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

Optimal. Leaf size=476 \[ -\frac{2 a^{11/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (77 \sqrt{a} e+65 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5005 b^{7/4} \sqrt{a+b x^4}}+\frac{4 a^{13/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^4}}-\frac{a^3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{32 b^{3/2}}-\frac{4 a^3 e x \sqrt{a+b x^4}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{4 a^2 c x \sqrt{a+b x^4}}{77 b}-\frac{a^2 d x^2 \sqrt{a+b x^4}}{32 b}+\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}-\frac{\left (a+b x^4\right )^{5/2} \left (12 a f-35 b d x^2\right )}{420 b^2}+\frac{1}{143} x^5 \left (a+b x^4\right )^{3/2} \left (13 c+11 e x^2\right )+\frac{2 a x^5 \sqrt{a+b x^4} \left (117 c+77 e x^2\right )}{3003}-\frac{a d x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b} \]

[Out]

(4*a^2*c*x*Sqrt[a + b*x^4])/(77*b) - (a^2*d*x^2*Sqrt[a + b*x^4])/(32*b) + (4*a^2
*e*x^3*Sqrt[a + b*x^4])/(195*b) - (4*a^3*e*x*Sqrt[a + b*x^4])/(65*b^(3/2)*(Sqrt[
a] + Sqrt[b]*x^2)) + (2*a*x^5*(117*c + 77*e*x^2)*Sqrt[a + b*x^4])/3003 - (a*d*x^
2*(a + b*x^4)^(3/2))/(48*b) + (x^5*(13*c + 11*e*x^2)*(a + b*x^4)^(3/2))/143 + (f
*x^4*(a + b*x^4)^(5/2))/(14*b) - ((12*a*f - 35*b*d*x^2)*(a + b*x^4)^(5/2))/(420*
b^2) - (a^3*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/(32*b^(3/2)) + (4*a^(13/4)
*e*(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])/(65*b^(7/4)*Sqrt[a + b*x^4]) - (2*a^(11/4)
*(65*Sqrt[b]*c + 77*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])/(5005*b^(7/4)*
Sqrt[a + b*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 1.31135, antiderivative size = 476, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{2 a^{11/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (77 \sqrt{a} e+65 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5005 b^{7/4} \sqrt{a+b x^4}}+\frac{4 a^{13/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^4}}-\frac{a^3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{32 b^{3/2}}-\frac{4 a^3 e x \sqrt{a+b x^4}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{4 a^2 c x \sqrt{a+b x^4}}{77 b}-\frac{a^2 d x^2 \sqrt{a+b x^4}}{32 b}+\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}-\frac{\left (a+b x^4\right )^{5/2} \left (12 a f-35 b d x^2\right )}{420 b^2}+\frac{1}{143} x^5 \left (a+b x^4\right )^{3/2} \left (13 c+11 e x^2\right )+\frac{2 a x^5 \sqrt{a+b x^4} \left (117 c+77 e x^2\right )}{3003}-\frac{a d x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b} \]

Antiderivative was successfully verified.

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

[Out]

(4*a^2*c*x*Sqrt[a + b*x^4])/(77*b) - (a^2*d*x^2*Sqrt[a + b*x^4])/(32*b) + (4*a^2
*e*x^3*Sqrt[a + b*x^4])/(195*b) - (4*a^3*e*x*Sqrt[a + b*x^4])/(65*b^(3/2)*(Sqrt[
a] + Sqrt[b]*x^2)) + (2*a*x^5*(117*c + 77*e*x^2)*Sqrt[a + b*x^4])/3003 - (a*d*x^
2*(a + b*x^4)^(3/2))/(48*b) + (x^5*(13*c + 11*e*x^2)*(a + b*x^4)^(3/2))/143 + (f
*x^4*(a + b*x^4)^(5/2))/(14*b) - ((12*a*f - 35*b*d*x^2)*(a + b*x^4)^(5/2))/(420*
b^2) - (a^3*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/(32*b^(3/2)) + (4*a^(13/4)
*e*(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])/(65*b^(7/4)*Sqrt[a + b*x^4]) - (2*a^(11/4)
*(65*Sqrt[b]*c + 77*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])/(5005*b^(7/4)*
Sqrt[a + b*x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 116.903, size = 442, normalized size = 0.93 \[ \frac{4 a^{\frac{13}{4}} e \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{65 b^{\frac{7}{4}} \sqrt{a + b x^{4}}} - \frac{2 a^{\frac{11}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) \left (77 \sqrt{a} e + 65 \sqrt{b} c\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5005 b^{\frac{7}{4}} \sqrt{a + b x^{4}}} - \frac{a^{3} d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a + b x^{4}}} \right )}}{32 b^{\frac{3}{2}}} - \frac{4 a^{3} e x \sqrt{a + b x^{4}}}{65 b^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} + \frac{4 a^{2} c x \sqrt{a + b x^{4}}}{77 b} - \frac{a^{2} d x^{2} \sqrt{a + b x^{4}}}{32 b} + \frac{4 a^{2} e x^{3} \sqrt{a + b x^{4}}}{195 b} + \frac{2 a x^{5} \sqrt{a + b x^{4}} \left (117 c + 77 e x^{2}\right )}{3003} - \frac{a d x^{2} \left (a + b x^{4}\right )^{\frac{3}{2}}}{48 b} + \frac{x^{5} \left (a + b x^{4}\right )^{\frac{3}{2}} \left (13 c + 11 e x^{2}\right )}{143} + \frac{f x^{4} \left (a + b x^{4}\right )^{\frac{5}{2}}}{14 b} - \frac{\left (a + b x^{4}\right )^{\frac{5}{2}} \left (12 a f - 35 b d x^{2}\right )}{420 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a**(13/4)*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)/(65*b**(7/4)*sqrt(a + b*x**4)
) - 2*a**(11/4)*sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b
)*x**2)*(77*sqrt(a)*e + 65*sqrt(b)*c)*elliptic_f(2*atan(b**(1/4)*x/a**(1/4)), 1/
2)/(5005*b**(7/4)*sqrt(a + b*x**4)) - a**3*d*atanh(sqrt(b)*x**2/sqrt(a + b*x**4)
)/(32*b**(3/2)) - 4*a**3*e*x*sqrt(a + b*x**4)/(65*b**(3/2)*(sqrt(a) + sqrt(b)*x*
*2)) + 4*a**2*c*x*sqrt(a + b*x**4)/(77*b) - a**2*d*x**2*sqrt(a + b*x**4)/(32*b)
+ 4*a**2*e*x**3*sqrt(a + b*x**4)/(195*b) + 2*a*x**5*sqrt(a + b*x**4)*(117*c + 77
*e*x**2)/3003 - a*d*x**2*(a + b*x**4)**(3/2)/(48*b) + x**5*(a + b*x**4)**(3/2)*(
13*c + 11*e*x**2)/143 + f*x**4*(a + b*x**4)**(5/2)/(14*b) - (a + b*x**4)**(5/2)*
(12*a*f - 35*b*d*x**2)/(420*b**2)

