Integrand size = 22, antiderivative size = 324 \[ \int \frac {x^{10} \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx=-\frac {(b c-a d) x^7}{2 b^2 \sqrt {a+b x^4}}+\frac {7 (9 b c-11 a d) x^3 \sqrt {a+b x^4}}{90 b^3}+\frac {d x^7 \sqrt {a+b x^4}}{9 b^2}-\frac {7 a (9 b c-11 a d) x \sqrt {a+b x^4}}{30 b^{7/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {7 a^{5/4} (9 b c-11 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{30 b^{15/4} \sqrt {a+b x^4}}-\frac {7 a^{5/4} (9 b c-11 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{60 b^{15/4} \sqrt {a+b x^4}} \] Output:
-1/2*(-a*d+b*c)*x^7/b^2/(b*x^4+a)^(1/2)+7/90*(-11*a*d+9*b*c)*x^3*(b*x^4+a) ^(1/2)/b^3+1/9*d*x^7*(b*x^4+a)^(1/2)/b^2-7/30*a*(-11*a*d+9*b*c)*x*(b*x^4+a )^(1/2)/b^(7/2)/(a^(1/2)+b^(1/2)*x^2)+7/30*a^(5/4)*(-11*a*d+9*b*c)*(a^(1/2 )+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*a rctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))/b^(15/4)/(b*x^4+a)^(1/2)-7/60*a^(5/ 4)*(-11*a*d+9*b*c)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^ 2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*x/a^(1/4)),1/2*2^(1/2))/b^(15/4) /(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.07 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.32 \[ \int \frac {x^{10} \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {x^3 \left (77 a^2 d+b^2 x^4 \left (9 c+5 d x^4\right )-a b \left (63 c+11 d x^4\right )+7 a (9 b c-11 a d) \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\frac {b x^4}{a}\right )\right )}{45 b^3 \sqrt {a+b x^4}} \] Input:
Integrate[(x^10*(c + d*x^4))/(a + b*x^4)^(3/2),x]
Output:
(x^3*(77*a^2*d + b^2*x^4*(9*c + 5*d*x^4) - a*b*(63*c + 11*d*x^4) + 7*a*(9* b*c - 11*a*d)*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((b*x^ 4)/a)]))/(45*b^3*Sqrt[a + b*x^4])
Time = 0.66 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {959, 817, 843, 834, 27, 761, 1510}
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.
| \(\displaystyle \int \frac {x^{10} \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \frac {(9 b c-11 a d) \int \frac {x^{10}}{\left (b x^4+a\right )^{3/2}}dx}{9 b}+\frac {d x^{11}}{9 b \sqrt {a+b x^4}}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {(9 b c-11 a d) \left (\frac {7 \int \frac {x^6}{\sqrt {b x^4+a}}dx}{2 b}-\frac {x^7}{2 b \sqrt {a+b x^4}}\right )}{9 b}+\frac {d x^{11}}{9 b \sqrt {a+b x^4}}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {(9 b c-11 a d) \left (\frac {7 \left (\frac {x^3 \sqrt {a+b x^4}}{5 b}-\frac {3 a \int \frac {x^2}{\sqrt {b x^4+a}}dx}{5 b}\right )}{2 b}-\frac {x^7}{2 b \sqrt {a+b x^4}}\right )}{9 b}+\frac {d x^{11}}{9 b \sqrt {a+b x^4}}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {(9 b c-11 a d) \left (\frac {7 \left (\frac {x^3 \sqrt {a+b x^4}}{5 b}-\frac {3 a \left (\frac {\sqrt {a} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {b}}-\frac {\sqrt {a} \int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {a} \sqrt {b x^4+a}}dx}{\sqrt {b}}\right )}{5 b}\right )}{2 b}-\frac {x^7}{2 b \sqrt {a+b x^4}}\right )}{9 b}+\frac {d x^{11}}{9 b \sqrt {a+b x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(9 b c-11 a d) \left (\frac {7 \left (\frac {x^3 \sqrt {a+b x^4}}{5 b}-\frac {3 a \left (\frac {\sqrt {a} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {b x^4+a}}dx}{\sqrt {b}}\right )}{5 b}\right )}{2 b}-\frac {x^7}{2 b \sqrt {a+b x^4}}\right )}{9 b}+\frac {d x^{11}}{9 b \sqrt {a+b x^4}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {(9 b c-11 a d) \left (\frac {7 \left (\frac {x^3 \sqrt {a+b x^4}}{5 b}-\frac {3 a \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^4}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {b x^4+a}}dx}{\sqrt {b}}\right )}{5 b}\right )}{2 b}-\frac {x^7}{2 b \sqrt {a+b x^4}}\right )}{9 b}+\frac {d x^{11}}{9 b \sqrt {a+b x^4}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {(9 b c-11 a d) \left (\frac {7 \left (\frac {x^3 \sqrt {a+b x^4}}{5 b}-\frac {3 a \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^4}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^4}}-\frac {x \sqrt {a+b x^4}}{\sqrt {a}+\sqrt {b} x^2}}{\sqrt {b}}\right )}{5 b}\right )}{2 b}-\frac {x^7}{2 b \sqrt {a+b x^4}}\right )}{9 b}+\frac {d x^{11}}{9 b \sqrt {a+b x^4}}\) |
