Integrand size = 28, antiderivative size = 409 \[ \int x \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\frac {4 a^2 f x \sqrt {a+b x^4}}{77 b}+\frac {3}{16} a c x^2 \sqrt {a+b x^4}+\frac {4 a^2 d x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {2 a x^3 \left (77 d+45 f x^2\right ) \sqrt {a+b x^4}}{1155}+\frac {1}{8} c x^2 \left (a+b x^4\right )^{3/2}+\frac {1}{99} x^3 \left (11 d+9 f x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {e \left (a+b x^4\right )^{5/2}}{10 b}+\frac {3 a^2 c \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 \sqrt {b}}-\frac {4 a^{9/4} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}+\frac {2 a^{9/4} \left (77 \sqrt {b} d-15 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{1155 b^{5/4} \sqrt {a+b x^4}} \]
1/8*c*x^2*(b*x^4+a)^(3/2)+1/99*x^3*(9*f*x^2+11*d)*(b*x^4+a)^(3/2)+1/10*e*( b*x^4+a)^(5/2)/b+3/16*a^2*c*arctanh(x^2*b^(1/2)/(b*x^4+a)^(1/2))/b^(1/2)+4 /77*a^2*f*x*(b*x^4+a)^(1/2)/b+3/16*a*c*x^2*(b*x^4+a)^(1/2)+2/1155*a*x^3*(4 5*f*x^2+77*d)*(b*x^4+a)^(1/2)+4/15*a^2*d*x*(b*x^4+a)^(1/2)/b^(1/2)/(a^(1/2 )+x^2*b^(1/2))-4/15*a^(9/4)*d*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/c os(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^(3/4)/(b*x^4+a)^(1/2)+2/1155*a^(9/4)*(cos(2*arctan(b^(1/4)*x/a^(1/4) ))^2)^(1/2)/cos(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))*(-15*f*a^(1/2)+77*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^(5/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.71 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.48 \[ \int x \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\frac {\sqrt {a+b x^4} \left (\frac {264 e \left (a+b x^4\right )^2}{b}+\frac {240 f x \left (a+b x^4\right )^2}{b}+165 c \left (5 a x^2+2 b x^6+\frac {3 a^{5/2} \sqrt {1+\frac {b x^4}{a}} \text {arcsinh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {b} \left (a+b x^4\right )}\right )-\frac {240 a^2 f x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^4}{a}\right )}{b \sqrt {1+\frac {b x^4}{a}}}+\frac {880 a d x^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^4}{a}\right )}{\sqrt {1+\frac {b x^4}{a}}}\right )}{2640} \]
(Sqrt[a + b*x^4]*((264*e*(a + b*x^4)^2)/b + (240*f*x*(a + b*x^4)^2)/b + 16 5*c*(5*a*x^2 + 2*b*x^6 + (3*a^(5/2)*Sqrt[1 + (b*x^4)/a]*ArcSinh[(Sqrt[b]*x ^2)/Sqrt[a]])/(Sqrt[b]*(a + b*x^4))) - (240*a^2*f*x*Hypergeometric2F1[-3/2 , 1/4, 5/4, -((b*x^4)/a)])/(b*Sqrt[1 + (b*x^4)/a]) + (880*a*d*x^3*Hypergeo metric2F1[-3/2, 3/4, 7/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a]))/2640
Time = 0.61 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, 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.
