5}{2}}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(5/2)/sqrt(c*x^4 + b*x^2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*x^(5/2)/sqrt(c*x^4 + b*x^2), x)