Input:
Int[(x^10*(c + d*x^4))/(a + b*x^4)^(3/2),x]
Output:
(d*x^11)/(9*b*Sqrt[a + b*x^4]) + ((9*b*c - 11*a*d)*(-1/2*x^7/(b*Sqrt[a + b *x^4]) + (7*((x^3*Sqrt[a + b*x^4])/(5*b) - (3*a*(-((-((x*Sqrt[a + b*x^4])/ (Sqrt[a] + Sqrt[b]*x^2)) + (a^(1/4)*(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 ])/(b^(1/4)*Sqrt[a + b*x^4]))/Sqrt[b]) + (a^(1/4)*(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])/(2*b^(3/4)*Sqrt[a + b*x^4])))/(5*b)))/(2*b)))/(9*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Result contains complex when optimal does not.
Time = 6.84 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.70
| method | result | size |
| elliptic | \(-\frac {a \,x^{3} \left (a d -c b \right )}{2 b^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {d \,x^{7} \sqrt {b \,x^{4}+a}}{9 b^{2}}+\frac {\left (-\frac {a d -c b}{b^{2}}-\frac {7 d a}{9 b^{2}}\right ) x^{3} \sqrt {b \,x^{4}+a}}{5 b}+\frac {i \left (\frac {3 a \left (a d -c b \right )}{2 b^{3}}-\frac {3 \left (-\frac {a d -c b}{b^{2}}-\frac {7 d a}{9 b^{2}}\right ) a}{5 b}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\) | \(226\) |
| default | \(c \left (\frac {a \,x^{3}}{2 b^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {x^{3} \sqrt {b \,x^{4}+a}}{5 b^{2}}-\frac {21 i a^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{10 b^{\frac {5}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (-\frac {a^{2} x^{3}}{2 b^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {x^{7} \sqrt {b \,x^{4}+a}}{9 b^{2}}-\frac {16 a \,x^{3} \sqrt {b \,x^{4}+a}}{45 b^{3}}+\frac {77 i a^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{30 b^{\frac {7}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(298\) |
| risch | \(-\frac {x^{3} \left (-5 d b \,x^{4}+16 a d -9 c b \right ) \sqrt {b \,x^{4}+a}}{45 b^{3}}+\frac {a \left (b \left (31 a d -24 c b \right ) \left (-\frac {x^{3}}{2 b \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {3 i \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+a \left (16 a d -9 c b \right ) \left (\frac {x^{3}}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )\right )}{15 b^{3}}\) | \(300\) |
Input:
int(x^10*(d*x^4+c)/(b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/2/b^3*a*x^3*(a*d-b*c)/((x^4+a/b)*b)^(1/2)+1/9*d*x^7*(b*x^4+a)^(1/2)/b^2 +1/5*(-1/b^2*(a*d-b*c)-7/9/b^2*d*a)/b*x^3*(b*x^4+a)^(1/2)+I*(3/2*a*(a*d-b* c)/b^3-3/5*(-1/b^2*(a*d-b*c)-7/9/b^2*d*a)/b*a)*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)*(EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x *(I/a^(1/2)*b^(1/2))^(1/2),I))
Time = 0.09 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.68 \[ \int \frac {x^{10} \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx=-\frac {21 \, {\left ({\left (9 \, a b^{2} c - 11 \, a^{2} b d\right )} x^{5} + {\left (9 \, a^{2} b c - 11 \, a^{3} d\right )} x\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 21 \, {\left ({\left (9 \, a b^{2} c - 11 \, a^{2} b d\right )} x^{5} + {\left (9 \, a^{2} b c - 11 \, a^{3} d\right )} x\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (10 \, b^{3} d x^{12} + 2 \, {\left (9 \, b^{3} c - 11 \, a b^{2} d\right )} x^{8} - 14 \, {\left (9 \, a b^{2} c - 11 \, a^{2} b d\right )} x^{4} - 189 \, a^{2} b c + 231 \, a^{3} d\right )} \sqrt {b x^{4} + a}}{90 \, {\left (b^{5} x^{5} + a b^{4} x\right )}} \] Input:
integrate(x^10*(d*x^4+c)/(b*x^4+a)^(3/2),x, algorithm="fricas")
Output:
-1/90*(21*((9*a*b^2*c - 11*a^2*b*d)*x^5 + (9*a^2*b*c - 11*a^3*d)*x)*sqrt(b )*(-a/b)^(3/4)*elliptic_e(arcsin((-a/b)^(1/4)/x), -1) - 21*((9*a*b^2*c - 1 1*a^2*b*d)*x^5 + (9*a^2*b*c - 11*a^3*d)*x)*sqrt(b)*(-a/b)^(3/4)*elliptic_f (arcsin((-a/b)^(1/4)/x), -1) - (10*b^3*d*x^12 + 2*(9*b^3*c - 11*a*b^2*d)*x ^8 - 14*(9*a*b^2*c - 11*a^2*b*d)*x^4 - 189*a^2*b*c + 231*a^3*d)*sqrt(b*x^4 + a))/(b^5*x^5 + a*b^4*x)
Result contains complex when optimal does not.
