3.355 \(\int \sqrt{x} \sqrt{b x^2+c x^4} \, dx\)

Optimal. Leaf size=146 \[ -\frac{2 b^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 c^{5/4} \sqrt{b x^2+c x^4}}+\frac{4 b \sqrt{b x^2+c x^4}}{21 c \sqrt{x}}+\frac{2}{7} x^{3/2} \sqrt{b x^2+c x^4} \]

[Out]

(4*b*Sqrt[b*x^2 + c*x^4])/(21*c*Sqrt[x]) + (2*x^(3/2)*Sqrt[b*x^2 + c*x^4])/7 - (
2*b^(7/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*Elli
pticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(21*c^(5/4)*Sqrt[b*x^2 + c*x^4]
)

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Rubi [A]  time = 0.355041, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{2 b^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 c^{5/4} \sqrt{b x^2+c x^4}}+\frac{4 b \sqrt{b x^2+c x^4}}{21 c \sqrt{x}}+\frac{2}{7} x^{3/2} \sqrt{b x^2+c x^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*Sqrt[b*x^2 + c*x^4],x]

[Out]

(4*b*Sqrt[b*x^2 + c*x^4])/(21*c*Sqrt[x]) + (2*x^(3/2)*Sqrt[b*x^2 + c*x^4])/7 - (
2*b^(7/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*Elli
pticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(21*c^(5/4)*Sqrt[b*x^2 + c*x^4]
)

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Rubi in Sympy [A]  time = 32.4169, size = 139, normalized size = 0.95 \[ - \frac{2 b^{\frac{7}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{21 c^{\frac{5}{4}} x \left (b + c x^{2}\right )} + \frac{4 b \sqrt{b x^{2} + c x^{4}}}{21 c \sqrt{x}} + \frac{2 x^{\frac{3}{2}} \sqrt{b x^{2} + c x^{4}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*b**(7/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*sq
rt(b*x**2 + c*x**4)*elliptic_f(2*atan(c**(1/4)*sqrt(x)/b**(1/4)), 1/2)/(21*c**(5
/4)*x*(b + c*x**2)) + 4*b*sqrt(b*x**2 + c*x**4)/(21*c*sqrt(x)) + 2*x**(3/2)*sqrt
(b*x**2 + c*x**4)/7

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Mathematica [C]  time = 0.427096, size = 120, normalized size = 0.82 \[ \frac{1}{21} \sqrt{x^2 \left (b+c x^2\right )} \left (-\frac{4 i b^2 \sqrt{\frac{b}{c x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{c \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right )}+\frac{4 b}{c \sqrt{x}}+6 x^{3/2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]*Sqrt[b*x^2 + c*x^4],x]

[Out]

(Sqrt[x^2*(b + c*x^2)]*((4*b)/(c*Sqrt[x]) + 6*x^(3/2) - ((4*I)*b^2*Sqrt[1 + b/(c
*x^2)]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[c]]/Sqrt[x]], -1])/(Sqrt[(I*Sqr
t[b])/Sqrt[c]]*c*(b + c*x^2))))/21

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Maple [A]  time = 0.032, size = 145, normalized size = 1. \[ -{\frac{2}{ \left ( 21\,c{x}^{2}+21\,b \right ){c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}} \left ({b}^{2}\sqrt{-bc}\sqrt{{1 \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-bc}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}},{\frac{\sqrt{2}}{2}} \right ) -3\,{c}^{3}{x}^{5}-5\,b{c}^{2}{x}^{3}-2\,{b}^{2}cx \right ){x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/21*(c*x^4+b*x^2)^(1/2)/x^(3/2)/(c*x^2+b)*(b^2*(-b*c)^(1/2)*((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))-
3*c^3*x^5-5*b*c^2*x^3-2*b^2*c*x)/c^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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