| \(\displaystyle \int x \left (a+b x^4\right )^{3/2} \left (c+d x+e x^2+f x^3\right ) \, dx\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \int \left (x \left (a+b x^4\right )^{3/2} \left (c+e x^2\right )+x^2 \left (a+b x^4\right )^{3/2} \left (d+f x^2\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (77 \sqrt {b} d-15 \sqrt {a} f\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{1155 b^{5/4} \sqrt {a+b x^4}}-\frac {4 a^{9/4} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}+\frac {3 a^2 c \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 \sqrt {b}}+\frac {4 a^2 d x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {4 a^2 f x \sqrt {a+b x^4}}{77 b}+\frac {1}{8} c x^2 \left (a+b x^4\right )^{3/2}+\frac {3}{16} a c x^2 \sqrt {a+b x^4}+\frac {1}{99} x^3 \left (a+b x^4\right )^{3/2} \left (11 d+9 f x^2\right )+\frac {2 a x^3 \sqrt {a+b x^4} \left (77 d+45 f x^2\right )}{1155}+\frac {e \left (a+b x^4\right )^{5/2}}{10 b}\) |
(4*a^2*f*x*Sqrt[a + b*x^4])/(77*b) + (3*a*c*x^2*Sqrt[a + b*x^4])/16 + (4*a ^2*d*x*Sqrt[a + b*x^4])/(15*Sqrt[b]*(Sqrt[a] + Sqrt[b]*x^2)) + (2*a*x^3*(7 7*d + 45*f*x^2)*Sqrt[a + b*x^4])/1155 + (c*x^2*(a + b*x^4)^(3/2))/8 + (x^3 *(11*d + 9*f*x^2)*(a + b*x^4)^(3/2))/99 + (e*(a + b*x^4)^(5/2))/(10*b) + ( 3*a^2*c*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/(16*Sqrt[b]) - (4*a^(9/4)* d*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Elli pticE[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]*d - 15*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])/(1155*b^(5/4)*Sqrt[a + b*x^4])
3.6.13.3.1 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 = 2.06 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.73
| method | result | size |
| risch | \(\frac {\left (5040 b^{2} f \,x^{9}+5544 b^{2} e \,x^{8}+6160 b^{2} d \,x^{7}+6930 b^{2} c \,x^{6}+9360 a b f \,x^{5}+11088 a b e \,x^{4}+13552 x^{3} a b d +17325 a b c \,x^{2}+2880 a^{2} f x +5544 a^{2} e \right ) \sqrt {b \,x^{4}+a}}{55440 b}-\frac {a^{2} \left (\frac {480 a f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {2464 i \sqrt {b}\, d \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3465 \sqrt {b}\, c \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{2}\right )}{9240 b}\) | \(300\) |
| default | \(f \left (\frac {b \,x^{9} \sqrt {b \,x^{4}+a}}{11}+\frac {13 a \,x^{5} \sqrt {b \,x^{4}+a}}{77}+\frac {4 a^{2} x \sqrt {b \,x^{4}+a}}{77 b}-\frac {4 a^{3} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+\frac {e \left (b \,x^{4}+a \right )^{\frac {5}{2}}}{10 b}+d \left (\frac {b \,x^{7} \sqrt {b \,x^{4}+a}}{9}+\frac {11 a \,x^{3} \sqrt {b \,x^{4}+a}}{45}+\frac {4 i a^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )+c \left (\frac {3 a^{2} \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{16 \sqrt {b}}+\frac {b \,x^{6} \sqrt {b \,x^{4}+a}}{8}+\frac {5 a \,x^{2} \sqrt {b \,x^{4}+a}}{16}\right )\) | \(332\) |
| elliptic | \(\frac {b f \,x^{9} \sqrt {b \,x^{4}+a}}{11}+\frac {b e \,x^{8} \sqrt {b \,x^{4}+a}}{10}+\frac {b d \,x^{7} \sqrt {b \,x^{4}+a}}{9}+\frac {b c \,x^{6} \sqrt {b \,x^{4}+a}}{8}+\frac {13 a f \,x^{5} \sqrt {b \,x^{4}+a}}{77}+\frac {a e \,x^{4} \sqrt {b \,x^{4}+a}}{5}+\frac {11 a d \,x^{3} \sqrt {b \,x^{4}+a}}{45}+\frac {5 a c \,x^{2} \sqrt {b \,x^{4}+a}}{16}+\frac {4 a^{2} f x \sqrt {b \,x^{4}+a}}{77 b}+\frac {a^{2} e \sqrt {b \,x^{4}+a}}{10 b}-\frac {4 a^{3} f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {3 a^{2} c \ln \left (2 x^{2} \sqrt {b}+2 \sqrt {b \,x^{4}+a}\right )}{16 \sqrt {b}}+\frac {4 i a^{\frac {5}{2}} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\) | \(371\) |
1/55440*(5040*b^2*f*x^9+5544*b^2*e*x^8+6160*b^2*d*x^7+6930*b^2*c*x^6+9360* a*b*f*x^5+11088*a*b*e*x^4+13552*a*b*d*x^3+17325*a*b*c*x^2+2880*a^2*f*x+554 4*a^2*e)/b*(b*x^4+a)^(1/2)-1/9240*a^2/b*(480*a*f/(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)-2464*I*b^(1/2)*d*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)-E llipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I))-3465/2*b^(1/2)*c*ln(x^2*b^(1/2)+( b*x^4+a)^(1/2)))
