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

Optimal. Leaf size=354 \[ \frac{a^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (21 \sqrt{b} d-5 \sqrt{a} f\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 b^{5/4} \sqrt{a+b x^4}}-\frac{2 a^{5/4} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+b x^4}}+\frac{1}{4} c x^2 \sqrt{a+b x^4}+\frac{a c \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 \sqrt{b}}+\frac{1}{35} x^3 \sqrt{a+b x^4} \left (7 d+5 f x^2\right )+\frac{2 a d x \sqrt{a+b x^4}}{5 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{e \left (a+b x^4\right )^{3/2}}{6 b}+\frac{2 a f x \sqrt{a+b x^4}}{21 b} \]

[Out]

(2*a*f*x*Sqrt[a + b*x^4])/(21*b) + (c*x^2*Sqrt[a + b*x^4])/4 + (2*a*d*x*Sqrt[a +
b*x^4])/(5*Sqrt[b]*(Sqrt[a] + Sqrt[b]*x^2)) + (x^3*(7*d + 5*f*x^2)*Sqrt[a + b*x
^4])/35 + (e*(a + b*x^4)^(3/2))/(6*b) + (a*c*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^
4]])/(4*Sqrt[b]) - (2*a^(5/4)*d*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a
] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(5*b^(3/4)*Sq
rt[a + b*x^4]) + (a^(5/4)*(21*Sqrt[b]*d - 5*Sqrt[a]*f)*(Sqrt[a] + Sqrt[b]*x^2)*S
qrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4
)], 1/2])/(105*b^(5/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.735569, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393 \[ \frac{a^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (21 \sqrt{b} d-5 \sqrt{a} f\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 b^{5/4} \sqrt{a+b x^4}}-\frac{2 a^{5/4} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+b x^4}}+\frac{1}{4} c x^2 \sqrt{a+b x^4}+\frac{a c \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 \sqrt{b}}+\frac{1}{35} x^3 \sqrt{a+b x^4} \left (7 d+5 f x^2\right )+\frac{2 a d x \sqrt{a+b x^4}}{5 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{e \left (a+b x^4\right )^{3/2}}{6 b}+\frac{2 a f x \sqrt{a+b x^4}}{21 b} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*f*x*Sqrt[a + b*x^4])/(21*b) + (c*x^2*Sqrt[a + b*x^4])/4 + (2*a*d*x*Sqrt[a +
b*x^4])/(5*Sqrt[b]*(Sqrt[a] + Sqrt[b]*x^2)) + (x^3*(7*d + 5*f*x^2)*Sqrt[a + b*x
^4])/35 + (e*(a + b*x^4)^(3/2))/(6*b) + (a*c*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^
4]])/(4*Sqrt[b]) - (2*a^(5/4)*d*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a
] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(5*b^(3/4)*Sq
rt[a + b*x^4]) + (a^(5/4)*(21*Sqrt[b]*d - 5*Sqrt[a]*f)*(Sqrt[a] + Sqrt[b]*x^2)*S
qrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4
)], 1/2])/(105*b^(5/4)*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 69.5724, size = 325, normalized size = 0.92 \[ - \frac{2 a^{\frac{5}{4}} d \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} \sqrt{a + b x^{4}}} - \frac{a^{\frac{5}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) \left (5 \sqrt{a} f - 21 \sqrt{b} d\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{105 b^{\frac{5}{4}} \sqrt{a + b x^{4}}} + \frac{2 a f x \sqrt{a + b x^{4}}}{21 b} + \frac{a c \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a + b x^{4}}} \right )}}{4 \sqrt{b}} + \frac{2 a d x \sqrt{a + b x^{4}}}{5 \sqrt{b} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} + \frac{c x^{2} \sqrt{a + b x^{4}}}{4} + \frac{x^{3} \sqrt{a + b x^{4}} \left (7 d + 5 f x^{2}\right )}{35} + \frac{e \left (a + b x^{4}\right )^{\frac{3}{2}}}{6 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(5/4)*d*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**(5/4)*sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*x*
*2)*(5*sqrt(a)*f - 21*sqrt(b)*d)*elliptic_f(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(1
05*b**(5/4)*sqrt(a + b*x**4)) + 2*a*f*x*sqrt(a + b*x**4)/(21*b) + a*c*atanh(sqrt
(b)*x**2/sqrt(a + b*x**4))/(4*sqrt(b)) + 2*a*d*x*sqrt(a + b*x**4)/(5*sqrt(b)*(sq
rt(a) + sqrt(b)*x**2)) + c*x**2*sqrt(a + b*x**4)/4 + x**3*sqrt(a + b*x**4)*(7*d
+ 5*f*x**2)/35 + e*(a + b*x**4)**(3/2)/(6*b)

