Integrand size = 30, antiderivative size = 427 \[ \int x^2 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\frac {4 a^2 e x \sqrt {a+b x^4}}{77 b}-\frac {a^2 f x^2 \sqrt {a+b x^4}}{32 b}+\frac {4 a^2 c x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {2 a x^3 \left (77 c+45 e x^2\right ) \sqrt {a+b x^4}}{1155}-\frac {a f x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac {1}{99} x^3 \left (11 c+9 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {\left (6 d+5 f x^2\right ) \left (a+b x^4\right )^{5/2}}{60 b}-\frac {a^3 f \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{32 b^{3/2}}-\frac {4 a^{9/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}+\frac {2 a^{9/4} \left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{1155 b^{5/4} \sqrt {a+b x^4}} \]
-1/48*a*f*x^2*(b*x^4+a)^(3/2)/b+1/99*x^3*(9*e*x^2+11*c)*(b*x^4+a)^(3/2)+1/ 60*(5*f*x^2+6*d)*(b*x^4+a)^(5/2)/b-1/32*a^3*f*arctanh(x^2*b^(1/2)/(b*x^4+a )^(1/2))/b^(3/2)+4/77*a^2*e*x*(b*x^4+a)^(1/2)/b-1/32*a^2*f*x^2*(b*x^4+a)^( 1/2)/b+2/1155*a*x^3*(45*e*x^2+77*c)*(b*x^4+a)^(1/2)+4/15*a^2*c*x*(b*x^4+a) ^(1/2)/b^(1/2)/(a^(1/2)+x^2*b^(1/2))-4/15*a^(9/4)*c*(cos(2*arctan(b^(1/4)* x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arct an(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1 /2)+x^2*b^(1/2))^2)^(1/2)/b^(3/4)/(b*x^4+a)^(1/2)+2/1155*a^(9/4)*(cos(2*ar ctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*Ellipti cF(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(-15*e*a^(1/2)+77*c*b^(1/ 2))*(a^(1/2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/b^(5/4 )/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.12 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.48 \[ \int x^2 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\frac {\sqrt {a+b x^4} \left (\frac {528 d \left (a+b x^4\right )^2}{b}+\frac {480 e x \left (a+b x^4\right )^2}{b}+\frac {55 f \left (\sqrt {b} x^2 \left (3 a^2+14 a b x^4+8 b^2 x^8\right )-\frac {3 a^{5/2} \text {arcsinh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {1+\frac {b x^4}{a}}}\right )}{b^{3/2}}-\frac {480 a^2 e x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^4}{a}\right )}{b \sqrt {1+\frac {b x^4}{a}}}+\frac {1760 a c x^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^4}{a}\right )}{\sqrt {1+\frac {b x^4}{a}}}\right )}{5280} \]
(Sqrt[a + b*x^4]*((528*d*(a + b*x^4)^2)/b + (480*e*x*(a + b*x^4)^2)/b + (5 5*f*(Sqrt[b]*x^2*(3*a^2 + 14*a*b*x^4 + 8*b^2*x^8) - (3*a^(5/2)*ArcSinh[(Sq rt[b]*x^2)/Sqrt[a]])/Sqrt[1 + (b*x^4)/a]))/b^(3/2) - (480*a^2*e*x*Hypergeo metric2F1[-3/2, 1/4, 5/4, -((b*x^4)/a)])/(b*Sqrt[1 + (b*x^4)/a]) + (1760*a *c*x^3*Hypergeometric2F1[-3/2, 3/4, 7/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a ]))/5280
Time = 0.68 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2372, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^2 \left (a+b x^4\right )^{3/2} \left (c+d x+e x^2+f x^3\right ) \, dx\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \int \left (x^2 \left (a+b x^4\right )^{3/2} \left (c+e x^2\right )+x^3 \left (a+b x^4\right )^{3/2} \left (d+f x^2\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{1155 b^{5/4} \sqrt {a+b x^4}}-\frac {4 a^{9/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}-\frac {a^3 f \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{32 b^{3/2}}+\frac {4 a^2 c x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {4 a^2 e x \sqrt {a+b x^4}}{77 b}-\frac {a^2 f x^2 \sqrt {a+b x^4}}{32 b}+\frac {2 a x^3 \sqrt {a+b x^4} \left (77 c+45 e x^2\right )}{1155}+\frac {1}{99} x^3 \left (a+b x^4\right )^{3/2} \left (11 c+9 e x^2\right )+\frac {d \left (a+b x^4\right )^{5/2}}{10 b}-\frac {a f x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac {f x^2 \left (a+b x^4\right )^{5/2}}{12 b}\) |
(4*a^2*e*x*Sqrt[a + b*x^4])/(77*b) - (a^2*f*x^2*Sqrt[a + b*x^4])/(32*b) + (4*a^2*c*x*Sqrt[a + b*x^4])/(15*Sqrt[b]*(Sqrt[a] + Sqrt[b]*x^2)) + (2*a*x^ 3*(77*c + 45*e*x^2)*Sqrt[a + b*x^4])/1155 - (a*f*x^2*(a + b*x^4)^(3/2))/(4 8*b) + (x^3*(11*c + 9*e*x^2)*(a + b*x^4)^(3/2))/99 + (d*(a + b*x^4)^(5/2)) /(10*b) + (f*x^2*(a + b*x^4)^(5/2))/(12*b) - (a^3*f*ArcTanh[(Sqrt[b]*x^2)/ Sqrt[a + b*x^4]])/(32*b^(3/2)) - (4*a^(9/4)*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])/(15*b^(3/4)*Sqrt[a + b*x^4]) + (2*a^(9/4)*(77*Sqrt[b]*c - 15* 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])/(1155*b^(5/4)*Sqrt[a + b*x^4])
3.6.12.3.1 Defintions of rubi rules used
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Mo dule[{q = Expon[Pq, x], j, k}, Int[Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0 ] && !PolyQ[Pq, x^(n/2)]
Result contains complex when optimal does not.
