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

Optimal. Leaf size=369 \[ \frac{a^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (21 \sqrt{b} c-5 \sqrt{a} e\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 b^{5/4} \sqrt{a+b x^4}}-\frac{2 a^{5/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+b x^4}}-\frac{a^2 f \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 b^{3/2}}+\frac{1}{35} x^3 \sqrt{a+b x^4} \left (7 c+5 e x^2\right )+\frac{2 a c x \sqrt{a+b x^4}}{5 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{\left (a+b x^4\right )^{3/2} \left (4 d+3 f x^2\right )}{24 b}+\frac{2 a e x \sqrt{a+b x^4}}{21 b}-\frac{a f x^2 \sqrt{a+b x^4}}{16 b} \]

[Out]

(2*a*e*x*Sqrt[a + b*x^4])/(21*b) - (a*f*x^2*Sqrt[a + b*x^4])/(16*b) + (2*a*c*x*S
qrt[a + b*x^4])/(5*Sqrt[b]*(Sqrt[a] + Sqrt[b]*x^2)) + (x^3*(7*c + 5*e*x^2)*Sqrt[
a + b*x^4])/35 + ((4*d + 3*f*x^2)*(a + b*x^4)^(3/2))/(24*b) - (a^2*f*ArcTanh[(Sq
rt[b]*x^2)/Sqrt[a + b*x^4]])/(16*b^(3/2)) - (2*a^(5/4)*c*(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)*Sqrt[a + b*x^4]) + (a^(5/4)*(21*Sqrt[b]*c - 5*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])/(105*b^(5/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.819695, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.367 \[ \frac{a^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (21 \sqrt{b} c-5 \sqrt{a} e\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 b^{5/4} \sqrt{a+b x^4}}-\frac{2 a^{5/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+b x^4}}-\frac{a^2 f \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 b^{3/2}}+\frac{1}{35} x^3 \sqrt{a+b x^4} \left (7 c+5 e x^2\right )+\frac{2 a c x \sqrt{a+b x^4}}{5 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{\left (a+b x^4\right )^{3/2} \left (4 d+3 f x^2\right )}{24 b}+\frac{2 a e x \sqrt{a+b x^4}}{21 b}-\frac{a f x^2 \sqrt{a+b x^4}}{16 b} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*e*x*Sqrt[a + b*x^4])/(21*b) - (a*f*x^2*Sqrt[a + b*x^4])/(16*b) + (2*a*c*x*S
qrt[a + b*x^4])/(5*Sqrt[b]*(Sqrt[a] + Sqrt[b]*x^2)) + (x^3*(7*c + 5*e*x^2)*Sqrt[
a + b*x^4])/35 + ((4*d + 3*f*x^2)*(a + b*x^4)^(3/2))/(24*b) - (a^2*f*ArcTanh[(Sq
rt[b]*x^2)/Sqrt[a + b*x^4]])/(16*b^(3/2)) - (2*a^(5/4)*c*(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)*Sqrt[a + b*x^4]) + (a^(5/4)*(21*Sqrt[b]*c - 5*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])/(105*b^(5/4)*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 75.3073, size = 338, normalized size = 0.92 \[ - \frac{2 a^{\frac{5}{4}} c \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} \sqrt{a + b x^{4}}} - \frac{a^{\frac{5}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) \left (5 \sqrt{a} e - 21 \sqrt{b} c\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{105 b^{\frac{5}{4}} \sqrt{a + b x^{4}}} - \frac{a^{2} f \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a + b x^{4}}} \right )}}{16 b^{\frac{3}{2}}} + \frac{2 a e x \sqrt{a + b x^{4}}}{21 b} - \frac{a f x^{2} \sqrt{a + b x^{4}}}{16 b} + \frac{2 a c x \sqrt{a + b x^{4}}}{5 \sqrt{b} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} + \frac{x^{3} \sqrt{a + b x^{4}} \left (7 c + 5 e x^{2}\right )}{35} + \frac{\left (a + b x^{4}\right )^{\frac{3}{2}} \left (4 d + 3 f x^{2}\right )}{24 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(5/4)*c*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)*e - 21*sqrt(b)*c)*elliptic_f(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(1
05*b**(5/4)*sqrt(a + b*x**4)) - a**2*f*atanh(sqrt(b)*x**2/sqrt(a + b*x**4))/(16*
b**(3/2)) + 2*a*e*x*sqrt(a + b*x**4)/(21*b) - a*f*x**2*sqrt(a + b*x**4)/(16*b) +
2*a*c*x*sqrt(a + b*x**4)/(5*sqrt(b)*(sqrt(a) + sqrt(b)*x**2)) + x**3*sqrt(a + b
*x**4)*(7*c + 5*e*x**2)/35 + (a + b*x**4)**(3/2)*(4*d + 3*f*x**2)/(24*b)

