Integrand size = 22, antiderivative size = 384 \[ \int \frac {x^{18} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=-\frac {(b c-a d) x^{15}}{10 b^2 \left (a+b x^4\right )^{5/2}}-\frac {(3 b c-5 a d) x^{11}}{12 b^3 \left (a+b x^4\right )^{3/2}}-\frac {(33 b c-67 a d) x^7}{24 b^4 \sqrt {a+b x^4}}+\frac {77 (9 b c-19 a d) x^3 \sqrt {a+b x^4}}{360 b^5}+\frac {d x^7 \sqrt {a+b x^4}}{9 b^4}-\frac {77 a (9 b c-19 a d) x \sqrt {a+b x^4}}{120 b^{11/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {77 a^{5/4} (9 b c-19 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{120 b^{23/4} \sqrt {a+b x^4}}-\frac {77 a^{5/4} (9 b c-19 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{240 b^{23/4} \sqrt {a+b x^4}} \] Output:
-1/10*(-a*d+b*c)*x^15/b^2/(b*x^4+a)^(5/2)-1/12*(-5*a*d+3*b*c)*x^11/b^3/(b* x^4+a)^(3/2)-1/24*(-67*a*d+33*b*c)*x^7/b^4/(b*x^4+a)^(1/2)+77/360*(-19*a*d +9*b*c)*x^3*(b*x^4+a)^(1/2)/b^5+1/9*d*x^7*(b*x^4+a)^(1/2)/b^4-77/120*a*(-1 9*a*d+9*b*c)*x*(b*x^4+a)^(1/2)/b^(11/2)/(a^(1/2)+b^(1/2)*x^2)+77/120*a^(5/ 4)*(-19*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*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))/b^(23/4)/ (b*x^4+a)^(1/2)-77/240*a^(5/4)*(-19*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^(23/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.15 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.40 \[ \int \frac {x^{18} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\frac {x^3 \left (1045 a^4 d-55 a^3 b \left (9 c-19 d x^4\right )+b^4 x^{12} \left (9 c+5 d x^4\right )+15 a^2 b^2 x^4 \left (-33 c+19 d x^4\right )-a b^3 x^8 \left (135 c+19 d x^4\right )-55 a (-9 b c+19 a d) \left (a+b x^4\right )^2 \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {7}{2},\frac {7}{4},-\frac {b x^4}{a}\right )\right )}{45 b^5 \left (a+b x^4\right )^{5/2}} \] Input:
Integrate[(x^18*(c + d*x^4))/(a + b*x^4)^(7/2),x]
Output:
(x^3*(1045*a^4*d - 55*a^3*b*(9*c - 19*d*x^4) + b^4*x^12*(9*c + 5*d*x^4) + 15*a^2*b^2*x^4*(-33*c + 19*d*x^4) - a*b^3*x^8*(135*c + 19*d*x^4) - 55*a*(- 9*b*c + 19*a*d)*(a + b*x^4)^2*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[3/4, 7 /2, 7/4, -((b*x^4)/a)]))/(45*b^5*(a + b*x^4)^(5/2))
Time = 0.75 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {959, 817, 817, 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.
| \(\displaystyle \int \frac {x^{18} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \frac {(9 b c-19 a d) \int \frac {x^{18}}{\left (b x^4+a\right )^{7/2}}dx}{9 b}+\frac {d x^{19}}{9 b \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {(9 b c-19 a d) \left (\frac {3 \int \frac {x^{14}}{\left (b x^4+a\right )^{5/2}}dx}{2 b}-\frac {x^{15}}{10 b \left (a+b x^4\right )^{5/2}}\right )}{9 b}+\frac {d x^{19}}{9 b \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {(9 b c-19 a d) \left (\frac {3 \left (\frac {11 \int \frac {x^{10}}{\left (b x^4+a\right )^{3/2}}dx}{6 b}-\frac {x^{11}}{6 b \left (a+b x^4\right )^{3/2}}\right )}{2 b}-\frac {x^{15}}{10 b \left (a+b x^4\right )^{5/2}}\right )}{9 b}+\frac {d x^{19}}{9 b \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {(9 b c-19 a d) \left (\frac {3 \left (\frac {11 \left (\frac {7 \int \frac {x^6}{\sqrt {b x^4+a}}dx}{2 b}-\frac {x^7}{2 b \sqrt {a+b x^4}}\right )}{6 b}-\frac {x^{11}}{6 b \left (a+b x^4\right )^{3/2}}\right )}{2 b}-\frac {x^{15}}{10 b \left (a+b x^4\right )^{5/2}}\right )}{9 b}+\frac {d x^{19}}{9 b \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {(9 b c-19 a d) \left (\frac {3 \left (\frac {11 \left (\frac {7 \left (\frac {x^3 \sqrt {a+b x^4}}{5 b}-\frac {3 a \int \frac {x^2}{\sqrt {b x^4+a}}dx}{5 b}\right )}{2 b}-\frac {x^7}{2 b \sqrt {a+b x^4}}\right )}{6 b}-\frac {x^{11}}{6 b \left (a+b x^4\right )^{3/2}}\right )}{2 b}-\frac {x^{15}}{10 b \left (a+b x^4\right )^{5/2}}\right )}{9 b}+\frac {d x^{19}}{9 b \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {(9 