Integrand size = 30, antiderivative size = 452 \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\frac {4 a^2 d x \sqrt {a+b x^4}}{77 b}-\frac {a^2 e x^2 \sqrt {a+b x^4}}{32 b}+\frac {4 a^2 f x^3 \sqrt {a+b x^4}}{195 b}-\frac {4 a^3 f x \sqrt {a+b x^4}}{65 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {2 a x^5 \left (117 d+77 f x^2\right ) \sqrt {a+b x^4}}{3003}-\frac {a e x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac {1}{143} x^5 \left (13 d+11 f x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {\left (6 c+5 e x^2\right ) \left (a+b x^4\right )^{5/2}}{60 b}-\frac {a^3 e \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{32 b^{3/2}}+\frac {4 a^{13/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{65 b^{7/4} \sqrt {a+b x^4}}-\frac {2 a^{11/4} \left (65 \sqrt {b} d+77 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{5005 b^{7/4} \sqrt {a+b x^4}} \]
-1/48*a*e*x^2*(b*x^4+a)^(3/2)/b+1/143*x^5*(11*f*x^2+13*d)*(b*x^4+a)^(3/2)+ 1/60*(5*e*x^2+6*c)*(b*x^4+a)^(5/2)/b-1/32*a^3*e*arctanh(x^2*b^(1/2)/(b*x^4 +a)^(1/2))/b^(3/2)+4/77*a^2*d*x*(b*x^4+a)^(1/2)/b-1/32*a^2*e*x^2*(b*x^4+a) ^(1/2)/b+4/195*a^2*f*x^3*(b*x^4+a)^(1/2)/b+2/3003*a*x^5*(77*f*x^2+117*d)*( b*x^4+a)^(1/2)-4/65*a^3*f*x*(b*x^4+a)^(1/2)/b^(3/2)/(a^(1/2)+x^2*b^(1/2))+ 4/65*a^(13/4)*f*(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*arctan(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^(7/4)/(b *x^4+a)^(1/2)-2/5005*a^(11/4)*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/c os(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))) ,1/2*2^(1/2))*(77*f*a^(1/2)+65*d*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^(7/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.75 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.53 \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\frac {\sqrt {a+b x^4} \left (6864 \sqrt {b} c \left (a+b x^4\right )^2+6240 \sqrt {b} d x \left (a+b x^4\right )^2+5280 \sqrt {b} f x^3 \left (a+b x^4\right )^2+715 e \left (\sqrt {b} x^2 \left (3 a^2+14 a b x^4+8 b^2 x^8\right )-\frac {3 a^{5/2} \text {arcsinh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {1+\frac {b x^4}{a}}}\right )-\frac {6240 a^2 \sqrt {b} d x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^4}{a}\right )}{\sqrt {1+\frac {b x^4}{a}}}-\frac {5280 a^2 \sqrt {b} f x^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^4}{a}\right )}{\sqrt {1+\frac {b x^4}{a}}}\right )}{68640 b^{3/2}} \]
(Sqrt[a + b*x^4]*(6864*Sqrt[b]*c*(a + b*x^4)^2 + 6240*Sqrt[b]*d*x*(a + b*x ^4)^2 + 5280*Sqrt[b]*f*x^3*(a + b*x^4)^2 + 715*e*(Sqrt[b]*x^2*(3*a^2 + 14* a*b*x^4 + 8*b^2*x^8) - (3*a^(5/2)*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]])/Sqrt[1 + (b*x^4)/a]) - (6240*a^2*Sqrt[b]*d*x*Hypergeometric2F1[-3/2, 1/4, 5/4, -(( b*x^4)/a)])/Sqrt[1 + (b*x^4)/a] - (5280*a^2*Sqrt[b]*f*x^3*Hypergeometric2F 1[-3/2, 3/4, 7/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a]))/(68640*b^(3/2))
Time = 0.73 (sec) , antiderivative size = 465, 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.
