3.487 \(\int \left (c+d x+e x^2+f x^3\right ) \sqrt{a+b x^4} \, dx\)

Optimal. Leaf size=331 \[ \frac{a^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (3 \sqrt{a} e+5 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^4}}-\frac{2 a^{5/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+b x^4}}+\frac{1}{15} x \sqrt{a+b x^4} \left (5 c+3 e x^2\right )+\frac{1}{4} d x^2 \sqrt{a+b x^4}+\frac{a d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 \sqrt{b}}+\frac{2 a e x \sqrt{a+b x^4}}{5 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{f \left (a+b x^4\right )^{3/2}}{6 b} \]

[Out]

(d*x^2*Sqrt[a + b*x^4])/4 + (2*a*e*x*Sqrt[a + b*x^4])/(5*Sqrt[b]*(Sqrt[a] + Sqrt
[b]*x^2)) + (x*(5*c + 3*e*x^2)*Sqrt[a + b*x^4])/15 + (f*(a + b*x^4)^(3/2))/(6*b)
+ (a*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/(4*Sqrt[b]) - (2*a^(5/4)*e*(Sqrt
[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTa
n[(b^(1/4)*x)/a^(1/4)], 1/2])/(5*b^(3/4)*Sqrt[a + b*x^4]) + (a^(3/4)*(5*Sqrt[b]*
c + 3*Sqrt[a]*e)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2
)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(15*b^(3/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.517732, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.37 \[ \frac{a^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (3 \sqrt{a} e+5 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^4}}-\frac{2 a^{5/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+b x^4}}+\frac{1}{15} x \sqrt{a+b x^4} \left (5 c+3 e x^2\right )+\frac{1}{4} d x^2 \sqrt{a+b x^4}+\frac{a d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 \sqrt{b}}+\frac{2 a e x \sqrt{a+b x^4}}{5 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{f \left (a+b x^4\right )^{3/2}}{6 b} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4],x]

[Out]

(d*x^2*Sqrt[a + b*x^4])/4 + (2*a*e*x*Sqrt[a + b*x^4])/(5*Sqrt[b]*(Sqrt[a] + Sqrt
[b]*x^2)) + (x*(5*c + 3*e*x^2)*Sqrt[a + b*x^4])/15 + (f*(a + b*x^4)^(3/2))/(6*b)
+ (a*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/(4*Sqrt[b]) - (2*a^(5/4)*e*(Sqrt
[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTa
n[(b^(1/4)*x)/a^(1/4)], 1/2])/(5*b^(3/4)*Sqrt[a + b*x^4]) + (a^(3/4)*(5*Sqrt[b]*
c + 3*Sqrt[a]*e)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2
)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(15*b^(3/4)*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 50.8368, size = 303, normalized size = 0.92 \[ - \frac{2 a^{\frac{5}{4}} e \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} \sqrt{a + b x^{4}}} + \frac{a^{\frac{3}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) \left (3 \sqrt{a} e + 5 \sqrt{b} c\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{3}{4}} \sqrt{a + b x^{4}}} + \frac{a d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a + b x^{4}}} \right )}}{4 \sqrt{b}} + \frac{2 a e x \sqrt{a + b x^{4}}}{5 \sqrt{b} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} + \frac{d x^{2} \sqrt{a + b x^{4}}}{4} + \frac{x \sqrt{a + b x^{4}} \left (5 c + 3 e x^{2}\right )}{15} + \frac{f \left (a + b x^{4}\right )^{\frac{3}{2}}}{6 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(1/2),x)

[Out]

-2*a**(5/4)*e*sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*
x**2)*elliptic_e(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(5*b**(3/4)*sqrt(a + b*x**4))
+ a**(3/4)*sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*x*
*2)*(3*sqrt(a)*e + 5*sqrt(b)*c)*elliptic_f(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(15
*b**(3/4)*sqrt(a + b*x**4)) + a*d*atanh(sqrt(b)*x**2/sqrt(a + b*x**4))/(4*sqrt(b
)) + 2*a*e*x*sqrt(a + b*x**4)/(5*sqrt(b)*(sqrt(a) + sqrt(b)*x**2)) + d*x**2*sqrt
(a + b*x**4)/4 + x*sqrt(a + b*x**4)*(5*c + 3*e*x**2)/15 + f*(a + b*x**4)**(3/2)/
(6*b)