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Mathematica [C]  time = 0.72535, size = 280, normalized size = 0.76 \[ \frac{32 i a^{3/2} \sqrt{b} \sqrt{\frac{b x^4}{a}+1} \left (5 \sqrt{a} e+21 i \sqrt{b} c\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+672 a^{3/2} b c \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (\sqrt{b} \left (a+b x^4\right ) \left (5 a (56 d+x (32 e+21 f x))+2 b x^3 (168 c+5 x (28 d+3 x (8 e+7 f x)))\right )-105 a^2 f \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )\right )}{1680 b^{3/2} \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(I*Sqrt[b])/Sqrt[a]]*(Sqrt[b]*(a + b*x^4)*(5*a*(56*d + x*(32*e + 21*f*x))
+ 2*b*x^3*(168*c + 5*x*(28*d + 3*x*(8*e + 7*f*x)))) - 105*a^2*f*Sqrt[a + b*x^4]*
ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]]) + 672*a^(3/2)*b*c*Sqrt[1 + (b*x^4)/a]*El
lipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] + (32*I)*a^(3/2)*Sqrt[b]*((2
1*I)*Sqrt[b]*c + 5*Sqrt[a]*e)*Sqrt[1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sq
rt[b])/Sqrt[a]]*x], -1])/(1680*Sqrt[(I*Sqrt[b])/Sqrt[a]]*b^(3/2)*Sqrt[a + b*x^4]
)

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Maple [C]  time = 0.015, size = 361, normalized size = 1. \[{\frac{d}{6\,b} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{c{x}^{3}}{5}\sqrt{b{x}^{4}+a}}+{{\frac{2\,i}{5}}c{a}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}}-{{\frac{2\,i}{5}}c{a}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}}+{\frac{e{x}^{5}}{7}\sqrt{b{x}^{4}+a}}+{\frac{2\,aex}{21\,b}\sqrt{b{x}^{4}+a}}-{\frac{2\,e{a}^{2}}{21\,b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{f{x}^{2}}{8\,b} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{x}^{2}af}{16\,b}\sqrt{b{x}^{4}+a}}-{\frac{{a}^{2}f}{16}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*d/b*(b*x^4+a)^(3/2)+1/5*c*x^3*(b*x^4+a)^(1/2)+2/5*I*c*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*c*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)+1/7*e*x^
5*(b*x^4+a)^(1/2)+2/21*a*e*x*(b*x^4+a)^(1/2)/b-2/21*e/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)+1/8*f*x^2*(b*x^4+a)^(3/2)/b-1/16*
a*f*x^2*(b*x^4+a)^(1/2)/b-1/16*f*a^2/b^(3/2)*ln(b^(1/2)*x^2+(b*x^4+a)^(1/2))

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

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^2,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (f x^{5} + e x^{4} + d x^{3} + c x^{2}\right )} \sqrt{b x^{4} + a}, x\right ) \]

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^2,x, algorithm="fricas")

[Out]

integral((f*x^5 + e*x^4 + d*x^3 + c*x^2)*sqrt(b*x^4 + a), x)

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Sympy [A]  time = 8.43041, size = 212, normalized size = 0.57 \[ \frac{a^{\frac{3}{2}} f x^{2}}{16 b \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} c x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{\sqrt{a} e x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{3 \sqrt{a} f x^{6}}{16 \sqrt{1 + \frac{b x^{4}}{a}}} - \frac{a^{2} f \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 b^{\frac{3}{2}}} + d \left (\begin{cases} \frac{\sqrt{a} x^{4}}{4} & \text{for}\: b = 0 \\\frac{\left (a + b x^{4}\right )^{\frac{3}{2}}}{6 b} & \text{otherwise} \end{cases}\right ) + \frac{b f x^{10}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{4}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/2)*f*x**2/(16*b*sqrt(1 + b*x**4/a)) + sqrt(a)*c*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)*e*x**5*gamma
(5/4)*hyper((-1/2, 5/4), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(9/4)) + 3*sq
rt(a)*f*x**6/(16*sqrt(1 + b*x**4/a)) - a**2*f*asinh(sqrt(b)*x**2/sqrt(a))/(16*b*
*(3/2)) + d*Piecewise((sqrt(a)*x**4/4, Eq(b, 0)), ((a + b*x**4)**(3/2)/(6*b), Tr
ue)) + b*f*x**10/(8*sqrt(a)*sqrt(1 + b*x**4/a))

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

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^2,x, algorithm="giac")

[Out]

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