| \(\displaystyle \int x^3 \left (a+b x^4\right )^{3/2} \left (c+d x+e x^2+f x^3\right ) \, dx\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \int \left (x^3 \left (a+b x^4\right )^{3/2} \left (c+e x^2\right )+x^4 \left (a+b x^4\right )^{3/2} \left (d+f x^2\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 a^{11/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (77 \sqrt {a} f+65 \sqrt {b} d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{5005 b^{7/4} \sqrt {a+b x^4}}+\frac {4 a^{13/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{65 b^{7/4} \sqrt {a+b x^4}}-\frac {a^3 e \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{32 b^{3/2}}-\frac {4 a^3 f x \sqrt {a+b x^4}}{65 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {4 a^2 d x \sqrt {a+b x^4}}{77 b}-\frac {a^2 e x^2 \sqrt {a+b x^4}}{32 b}+\frac {4 a^2 f x^3 \sqrt {a+b x^4}}{195 b}+\frac {c \left (a+b x^4\right )^{5/2}}{10 b}+\frac {1}{143} x^5 \left (a+b x^4\right )^{3/2} \left (13 d+11 f x^2\right )+\frac {2 a x^5 \sqrt {a+b x^4} \left (117 d+77 f x^2\right )}{3003}+\frac {e x^2 \left (a+b x^4\right )^{5/2}}{12 b}-\frac {a e x^2 \left (a+b x^4\right )^{3/2}}{48 b}\) |
(4*a^2*d*x*Sqrt[a + b*x^4])/(77*b) - (a^2*e*x^2*Sqrt[a + b*x^4])/(32*b) + (4*a^2*f*x^3*Sqrt[a + b*x^4])/(195*b) - (4*a^3*f*x*Sqrt[a + b*x^4])/(65*b^ (3/2)*(Sqrt[a] + Sqrt[b]*x^2)) + (2*a*x^5*(117*d + 77*f*x^2)*Sqrt[a + b*x^ 4])/3003 - (a*e*x^2*(a + b*x^4)^(3/2))/(48*b) + (x^5*(13*d + 11*f*x^2)*(a + b*x^4)^(3/2))/143 + (c*(a + b*x^4)^(5/2))/(10*b) + (e*x^2*(a + b*x^4)^(5 /2))/(12*b) - (a^3*e*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/(32*b^(3/2)) + (4*a^(13/4)*f*(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]*d + 77*Sqrt[a]*f)*(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])
3.6.11.3.1 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 = 1.99 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(\frac {\left (36960 b^{2} f \,x^{11}+40040 b^{2} e \,x^{10}+43680 b^{2} d \,x^{9}+48048 b^{2} c \,x^{8}+61600 a b f \,x^{7}+70070 a e b \,x^{6}+81120 x^{5} d b a +96096 a b c \,x^{4}+9856 a^{2} f \,x^{3}+15015 a^{2} e \,x^{2}+24960 a^{2} d x +48048 a^{2} c \right ) \sqrt {b \,x^{4}+a}}{480480 b}-\frac {a^{3} \left (\frac {4160 d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {4928 i f \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}+\frac {5005 e \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{2 \sqrt {b}}\right )}{80080 b}\) | \(317\) |
| default | \(f \left (\frac {b \,x^{11} \sqrt {b \,x^{4}+a}}{13}+\frac {5 a \,x^{7} \sqrt {b \,x^{4}+a}}{39}+\frac {4 a^{2} x^{3} \sqrt {b \,x^{4}+a}}{195 b}-\frac {4 i a^{\frac {7}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{65 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+e \left (\frac {b \,x^{10} \sqrt {b \,x^{4}+a}}{12}+\frac {7 a \,x^{6} \sqrt {b \,x^{4}+a}}{48}+\frac {a^{2} x^{2} \sqrt {b \,x^{4}+a}}{32 b}-\frac {a^{3} \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{32 b^{\frac {3}{2}}}\right )+d \left (\frac {b \,x^{9} \sqrt {b \,x^{4}+a}}{11}+\frac {13 a \,x^{5} \sqrt {b \,x^{4}+a}}{77}+\frac {4 a^{2} x \sqrt {b \,x^{4}+a}}{77 b}-\frac {4 a^{3} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+\frac {c \left (b \,x^{4}+a \right )^{\frac {5}{2}}}{10 b}\) | \(372\) |
| elliptic | \(\frac {b f \,x^{11} \sqrt {b \,x^{4}+a}}{13}+\frac {b e \,x^{10} \sqrt {b \,x^{4}+a}}{12}+\frac {b d \,x^{9} \sqrt {b \,x^{4}+a}}{11}+\frac {b c \,x^{8} \sqrt {b \,x^{4}+a}}{10}+\frac {5 a f \,x^{7} \sqrt {b \,x^{4}+a}}{39}+\frac {7 a e \,x^{6} \sqrt {b \,x^{4}+a}}{48}+\frac {13 a d \,x^{5} \sqrt {b \,x^{4}+a}}{77}+\frac {a c \,x^{4} \sqrt {b \,x^{4}+a}}{5}+\frac {4 a^{2} f \,x^{3} \sqrt {b \,x^{4}+a}}{195 b}+\frac {a^{2} e \,x^{2} \sqrt {b \,x^{4}+a}}{32 b}+\frac {4 a^{2} d x \sqrt {b \,x^{4}+a}}{77 b}+\frac {a^{2} c \sqrt {b \,x^{4}+a}}{10 b}-\frac {4 a^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {e \,a^{3} \ln \left (2 x^{2} \sqrt {b}+2 \sqrt {b \,x^{4}+a}\right )}{32 b^{\frac {3}{2}}}-\frac {4 i a^{\frac {7}{2}} f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{65 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(413\) |