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Mathematica [C]  time = 0.719077, size = 266, normalized size = 0.75 \[ \frac{8 i a^{3/2} \sqrt{\frac{b x^4}{a}+1} \left (5 \sqrt{a} f+21 i \sqrt{b} d\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+168 a^{3/2} \sqrt{b} d \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (\left (a+b x^4\right ) \left (10 a (7 e+4 f x)+b x^2 \left (105 c+84 d x+70 e x^2+60 f x^3\right )\right )+105 a \sqrt{b} c \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )\right )}{420 b \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(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*(7*e + 4*f*x) + b*x^2*(105*c + 84*
d*x + 70*e*x^2 + 60*f*x^3)) + 105*a*Sqrt[b]*c*Sqrt[a + b*x^4]*ArcTanh[(Sqrt[b]*x
^2)/Sqrt[a + b*x^4]]) + 168*a^(3/2)*Sqrt[b]*d*Sqrt[1 + (b*x^4)/a]*EllipticE[I*Ar
cSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] + (8*I)*a^(3/2)*((21*I)*Sqrt[b]*d + 5*Sq
rt[a]*f)*Sqrt[1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -
1])/(420*Sqrt[(I*Sqrt[b])/Sqrt[a]]*b*Sqrt[a + b*x^4])

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Maple [C]  time = 0.013, size = 337, normalized size = 1. \[{\frac{d{x}^{3}}{5}\sqrt{b{x}^{4}+a}}+{{\frac{2\,i}{5}}d{a}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}}-{{\frac{2\,i}{5}}d{a}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}}+{\frac{c{x}^{2}}{4}\sqrt{b{x}^{4}+a}}+{\frac{ac}{4}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{e}{6\,b} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{f{x}^{5}}{7}\sqrt{b{x}^{4}+a}}+{\frac{2\,afx}{21\,b}\sqrt{b{x}^{4}+a}}-{\frac{2\,{a}^{2}f}{21\,b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*x^3*d*(b*x^4+a)^(1/2)+2/5*I*d*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)*Elli
pticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-2/5*I*d*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)+1/4*c*x^2*(b*x^4+a)^(1/2)+1/4*c*
a/b^(1/2)*ln(b^(1/2)*x^2+(b*x^4+a)^(1/2))+1/6*e*(b*x^4+a)^(3/2)/b+1/7*f*x^5*(b*x
^4+a)^(1/2)+2/21*a*f*x*(b*x^4+a)^(1/2)/b-2/21*f/b*a^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)*
EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

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,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.44284, size = 158, normalized size = 0.45 \[ \frac{\sqrt{a} c x^{2} \sqrt{1 + \frac{b x^{4}}{a}}}{4} + \frac{\sqrt{a} d x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{\sqrt{a} f x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{a c \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 \sqrt{b}} + e \left (\begin{cases} \frac{\sqrt{a} x^{4}}{4} & \text{for}\: b = 0 \\\frac{\left (a + b x^{4}\right )^{\frac{3}{2}}}{6 b} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*c*x**2*sqrt(1 + b*x**4/a)/4 + sqrt(a)*d*x**3*gamma(3/4)*hyper((-1/2, 3/4
), (7/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(7/4)) + sqrt(a)*f*x**5*gamma(5/4)*
hyper((-1/2, 5/4), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(9/4)) + a*c*asinh(
sqrt(b)*x**2/sqrt(a))/(4*sqrt(b)) + e*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. \[ \int \sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )} x\,{d x} \]

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,x, algorithm="giac")

[Out]

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