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

Optimal. Leaf size=204 \[ \frac{2 b^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-11 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{4 b \sqrt{b x^2+c x^4} (5 b B-11 A c)}{231 c^2 \sqrt{x}}-\frac{2 x^{3/2} \sqrt{b x^2+c x^4} (5 b B-11 A c)}{77 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt{x}} \]

[Out]

(-4*b*(5*b*B - 11*A*c)*Sqrt[b*x^2 + c*x^4])/(231*c^2*Sqrt[x]) - (2*(5*b*B - 11*A
*c)*x^(3/2)*Sqrt[b*x^2 + c*x^4])/(77*c) + (2*B*(b*x^2 + c*x^4)^(3/2))/(11*c*Sqrt
[x]) + (2*b^(7/4)*(5*b*B - 11*A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqr
t[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(231*c
^(9/4)*Sqrt[b*x^2 + c*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.526323, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{2 b^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-11 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{4 b \sqrt{b x^2+c x^4} (5 b B-11 A c)}{231 c^2 \sqrt{x}}-\frac{2 x^{3/2} \sqrt{b x^2+c x^4} (5 b B-11 A c)}{77 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*b*(5*b*B - 11*A*c)*Sqrt[b*x^2 + c*x^4])/(231*c^2*Sqrt[x]) - (2*(5*b*B - 11*A
*c)*x^(3/2)*Sqrt[b*x^2 + c*x^4])/(77*c) + (2*B*(b*x^2 + c*x^4)^(3/2))/(11*c*Sqrt
[x]) + (2*b^(7/4)*(5*b*B - 11*A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqr
t[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(231*c
^(9/4)*Sqrt[b*x^2 + c*x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 44.3876, size = 199, normalized size = 0.98 \[ \frac{2 B \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{11 c \sqrt{x}} - \frac{2 b^{\frac{7}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (11 A c - 5 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{231 c^{\frac{9}{4}} x \left (b + c x^{2}\right )} + \frac{4 b \left (11 A c - 5 B b\right ) \sqrt{b x^{2} + c x^{4}}}{231 c^{2} \sqrt{x}} + \frac{2 x^{\frac{3}{2}} \left (11 A c - 5 B b\right ) \sqrt{b x^{2} + c x^{4}}}{77 c} \]

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)**(1/2),x)

[Out]

2*B*(b*x**2 + c*x**4)**(3/2)/(11*c*sqrt(x)) - 2*b**(7/4)*sqrt((b + c*x**2)/(sqrt
(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*(11*A*c - 5*B*b)*sqrt(b*x**2 + c*x**4
)*elliptic_f(2*atan(c**(1/4)*sqrt(x)/b**(1/4)), 1/2)/(231*c**(9/4)*x*(b + c*x**2
)) + 4*b*(11*A*c - 5*B*b)*sqrt(b*x**2 + c*x**4)/(231*c**2*sqrt(x)) + 2*x**(3/2)*
(11*A*c - 5*B*b)*sqrt(b*x**2 + c*x**4)/(77*c)

_______________________________________________________________________________________

Mathematica [C]  time = 0.720126, size = 159, normalized size = 0.78 \[ \frac{1}{231} \sqrt{x^2 \left (b+c x^2\right )} \left (\frac{4 b c \left (11 A+3 B x^2\right )+6 c^2 x^2 \left (11 A+7 B x^2\right )-20 b^2 B}{c^2 \sqrt{x}}+\frac{4 i b^2 \sqrt{\frac{b}{c x^2}+1} (5 b B-11 A c) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{c^2 \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right )}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x^2*(b + c*x^2)]*((-20*b^2*B + 4*b*c*(11*A + 3*B*x^2) + 6*c^2*x^2*(11*A +
7*B*x^2))/(c^2*Sqrt[x]) + ((4*I)*b^2*(5*b*B - 11*A*c)*Sqrt[1 + b/(c*x^2)]*Ellipt
icF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[c]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[b])/Sqrt[c]
]*c^2*(b + c*x^2))))/231

_______________________________________________________________________________________

Maple [A]  time = 0.038, size = 283, normalized size = 1.4 \[ -{\frac{2}{ \left ( 231\,c{x}^{2}+231\,b \right ){c}^{3}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( -21\,B{x}^{7}{c}^{4}+11\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{b}^{2}c-33\,A{x}^{5}{c}^{4}-5\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{b}^{3}-27\,B{x}^{5}b{c}^{3}-55\,A{x}^{3}b{c}^{3}+4\,B{x}^{3}{b}^{2}{c}^{2}-22\,Ax{b}^{2}{c}^{2}+10\,Bx{b}^{3}c \right ){x}^{-{\frac{3}{2}}}} \]

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)^(1/2),x)

[Out]

-2/231*(c*x^4+b*x^2)^(1/2)/x^(3/2)/(c*x^2+b)*(-21*B*x^7*c^4+11*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)*b^2*c-33*A*x^5*c^4-5*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)*Ellipt
icF(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*(-b*c)^(1/2)*b^3-27*B*x
^5*b*c^3-55*A*x^3*b*c^3+4*B*x^3*b^2*c^2-22*A*x*b^2*c^2+10*B*x*b^3*c)/c^3

_______________________________________________________________________________________

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

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. \[{\rm integral}\left (\sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )} \sqrt{x}, x\right ) \]

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. \[ \int \sqrt{x} \sqrt{x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )\, dx \]

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)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

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

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)