Time = 2.09 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.73
| method | result | size |
| risch | \(\frac {\left (9240 b^{2} f \,x^{10}+10080 b^{2} e \,x^{9}+11088 b^{2} d \,x^{8}+12320 b^{2} c \,x^{7}+16170 a b f \,x^{6}+18720 a b e \,x^{5}+22176 a b d \,x^{4}+27104 a b c \,x^{3}+3465 a^{2} f \,x^{2}+5760 a^{2} e x +11088 a^{2} d \right ) \sqrt {b \,x^{4}+a}}{110880 b}-\frac {a^{2} \left (\frac {960 a e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {4928 i \sqrt {b}\, c \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {1155 a f \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{2 \sqrt {b}}\right )}{18480 b}\) | \(310\) |
| default | \(f \left (\frac {b \,x^{10} \sqrt {b \,x^{4}+a}}{12}+\frac {7 a \,x^{6} \sqrt {b \,x^{4}+a}}{48}+\frac {a^{2} x^{2} \sqrt {b \,x^{4}+a}}{32 b}-\frac {a^{3} \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{32 b^{\frac {3}{2}}}\right )+e \left (\frac {b \,x^{9} \sqrt {b \,x^{4}+a}}{11}+\frac {13 a \,x^{5} \sqrt {b \,x^{4}+a}}{77}+\frac {4 a^{2} x \sqrt {b \,x^{4}+a}}{77 b}-\frac {4 a^{3} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+\frac {d \left (b \,x^{4}+a \right )^{\frac {5}{2}}}{10 b}+c \left (\frac {b \,x^{7} \sqrt {b \,x^{4}+a}}{9}+\frac {11 a \,x^{3} \sqrt {b \,x^{4}+a}}{45}+\frac {4 i a^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )\) | \(352\) |
| elliptic | \(\frac {b f \,x^{10} \sqrt {b \,x^{4}+a}}{12}+\frac {b e \,x^{9} \sqrt {b \,x^{4}+a}}{11}+\frac {b d \,x^{8} \sqrt {b \,x^{4}+a}}{10}+\frac {b c \,x^{7} \sqrt {b \,x^{4}+a}}{9}+\frac {7 a f \,x^{6} \sqrt {b \,x^{4}+a}}{48}+\frac {13 a e \,x^{5} \sqrt {b \,x^{4}+a}}{77}+\frac {a d \,x^{4} \sqrt {b \,x^{4}+a}}{5}+\frac {11 a c \,x^{3} \sqrt {b \,x^{4}+a}}{45}+\frac {a^{2} f \,x^{2} \sqrt {b \,x^{4}+a}}{32 b}+\frac {4 a^{2} e x \sqrt {b \,x^{4}+a}}{77 b}+\frac {a^{2} d \sqrt {b \,x^{4}+a}}{10 b}-\frac {4 e \,a^{3} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {a^{3} f \ln \left (2 x^{2} \sqrt {b}+2 \sqrt {b \,x^{4}+a}\right )}{32 b^{\frac {3}{2}}}+\frac {4 i a^{\frac {5}{2}} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\) | \(392\) |
1/110880*(9240*b^2*f*x^10+10080*b^2*e*x^9+11088*b^2*d*x^8+12320*b^2*c*x^7+ 16170*a*b*f*x^6+18720*a*b*e*x^5+22176*a*b*d*x^4+27104*a*b*c*x^3+3465*a^2*f *x^2+5760*a^2*e*x+11088*a^2*d)/b*(b*x^4+a)^(1/2)-1/18480*a^2/b*(960*a*e/(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)-492 8*I*b^(1/2)*c*a^(1/2)/(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)-EllipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I))+1155/2*a*f *ln(x^2*b^(1/2)+(b*x^4+a)^(1/2))/b^(1/2))
Time = 0.15 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.58 \[ \int x^2 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\frac {59136 \, a^{2} b^{\frac {3}{2}} c x \left (-\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 3465 \, a^{3} \sqrt {b} f x \log \left (-2 \, b x^{4} + 2 \, \sqrt {b x^{4} + a} \sqrt {b} x^{2} - a\right ) - 768 \, {\left (77 \, a^{2} b c + 15 \, a^{2} b e\right )} \sqrt {b} x \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 2 \, {\left (9240 \, b^{3} f x^{11} + 10080 \, b^{3} e x^{10} + 11088 \, b^{3} d x^{9} + 12320 \, b^{3} c x^{8} + 16170 \, a b^{2} f x^{7} + 18720 \, a b^{2} e x^{6} + 22176 \, a b^{2} d x^{5} + 27104 \, a b^{2} c x^{4} + 3465 \, a^{2} b f x^{3} + 5760 \, a^{2} b e x^{2} + 11088 \, a^{2} b d x + 29568 \, a^{2} b c\right )} \sqrt {b x^{4} + a}}{221760 \, b^{2} x} \]