_______________________________________________________________________________________

Mathematica [C]  time = 1.09568, size = 327, normalized size = 0.69 \[ \frac{-29568 a^{7/2} \sqrt{b} e \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+384 a^3 \sqrt{b} \sqrt{\frac{b x^4}{a}+1} \left (77 \sqrt{a} e+65 i \sqrt{b} c\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (-15015 a^3 \sqrt{b} d \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )-\left (a+b x^4\right ) \left (13728 a^3 f-a^2 b x \left (24960 c+11 x \left (1365 d+896 e x+624 f x^2\right )\right )-2 a b^2 x^5 \left (40560 c+11 x \left (3185 d+2800 e x+2496 f x^2\right )\right )-40 b^3 x^9 \left (1092 c+11 x \left (91 d+84 e x+78 f x^2\right )\right )\right )\right )}{480480 b^2 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(I*Sqrt[b])/Sqrt[a]]*(-((a + b*x^4)*(13728*a^3*f - 40*b^3*x^9*(1092*c + 11
*x*(91*d + 84*e*x + 78*f*x^2)) - a^2*b*x*(24960*c + 11*x*(1365*d + 896*e*x + 624
*f*x^2)) - 2*a*b^2*x^5*(40560*c + 11*x*(3185*d + 2800*e*x + 2496*f*x^2)))) - 150
15*a^3*Sqrt[b]*d*Sqrt[a + b*x^4]*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]]) - 29568
*a^(7/2)*Sqrt[b]*e*Sqrt[1 + (b*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt
[a]]*x], -1] + 384*a^3*Sqrt[b]*((65*I)*Sqrt[b]*c + 77*Sqrt[a]*e)*Sqrt[1 + (b*x^4
)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(480480*Sqrt[(I*Sqrt
[b])/Sqrt[a]]*b^2*Sqrt[a + b*x^4])