Time = 0.13 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.55 \[ \int x \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\frac {29568 \, a^{2} \sqrt {b} d x \left (-\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 10395 \, a^{2} \sqrt {b} c x \log \left (-2 \, b x^{4} - 2 \, \sqrt {b x^{4} + a} \sqrt {b} x^{2} - a\right ) - 384 \, {\left (77 \, a^{2} d + 15 \, a^{2} f\right )} \sqrt {b} x \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 2 \, {\left (5040 \, b^{2} f x^{10} + 5544 \, b^{2} e x^{9} + 6160 \, b^{2} d x^{8} + 6930 \, b^{2} c x^{7} + 9360 \, a b f x^{6} + 11088 \, a b e x^{5} + 13552 \, a b d x^{4} + 17325 \, a b c x^{3} + 2880 \, a^{2} f x^{2} + 5544 \, a^{2} e x + 14784 \, a^{2} d\right )} \sqrt {b x^{4} + a}}{110880 \, b x} \]
1/110880*(29568*a^2*sqrt(b)*d*x*(-a/b)^(3/4)*elliptic_e(arcsin((-a/b)^(1/4 )/x), -1) + 10395*a^2*sqrt(b)*c*x*log(-2*b*x^4 - 2*sqrt(b*x^4 + a)*sqrt(b) *x^2 - a) - 384*(77*a^2*d + 15*a^2*f)*sqrt(b)*x*(-a/b)^(3/4)*elliptic_f(ar csin((-a/b)^(1/4)/x), -1) + 2*(5040*b^2*f*x^10 + 5544*b^2*e*x^9 + 6160*b^2 *d*x^8 + 6930*b^2*c*x^7 + 9360*a*b*f*x^6 + 11088*a*b*e*x^5 + 13552*a*b*d*x ^4 + 17325*a*b*c*x^3 + 2880*a^2*f*x^2 + 5544*a^2*e*x + 14784*a^2*d)*sqrt(b *x^4 + a))/(b*x)
Time = 5.19 (sec) , antiderivative size = 396, normalized size of antiderivative = 0.97 \[ \int x \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\frac {a^{\frac {3}{2}} c x^{2} \sqrt {1 + \frac {b x^{4}}{a}}}{4} + \frac {a^{\frac {3}{2}} c x^{2}}{16 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {a^{\frac {3}{2}} d x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {a^{\frac {3}{2}} f x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {3 \sqrt {a} b c x^{6}}{16 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} b d x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} + \frac {\sqrt {a} b f x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} + \frac {3 a^{2} c \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{16 \sqrt {b}} + a e \left (\begin {cases} \frac {\sqrt {a} x^{4}}{4} & \text {for}\: b = 0 \\\frac {\left (a + b x^{4}\right )^{\frac {3}{2}}}{6 b} & \text {otherwise} \end {cases}\right ) + b e \left (\begin {cases} - \frac {a^{2} \sqrt {a + b x^{4}}}{15 b^{2}} + \frac {a x^{4} \sqrt {a + b x^{4}}}{30 b} + \frac {x^{8} \sqrt {a + b x^{4}}}{10} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{8}}{8} & \text {otherwise} \end {cases}\right ) + \frac {b^{2} c x^{10}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \]
a**(3/2)*c*x**2*sqrt(1 + b*x**4/a)/4 + a**(3/2)*c*x**2/(16*sqrt(1 + b*x**4 /a)) + a**(3/2)*d*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), b*x**4*exp_po lar(I*pi)/a)/(4*gamma(7/4)) + a**(3/2)*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)) + 3*sqrt(a)*b*c*x**6/(1 6*sqrt(1 + b*x**4/a)) + sqrt(a)*b*d*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*f*x**9*gamma(9 /4)*hyper((-1/2, 9/4), (13/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(13/4)) + 3*a**2*c*asinh(sqrt(b)*x**2/sqrt(a))/(16*sqrt(b)) + a*e*Piecewise((sqrt( a)*x**4/4, Eq(b, 0)), ((a + b*x**4)**(3/2)/(6*b), True)) + b*e*Piecewise(( -a**2*sqrt(a + b*x**4)/(15*b**2) + a*x**4*sqrt(a + b*x**4)/(30*b) + x**8*s qrt(a + b*x**4)/10, Ne(b, 0)), (sqrt(a)*x**8/8, True)) + b**2*c*x**10/(8*s qrt(a)*sqrt(1 + b*x**4/a))
\[ \int x \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )} x \,d x } \]
-1/32*(3*a^2*log(-(sqrt(b) - sqrt(b*x^4 + a)/x^2)/(sqrt(b) + sqrt(b*x^4 + a)/x^2))/sqrt(b) + 2*(3*sqrt(b*x^4 + a)*a^2*b/x^2 - 5*(b*x^4 + a)^(3/2)*a^ 2/x^6)/(b^2 - 2*(b*x^4 + a)*b/x^4 + (b*x^4 + a)^2/x^8))*c + integrate((b*f *x^8 + b*e*x^7 + b*d*x^6 + a*f*x^4 + a*e*x^3 + a*d*x^2)*sqrt(b*x^4 + a), x )
\[ \int x \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )} x \,d x } \]
Timed out. \[ \int x \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\int x\,{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]