3.252 \(\int \frac{A+B x^2}{x^{3/2} \sqrt{b x^2+c x^4}} \, dx\)

Optimal. Leaf size=131 \[ \frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 b^{5/4} \sqrt [4]{c} \sqrt{b x^2+c x^4}}-\frac{2 A \sqrt{b x^2+c x^4}}{3 b x^{5/2}} \]

[Out]

(-2*A*Sqrt[b*x^2 + c*x^4])/(3*b*x^(5/2)) + ((3*b*B - 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])/(3*b^(5/4)*c^(1/4)*Sqrt[b*x^2 + c*x^4])

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Rubi [A]  time = 0.370283, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 b^{5/4} \sqrt [4]{c} \sqrt{b x^2+c x^4}}-\frac{2 A \sqrt{b x^2+c x^4}}{3 b x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^(3/2)*Sqrt[b*x^2 + c*x^4]),x]

[Out]

(-2*A*Sqrt[b*x^2 + c*x^4])/(3*b*x^(5/2)) + ((3*b*B - 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])/(3*b^(5/4)*c^(1/4)*Sqrt[b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 28.2331, size = 126, normalized size = 0.96 \[ - \frac{2 A \sqrt{b x^{2} + c x^{4}}}{3 b x^{\frac{5}{2}}} - \frac{\sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (A c - 3 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{3 b^{\frac{5}{4}} \sqrt [4]{c} x \left (b + c x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*sqrt(b*x**2 + c*x**4)/(3*b*x**(5/2)) - sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)
*x)**2)*(sqrt(b) + sqrt(c)*x)*(A*c - 3*B*b)*sqrt(b*x**2 + c*x**4)*elliptic_f(2*a
tan(c**(1/4)*sqrt(x)/b**(1/4)), 1/2)/(3*b**(5/4)*c**(1/4)*x*(b + c*x**2))

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Mathematica [C]  time = 0.401905, size = 119, normalized size = 0.91 \[ \frac{2 \left (-A \left (b+c x^2\right )+\frac{i x^{5/2} \sqrt{\frac{b}{c x^2}+1} (3 b B-A c) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}\right )}{3 b \sqrt{x} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^(3/2)*Sqrt[b*x^2 + c*x^4]),x]

[Out]

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

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Maple [A]  time = 0.026, size = 219, normalized size = 1.7 \[ -{\frac{1}{3\,bc} \left ( A\sqrt{{1 \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-bc}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-bc}xc-3\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}xb+2\,A{x}^{2}{c}^{2}+2\,Abc \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/(c*x^4+b*x^2)^(1/2)/x^(1/2)*(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)*EllipticF
(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*(-b*c)^(1/2)*x*c-3*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)*x*b+2*A*x^2*c^2+2*A*b*c)/c/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{\sqrt{c x^{4} + b x^{2}} x^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(3/2)),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x^{2} + A}{\sqrt{c x^{4} + b x^{2}} x^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(3/2)),x, algorithm="fricas")

[Out]

integral((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(3/2)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x^{2}}{x^{\frac{3}{2}} \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x**2)/(x**(3/2)*sqrt(x**2*(b + c*x**2))), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{\sqrt{c x^{4} + b x^{2}} x^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(3/2)),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(3/2)), x)