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

Optimal. Leaf size=167 \[ \frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-5 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{6 b^{9/4} \sqrt [4]{c} \sqrt{b x^2+c x^4}}+\frac{x^{3/2} (3 b B-5 A c)}{3 b^2 \sqrt{b x^2+c x^4}}-\frac{2 A}{3 b \sqrt{x} \sqrt{b x^2+c x^4}} \]

[Out]

(-2*A)/(3*b*Sqrt[x]*Sqrt[b*x^2 + c*x^4]) + ((3*b*B - 5*A*c)*x^(3/2))/(3*b^2*Sqrt
[b*x^2 + c*x^4]) + ((3*b*B - 5*A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sq
rt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(6*b^
(9/4)*c^(1/4)*Sqrt[b*x^2 + c*x^4])

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Rubi [A]  time = 0.463581, 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 \[ \frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-5 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{6 b^{9/4} \sqrt [4]{c} \sqrt{b x^2+c x^4}}+\frac{x^{3/2} (3 b B-5 A c)}{3 b^2 \sqrt{b x^2+c x^4}}-\frac{2 A}{3 b \sqrt{x} \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A)/(3*b*Sqrt[x]*Sqrt[b*x^2 + c*x^4]) + ((3*b*B - 5*A*c)*x^(3/2))/(3*b^2*Sqrt
[b*x^2 + c*x^4]) + ((3*b*B - 5*A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sq
rt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(6*b^
(9/4)*c^(1/4)*Sqrt[b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 37.1729, size = 162, normalized size = 0.97 \[ - \frac{2 A}{3 b \sqrt{x} \sqrt{b x^{2} + c x^{4}}} - \frac{x^{\frac{3}{2}} \left (5 A c - 3 B b\right )}{3 b^{2} \sqrt{b x^{2} + c x^{4}}} - \frac{\sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (5 A c - 3 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{6 b^{\frac{9}{4}} \sqrt [4]{c} x \left (b + c x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/(3*b*sqrt(x)*sqrt(b*x**2 + c*x**4)) - x**(3/2)*(5*A*c - 3*B*b)/(3*b**2*sqrt
(b*x**2 + c*x**4)) - sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt
(c)*x)*(5*A*c - 3*B*b)*sqrt(b*x**2 + c*x**4)*elliptic_f(2*atan(c**(1/4)*sqrt(x)/
b**(1/4)), 1/2)/(6*b**(9/4)*c**(1/4)*x*(b + c*x**2))

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Mathematica [C]  time = 0.194249, size = 147, normalized size = 0.88 \[ \frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (-2 A b-5 A c x^2+3 b B x^2\right )-i x^{5/2} \sqrt{\frac{b}{c x^2}+1} (5 A c-3 b B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{3 b^2 \sqrt{x} \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.032, size = 235, normalized size = 1.4 \[ -{\frac{c{x}^{2}+b}{6\,{b}^{2}c}{x}^{{\frac{3}{2}}} \left ( 5\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}xc-3\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}xb+10\,A{x}^{2}{c}^{2}-6\,B{x}^{2}bc+4\,Abc \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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