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Mathematica [C]  time = 0.763197, size = 257, normalized size = 0.78 \[ \frac{24 a^{3/2} \sqrt{b} e \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (\left (a+b x^4\right ) (10 a f+b x (20 c+x (15 d+2 x (6 e+5 f x))))+15 a \sqrt{b} d \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )\right )-8 a \sqrt{b} \sqrt{\frac{b x^4}{a}+1} \left (3 \sqrt{a} e+5 i \sqrt{b} c\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{60 b \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4],x]

[Out]

(Sqrt[(I*Sqrt[b])/Sqrt[a]]*((a + b*x^4)*(10*a*f + b*x*(20*c + x*(15*d + 2*x*(6*e
+ 5*f*x)))) + 15*a*Sqrt[b]*d*Sqrt[a + b*x^4]*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x
^4]]) + 24*a^(3/2)*Sqrt[b]*e*Sqrt[1 + (b*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqr
t[b])/Sqrt[a]]*x], -1] - 8*a*Sqrt[b]*((5*I)*Sqrt[b]*c + 3*Sqrt[a]*e)*Sqrt[1 + (b
*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(60*Sqrt[(I*Sqrt
[b])/Sqrt[a]]*b*Sqrt[a + b*x^4])

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Maple [C]  time = 0.011, size = 313, normalized size = 1. \[{\frac{cx}{3}\sqrt{b{x}^{4}+a}}+{\frac{2\,ac}{3}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{d{x}^{2}}{4}\sqrt{b{x}^{4}+a}}+{\frac{ad}{4}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{e{x}^{3}}{5}\sqrt{b{x}^{4}+a}}+{{\frac{2\,i}{5}}e{a}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}}-{{\frac{2\,i}{5}}e{a}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}}+{\frac{f}{6\,b} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2),x)

[Out]

1/3*c*x*(b*x^4+a)^(1/2)+2/3*c*a/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*x
^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)
*b^(1/2))^(1/2),I)+1/4*d*x^2*(b*x^4+a)^(1/2)+1/4*d*a/b^(1/2)*ln(b^(1/2)*x^2+(b*x
^4+a)^(1/2))+1/5*e*x^3*(b*x^4+a)^(1/2)+2/5*I*e*a^(3/2)/(I/a^(1/2)*b^(1/2))^(1/2)
*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)
/b^(1/2)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-2/5*I*e*a^(3/2)/(I/a^(1/2)*b^(
1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x
^4+a)^(1/2)/b^(1/2)*EllipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I)+1/6*f*(b*x^4+a)^(3/
2)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c),x, algorithm="maxima")

[Out]

integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c),x, algorithm="fricas")

[Out]

integral(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c), x)

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Sympy [A]  time = 5.15827, size = 156, normalized size = 0.47 \[ \frac{\sqrt{a} c x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} + \frac{\sqrt{a} d x^{2} \sqrt{1 + \frac{b x^{4}}{a}}}{4} + \frac{\sqrt{a} e x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{a d \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 \sqrt{b}} + f \left (\begin{cases} \frac{\sqrt{a} x^{4}}{4} & \text{for}\: b = 0 \\\frac{\left (a + b x^{4}\right )^{\frac{3}{2}}}{6 b} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(1/2),x)

[Out]

sqrt(a)*c*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), b*x**4*exp_polar(I*pi)/a)/(4*g
amma(5/4)) + sqrt(a)*d*x**2*sqrt(1 + b*x**4/a)/4 + sqrt(a)*e*x**3*gamma(3/4)*hyp
er((-1/2, 3/4), (7/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(7/4)) + a*d*asinh(sqr
t(b)*x**2/sqrt(a))/(4*sqrt(b)) + f*Piecewise((sqrt(a)*x**4/4, Eq(b, 0)), ((a + b
*x**4)**(3/2)/(6*b), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c),x, algorithm="giac")

[Out]

integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c), x)