1/480480*(36960*b^2*f*x^11+40040*b^2*e*x^10+43680*b^2*d*x^9+48048*b^2*c*x^ 8+61600*a*b*f*x^7+70070*a*b*e*x^6+81120*a*b*d*x^5+96096*a*b*c*x^4+9856*a^2 *f*x^3+15015*a^2*e*x^2+24960*a^2*d*x+48048*a^2*c)/b*(b*x^4+a)^(1/2)-1/8008 0/b*a^3*(4160*d/(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)+4928*I*f*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)/b^(1/2)*(E llipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*b^(1/2))^(1 /2),I))+5005/2*e*ln(x^2*b^(1/2)+(b*x^4+a)^(1/2))/b^(1/2))
Time = 0.16 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.56 \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=-\frac {59136 \, a^{3} \sqrt {b} f x \left (-\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 15015 \, a^{3} \sqrt {b} e x \log \left (-2 \, b x^{4} + 2 \, \sqrt {b x^{4} + a} \sqrt {b} x^{2} - a\right ) + 768 \, {\left (65 \, a^{2} b d - 77 \, a^{3} f\right )} \sqrt {b} x \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 2 \, {\left (36960 \, b^{3} f x^{12} + 40040 \, b^{3} e x^{11} + 43680 \, b^{3} d x^{10} + 48048 \, b^{3} c x^{9} + 61600 \, a b^{2} f x^{8} + 70070 \, a b^{2} e x^{7} + 81120 \, a b^{2} d x^{6} + 96096 \, a b^{2} c x^{5} + 9856 \, a^{2} b f x^{4} + 15015 \, a^{2} b e x^{3} + 24960 \, a^{2} b d x^{2} + 48048 \, a^{2} b c x - 29568 \, a^{3} f\right )} \sqrt {b x^{4} + a}}{960960 \, b^{2} x} \]
-1/960960*(59136*a^3*sqrt(b)*f*x*(-a/b)^(3/4)*elliptic_e(arcsin((-a/b)^(1/ 4)/x), -1) - 15015*a^3*sqrt(b)*e*x*log(-2*b*x^4 + 2*sqrt(b*x^4 + a)*sqrt(b )*x^2 - a) + 768*(65*a^2*b*d - 77*a^3*f)*sqrt(b)*x*(-a/b)^(3/4)*elliptic_f (arcsin((-a/b)^(1/4)/x), -1) - 2*(36960*b^3*f*x^12 + 40040*b^3*e*x^11 + 43 680*b^3*d*x^10 + 48048*b^3*c*x^9 + 61600*a*b^2*f*x^8 + 70070*a*b^2*e*x^7 + 81120*a*b^2*d*x^6 + 96096*a*b^2*c*x^5 + 9856*a^2*b*f*x^4 + 15015*a^2*b*e* x^3 + 24960*a^2*b*d*x^2 + 48048*a^2*b*c*x - 29568*a^3*f)*sqrt(b*x^4 + a))/ (b^2*x)
Time = 9.14 (sec) , antiderivative size = 398, normalized size of antiderivative = 0.88 \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\frac {a^{\frac {5}{2}} e x^{2}}{32 b \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {a^{\frac {3}{2}} d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {17 a^{\frac {3}{2}} e x^{6}}{96 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {a^{\frac {3}{2}} f x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} + \frac {\sqrt {a} b d x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} + \frac {11 \sqrt {a} b e x^{10}}{48 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} b f x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {15}{4}\right )} - \frac {a^{3} e \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{32 b^{\frac {3}{2}}} + a c \left (\begin {cases} \frac {\sqrt {a} x^{4}}{4} & \text {for}\: b = 0 \\\frac {\left (a + b x^{4}\right )^{\frac {3}{2}}}{6 b} & \text {otherwise} \end {cases}\right ) + b c \left (\begin {cases} - \frac {a^{2} \sqrt {a + b x^{4}}}{15 b^{2}} + \frac {a x^{4} \sqrt {a + b x^{4}}}{30 b} + \frac {x^{8} \sqrt {a + b x^{4}}}{10} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{8}}{8} & \text {otherwise} \end {cases}\right ) + \frac {b^{2} e x^{14}}{12 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \]
a**(5/2)*e*x**2/(32*b*sqrt(1 + b*x**4/a)) + a**(3/2)*d*x**5*gamma(5/4)*hyp er((-1/2, 5/4), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(9/4)) + 17*a**( 3/2)*e*x**6/(96*sqrt(1 + b*x**4/a)) + a**(3/2)*f*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*d *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*e*x**10/(48*sqrt(1 + b*x**4/a)) + sqrt(a)*b*f* 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*e*asinh(sqrt(b)*x**2/sqrt(a))/(32*b**(3/2)) + a*c*Pi ecewise((sqrt(a)*x**4/4, Eq(b, 0)), ((a + b*x**4)**(3/2)/(6*b), True)) + b *c*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** 2*e*x**14/(12*sqrt(a)*sqrt(1 + b*x**4/a))
\[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )} x^{3} \,d x } \]
1/10*(b*x^4 + a)^(5/2)*c/b + integrate((b*f*x^10 + b*e*x^9 + b*d*x^8 + a*f *x^6 + a*e*x^5 + a*d*x^4)*sqrt(b*x^4 + a), x)
\[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )} x^{3} \,d x } \]
Timed out. \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\int x^3\,{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]