Time = 19.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.25 \[ \int \frac {x^{10} \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {c x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {15}{4}\right )} + \frac {d x^{15} \Gamma \left (\frac {15}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {15}{4} \\ \frac {19}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {19}{4}\right )} \] Input:
integrate(x**10*(d*x**4+c)/(b*x**4+a)**(3/2),x)
Output:
c*x**11*gamma(11/4)*hyper((3/2, 11/4), (15/4,), b*x**4*exp_polar(I*pi)/a)/ (4*a**(3/2)*gamma(15/4)) + d*x**15*gamma(15/4)*hyper((3/2, 15/4), (19/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3/2)*gamma(19/4))
\[ \int \frac {x^{10} \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{10}}{{\left (b x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^10*(d*x^4+c)/(b*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((d*x^4 + c)*x^10/(b*x^4 + a)^(3/2), x)
\[ \int \frac {x^{10} \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{10}}{{\left (b x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^10*(d*x^4+c)/(b*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((d*x^4 + c)*x^10/(b*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {x^{10} \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx=\int \frac {x^{10}\,\left (d\,x^4+c\right )}{{\left (b\,x^4+a\right )}^{3/2}} \,d x \] Input:
int((x^10*(c + d*x^4))/(a + b*x^4)^(3/2),x)
Output:
int((x^10*(c + d*x^4))/(a + b*x^4)^(3/2), x)
\[ \int \frac {x^{10} \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {77 \sqrt {b \,x^{4}+a}\, a^{2} d \,x^{3}-63 \sqrt {b \,x^{4}+a}\, a b c \,x^{3}-11 \sqrt {b \,x^{4}+a}\, a b d \,x^{7}+9 \sqrt {b \,x^{4}+a}\, b^{2} c \,x^{7}+5 \sqrt {b \,x^{4}+a}\, b^{2} d \,x^{11}-231 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{2} x^{8}+2 a b \,x^{4}+a^{2}}d x \right ) a^{4} d +189 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{2} x^{8}+2 a b \,x^{4}+a^{2}}d x \right ) a^{3} b c -231 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{2} x^{8}+2 a b \,x^{4}+a^{2}}d x \right ) a^{3} b d \,x^{4}+189 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{2} x^{8}+2 a b \,x^{4}+a^{2}}d x \right ) a^{2} b^{2} c \,x^{4}}{45 b^{3} \left (b \,x^{4}+a \right )} \] Input:
int(x^10*(d*x^4+c)/(b*x^4+a)^(3/2),x)
Output:
(77*sqrt(a + b*x**4)*a**2*d*x**3 - 63*sqrt(a + b*x**4)*a*b*c*x**3 - 11*sqr t(a + b*x**4)*a*b*d*x**7 + 9*sqrt(a + b*x**4)*b**2*c*x**7 + 5*sqrt(a + b*x **4)*b**2*d*x**11 - 231*int((sqrt(a + b*x**4)*x**2)/(a**2 + 2*a*b*x**4 + b **2*x**8),x)*a**4*d + 189*int((sqrt(a + b*x**4)*x**2)/(a**2 + 2*a*b*x**4 + b**2*x**8),x)*a**3*b*c - 231*int((sqrt(a + b*x**4)*x**2)/(a**2 + 2*a*b*x* *4 + b**2*x**8),x)*a**3*b*d*x**4 + 189*int((sqrt(a + b*x**4)*x**2)/(a**2 + 2*a*b*x**4 + b**2*x**8),x)*a**2*b**2*c*x**4)/(45*b**3*(a + b*x**4))