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

Optimal. Leaf size=326 \[ -\frac{2 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (b B-3 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{4 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (b B-3 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{4 b x^{3/2} \left (b+c x^2\right ) (b B-3 A c)}{15 c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{x} \sqrt{b x^2+c x^4} (b B-3 A c)}{15 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}} \]

[Out]

(-4*b*(b*B - 3*A*c)*x^(3/2)*(b + c*x^2))/(15*c^(3/2)*(Sqrt[b] + Sqrt[c]*x)*Sqrt[
b*x^2 + c*x^4]) - (2*(b*B - 3*A*c)*Sqrt[x]*Sqrt[b*x^2 + c*x^4])/(15*c) + (2*B*(b
*x^2 + c*x^4)^(3/2))/(9*c*x^(3/2)) + (4*b^(5/4)*(b*B - 3*A*c)*x*(Sqrt[b] + Sqrt[
c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt
[x])/b^(1/4)], 1/2])/(15*c^(7/4)*Sqrt[b*x^2 + c*x^4]) - (2*b^(5/4)*(b*B - 3*A*c)
*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[2*A
rcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(15*c^(7/4)*Sqrt[b*x^2 + c*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.697145, antiderivative size = 326, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (b B-3 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{4 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (b B-3 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{4 b x^{3/2} \left (b+c x^2\right ) (b B-3 A c)}{15 c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{x} \sqrt{b x^2+c x^4} (b B-3 A c)}{15 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*b*(b*B - 3*A*c)*x^(3/2)*(b + c*x^2))/(15*c^(3/2)*(Sqrt[b] + Sqrt[c]*x)*Sqrt[
b*x^2 + c*x^4]) - (2*(b*B - 3*A*c)*Sqrt[x]*Sqrt[b*x^2 + c*x^4])/(15*c) + (2*B*(b
*x^2 + c*x^4)^(3/2))/(9*c*x^(3/2)) + (4*b^(5/4)*(b*B - 3*A*c)*x*(Sqrt[b] + Sqrt[
c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt
[x])/b^(1/4)], 1/2])/(15*c^(7/4)*Sqrt[b*x^2 + c*x^4]) - (2*b^(5/4)*(b*B - 3*A*c)
*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[2*A
rcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(15*c^(7/4)*Sqrt[b*x^2 + c*x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 60.1155, size = 309, normalized size = 0.95 \[ \frac{2 B \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{9 c x^{\frac{3}{2}}} - \frac{4 b^{\frac{5}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (3 A c - B b\right ) \sqrt{b x^{2} + c x^{4}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{7}{4}} x \left (b + c x^{2}\right )} + \frac{2 b^{\frac{5}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (3 A c - B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{7}{4}} x \left (b + c x^{2}\right )} + \frac{4 b \left (3 A c - B b\right ) \sqrt{b x^{2} + c x^{4}}}{15 c^{\frac{3}{2}} \sqrt{x} \left (\sqrt{b} + \sqrt{c} x\right )} + \frac{2 \sqrt{x} \left (3 A c - B b\right ) \sqrt{b x^{2} + c x^{4}}}{15 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(b*x**2 + c*x**4)**(3/2)/(9*c*x**(3/2)) - 4*b**(5/4)*sqrt((b + c*x**2)/(sqrt
(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*(3*A*c - B*b)*sqrt(b*x**2 + c*x**4)*e
lliptic_e(2*atan(c**(1/4)*sqrt(x)/b**(1/4)), 1/2)/(15*c**(7/4)*x*(b + c*x**2)) +
2*b**(5/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*(3
*A*c - B*b)*sqrt(b*x**2 + c*x**4)*elliptic_f(2*atan(c**(1/4)*sqrt(x)/b**(1/4)),
1/2)/(15*c**(7/4)*x*(b + c*x**2)) + 4*b*(3*A*c - B*b)*sqrt(b*x**2 + c*x**4)/(15*
c**(3/2)*sqrt(x)*(sqrt(b) + sqrt(c)*x)) + 2*sqrt(x)*(3*A*c - B*b)*sqrt(b*x**2 +
c*x**4)/(15*c)

_______________________________________________________________________________________

Mathematica [C]  time = 0.786245, size = 247, normalized size = 0.76 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (\frac{2 x^{3/2} \left (9 A c+2 b B+5 B c x^2\right )}{3 c}-\frac{4 b (b B-3 A c) \left (\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right )+\sqrt{b} \sqrt{c} x^{3/2} \sqrt{\frac{b}{c x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{b} \sqrt{c} x^{3/2} \sqrt{\frac{b}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{c^2 \sqrt{x} \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right )}\right )}{15 x} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x^2*(b + c*x^2)]*((2*x^(3/2)*(2*b*B + 9*A*c + 5*B*c*x^2))/(3*c) - (4*b*(b*
B - 3*A*c)*(Sqrt[(I*Sqrt[b])/Sqrt[c]]*(b + c*x^2) - Sqrt[b]*Sqrt[c]*Sqrt[1 + b/(
c*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[c]]/Sqrt[x]], -1] + Sq
rt[b]*Sqrt[c]*Sqrt[1 + b/(c*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/S
qrt[c]]/Sqrt[x]], -1]))/(Sqrt[(I*Sqrt[b])/Sqrt[c]]*c^2*Sqrt[x]*(b + c*x^2))))/(1
5*x)

_______________________________________________________________________________________

Maple [A]  time = 0.039, size = 422, normalized size = 1.3 \[{\frac{2}{ \left ( 45\,c{x}^{2}+45\,b \right ){c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( 5\,B{c}^{3}{x}^{6}+18\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{2}c-9\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{2}c-6\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{3}+3\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{3}+9\,A{x}^{4}{c}^{3}+7\,B{x}^{4}b{c}^{2}+9\,A{x}^{2}b{c}^{2}+2\,B{x}^{2}{b}^{2}c \right ){x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45*(c*x^4+b*x^2)^(1/2)/x^(3/2)/(c*x^2+b)/c^2*(5*B*c^3*x^6+18*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)*EllipticE(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2
))*b^2*c-9*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^2*c-6*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)*Elli
pticE(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*b^3+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^3+9*A*x^4*c^3+7*B*x^4*b*c^2+9*A*x^2*b*c^2+2*B*x^2*b^2*c)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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