1/221760*(59136*a^2*b^(3/2)*c*x*(-a/b)^(3/4)*elliptic_e(arcsin((-a/b)^(1/4 )/x), -1) + 3465*a^3*sqrt(b)*f*x*log(-2*b*x^4 + 2*sqrt(b*x^4 + a)*sqrt(b)* x^2 - a) - 768*(77*a^2*b*c + 15*a^2*b*e)*sqrt(b)*x*(-a/b)^(3/4)*elliptic_f (arcsin((-a/b)^(1/4)/x), -1) + 2*(9240*b^3*f*x^11 + 10080*b^3*e*x^10 + 110 88*b^3*d*x^9 + 12320*b^3*c*x^8 + 16170*a*b^2*f*x^7 + 18720*a*b^2*e*x^6 + 2 2176*a*b^2*d*x^5 + 27104*a*b^2*c*x^4 + 3465*a^2*b*f*x^3 + 5760*a^2*b*e*x^2 + 11088*a^2*b*d*x + 29568*a^2*b*c)*sqrt(b*x^4 + a))/(b^2*x)
Time = 9.12 (sec) , antiderivative size = 398, normalized size of antiderivative = 0.93 \[ \int x^2 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\frac {a^{\frac {5}{2}} f x^{2}}{32 b \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {a^{\frac {3}{2}} c x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {a^{\frac {3}{2}} e x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {17 a^{\frac {3}{2}} f x^{6}}{96 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} b c x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} + \frac {\sqrt {a} b e x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} + \frac {11 \sqrt {a} b f x^{10}}{48 \sqrt {1 + \frac {b x^{4}}{a}}} - \frac {a^{3} f \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{32 b^{\frac {3}{2}}} + a d \left (\begin {cases} \frac {\sqrt {a} x^{4}}{4} & \text {for}\: b = 0 \\\frac {\left (a + b x^{4}\right )^{\frac {3}{2}}}{6 b} & \text {otherwise} \end {cases}\right ) + b d \left (\begin {cases} - \frac {a^{2} \sqrt {a + b x^{4}}}{15 b^{2}} + \frac {a x^{4} \sqrt {a + b x^{4}}}{30 b} + \frac {x^{8} \sqrt {a + b x^{4}}}{10} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{8}}{8} & \text {otherwise} \end {cases}\right ) + \frac {b^{2} f x^{14}}{12 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \]
a**(5/2)*f*x**2/(32*b*sqrt(1 + b*x**4/a)) + a**(3/2)*c*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**(3/2 )*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)) + 17*a**(3/2)*f*x**6/(96*sqrt(1 + b*x**4/a)) + sqrt(a)*b*c*x **7*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), b*x**4*exp_polar(I*pi)/a)/(4*ga mma(11/4)) + sqrt(a)*b*e*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*f*x**10/(48*sqrt(1 + b *x**4/a)) - a**3*f*asinh(sqrt(b)*x**2/sqrt(a))/(32*b**(3/2)) + a*d*Piecewi se((sqrt(a)*x**4/4, Eq(b, 0)), ((a + b*x**4)**(3/2)/(6*b), True)) + b*d*Pi ecewise((-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**2*f*x **14/(12*sqrt(a)*sqrt(1 + b*x**4/a))
\[ \int x^2 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )} x^{2} \,d x } \]
\[ \int x^2 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\int x^2\,{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]