_______________________________________________________________________________________

Maple [C]  time = 0.047, size = 462, normalized size = 1. \[{\frac{bc{x}^{9}}{11}\sqrt{b{x}^{4}+a}}+{\frac{13\,ac{x}^{5}}{77}\sqrt{b{x}^{4}+a}}+{\frac{4\,{a}^{2}cx}{77\,b}\sqrt{b{x}^{4}+a}}-{\frac{4\,{a}^{3}c}{77\,b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{{a}^{2}d{x}^{2}}{32\,b}\sqrt{b{x}^{4}+a}}-{\frac{{a}^{3}d}{32}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{bd{x}^{10}}{12}\sqrt{b{x}^{4}+a}}+{\frac{7\,ad{x}^{6}}{48}\sqrt{b{x}^{4}+a}}+{\frac{be{x}^{11}}{13}\sqrt{b{x}^{4}+a}}+{\frac{5\,ae{x}^{7}}{39}\sqrt{b{x}^{4}+a}}+{\frac{4\,{a}^{2}e{x}^{3}}{195\,b}\sqrt{b{x}^{4}+a}}-{{\frac{4\,i}{65}}e{a}^{{\frac{7}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{{\frac{4\,i}{65}}e{a}^{{\frac{7}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{f \left ( -5\,b{x}^{4}+2\,a \right ) \left ({b}^{2}{x}^{8}+2\,ab{x}^{4}+{a}^{2} \right ) }{70\,{b}^{2}}\sqrt{b{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*c*b*x^9*(b*x^4+a)^(1/2)+13/77*c*a*x^5*(b*x^4+a)^(1/2)+4/77*a^2*c*x*(b*x^4+a
)^(1/2)/b-4/77*c/b*a^3/(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)+1/32*a^2*d*x^2*(b*x^4+a)^(1/2)/b-1/32*d*a^3/b^(3/2)*ln(b^(1/2)*x^2+(b*
x^4+a)^(1/2))+1/12*d*b*x^10*(b*x^4+a)^(1/2)+7/48*d*a*x^6*(b*x^4+a)^(1/2)+1/13*e*
b*x^11*(b*x^4+a)^(1/2)+5/39*e*a*x^7*(b*x^4+a)^(1/2)+4/195*a^2*e*x^3*(b*x^4+a)^(1
/2)/b-4/65*I*e/b^(3/2)*a^(7/2)/(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)+4/65*I*e/b^(3/2)*a^(7/2)/(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)*EllipticE(x
*(I/a^(1/2)*b^(1/2))^(1/2),I)-1/70*f*(b*x^4+a)^(1/2)*(-5*b*x^4+2*a)*(b^2*x^8+2*a
*b*x^4+a^2)/b^2

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b f x^{11} + b e x^{10} + b d x^{9} + b c x^{8} + a f x^{7} + a e x^{6} + a d x^{5} + a c x^{4}\right )} \sqrt{b x^{4} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*f*x^11 + b*e*x^10 + b*d*x^9 + b*c*x^8 + a*f*x^7 + a*e*x^6 + a*d*x^5
+ a*c*x^4)*sqrt(b*x^4 + a), x)

_______________________________________________________________________________________

Sympy [A]  time = 28.9211, size = 462, normalized size = 0.97 \[ \frac{a^{\frac{5}{2}} d x^{2}}{32 b \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{a^{\frac{3}{2}} c x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{17 a^{\frac{3}{2}} d x^{6}}{96 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{a^{\frac{3}{2}} e x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} + \frac{\sqrt{a} b c x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{13}{4}\right )} + \frac{11 \sqrt{a} b d x^{10}}{48 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} b e x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{15}{4}\right )} - \frac{a^{3} d \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{32 b^{\frac{3}{2}}} + a f \left (\begin{cases} - \frac{a^{2} \sqrt{a + b x^{4}}}{15 b^{2}} + \frac{a x^{4} \sqrt{a + b x^{4}}}{30 b} + \frac{x^{8} \sqrt{a + b x^{4}}}{10} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{8}}{8} & \text{otherwise} \end{cases}\right ) + b f \left (\begin{cases} \frac{4 a^{3} \sqrt{a + b x^{4}}}{105 b^{3}} - \frac{2 a^{2} x^{4} \sqrt{a + b x^{4}}}{105 b^{2}} + \frac{a x^{8} \sqrt{a + b x^{4}}}{70 b} + \frac{x^{12} \sqrt{a + b x^{4}}}{14} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{12}}{12} & \text{otherwise} \end{cases}\right ) + \frac{b^{2} d x^{14}}{12 \sqrt{a} \sqrt{1 + \frac{b x^{4}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(5/2)*d*x**2/(32*b*sqrt(1 + b*x**4/a)) + a**(3/2)*c*x**5*gamma(5/4)*hyper((-1
/2, 5/4), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(9/4)) + 17*a**(3/2)*d*x**6/
(96*sqrt(1 + b*x**4/a)) + a**(3/2)*e*x**7*gamma(7/4)*hyper((-1/2, 7/4), (11/4,),
b*x**4*exp_polar(I*pi)/a)/(4*gamma(11/4)) + sqrt(a)*b*c*x**9*gamma(9/4)*hyper((
-1/2, 9/4), (13/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(13/4)) + 11*sqrt(a)*b*d*
x**10/(48*sqrt(1 + b*x**4/a)) + sqrt(a)*b*e*x**11*gamma(11/4)*hyper((-1/2, 11/4)
, (15/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(15/4)) - a**3*d*asinh(sqrt(b)*x**2
/sqrt(a))/(32*b**(3/2)) + a*f*Piecewise((-a**2*sqrt(a + b*x**4)/(15*b**2) + a*x*
*4*sqrt(a + b*x**4)/(30*b) + x**8*sqrt(a + b*x**4)/10, Ne(b, 0)), (sqrt(a)*x**8/
8, True)) + b*f*Piecewise((4*a**3*sqrt(a + b*x**4)/(105*b**3) - 2*a**2*x**4*sqrt
(a + b*x**4)/(105*b**2) + a*x**8*sqrt(a + b*x**4)/(70*b) + x**12*sqrt(a + b*x**4
)/14, Ne(b, 0)), (sqrt(a)*x**12/12, True)) + b**2*d*x**14/(12*sqrt(a)*sqrt(1 + b
*x**4/a))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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