b c-19 a d) \left (\frac {3 \left (\frac {11 \left (\frac {7 \left (\frac {x^3 \sqrt {a+b x^4}}{5 b}-\frac {3 a \left (\frac {\sqrt {a} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {b}}-\frac {\sqrt {a} \int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {a} \sqrt {b x^4+a}}dx}{\sqrt {b}}\right )}{5 b}\right )}{2 b}-\frac {x^7}{2 b \sqrt {a+b x^4}}\right )}{6 b}-\frac {x^{11}}{6 b \left (a+b x^4\right )^{3/2}}\right )}{2 b}-\frac {x^{15}}{10 b \left (a+b x^4\right )^{5/2}}\right )}{9 b}+\frac {d x^{19}}{9 b \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(9 b c-19 a d) \left (\frac {3 \left (\frac {11 \left (\frac {7 \left (\frac {x^3 \sqrt {a+b x^4}}{5 b}-\frac {3 a \left (\frac {\sqrt {a} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {b x^4+a}}dx}{\sqrt {b}}\right )}{5 b}\right )}{2 b}-\frac {x^7}{2 b \sqrt {a+b x^4}}\right )}{6 b}-\frac {x^{11}}{6 b \left (a+b x^4\right )^{3/2}}\right )}{2 b}-\frac {x^{15}}{10 b \left (a+b x^4\right )^{5/2}}\right )}{9 b}+\frac {d x^{19}}{9 b \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {(9 b c-19 a d) \left (\frac {3 \left (\frac {11 \left (\frac {7 \left (\frac {x^3 \sqrt {a+b x^4}}{5 b}-\frac {3 a \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^4}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {b x^4+a}}dx}{\sqrt {b}}\right )}{5 b}\right )}{2 b}-\frac {x^7}{2 b \sqrt {a+b x^4}}\right )}{6 b}-\frac {x^{11}}{6 b \left (a+b x^4\right )^{3/2}}\right )}{2 b}-\frac {x^{15}}{10 b \left (a+b x^4\right )^{5/2}}\right )}{9 b}+\frac {d x^{19}}{9 b \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {(9 b c-19 a d) \left (\frac {3 \left (\frac {11 \left (\frac {7 \left (\frac {x^3 \sqrt {a+b x^4}}{5 b}-\frac {3 a \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^4}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^4}}-\frac {x \sqrt {a+b x^4}}{\sqrt {a}+\sqrt {b} x^2}}{\sqrt {b}}\right )}{5 b}\right )}{2 b}-\frac {x^7}{2 b \sqrt {a+b x^4}}\right )}{6 b}-\frac {x^{11}}{6 b \left (a+b x^4\right )^{3/2}}\right )}{2 b}-\frac {x^{15}}{10 b \left (a+b x^4\right )^{5/2}}\right )}{9 b}+\frac {d x^{19}}{9 b \left (a+b x^4\right )^{5/2}}\) |
Input:
Int[(x^18*(c + d*x^4))/(a + b*x^4)^(7/2),x]
Output:
(d*x^19)/(9*b*(a + b*x^4)^(5/2)) + ((9*b*c - 19*a*d)*(-1/10*x^15/(b*(a + b *x^4)^(5/2)) + (3*(-1/6*x^11/(b*(a + b*x^4)^(3/2)) + (11*(-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)))/(6*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 = 10.99 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.84
| method | result | size |
| elliptic | \(-\frac {a^{3} x^{3} \left (a d -c b \right ) \sqrt {b \,x^{4}+a}}{10 b^{8} \left (x^{4}+\frac {a}{b}\right )^{3}}+\frac {a^{2} x^{3} \left (43 a d -33 c b \right ) \sqrt {b \,x^{4}+a}}{60 b^{7} \left (x^{4}+\frac {a}{b}\right )^{2}}-\frac {a \,x^{3} \left (157 a d -87 c b \right )}{40 b^{5} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {d \,x^{7} \sqrt {b \,x^{4}+a}}{9 b^{4}}+\frac {\left (-\frac {3 a d -c b}{b^{4}}-\frac {7 d a}{9 b^{4}}\right ) x^{3} \sqrt {b \,x^{4}+a}}{5 b}+\frac {i \left (\frac {3 a \left (2 a d -c b \right )}{b^{5}}+\frac {a \left (157 a d -87 c b \right )}{40 b^{5}}-\frac {3 \left (-\frac {3 a d -c b}{b^{4}}-\frac {7 d a}{9 b^{4}}\right ) a}{5 b}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\) | \(324\) |
| default | \(c \left (\frac {a^{3} x^{3} \sqrt {b \,x^{4}+a}}{10 b^{7} \left (x^{4}+\frac {a}{b}\right )^{3}}-\frac {11 a^{2} x^{3} \sqrt {b \,x^{4}+a}}{20 b^{6} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {87 a \,x^{3}}{40 b^{4} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {x^{3} \sqrt {b \,x^{4}+a}}{5 b^{4}}-\frac {231 i a^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{40 b^{\frac {9}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (-\frac {a^{4} x^{3} \sqrt {b \,x^{4}+a}}{10 b^{8} \left (x^{4}+\frac {a}{b}\right )^{3}}+\frac {43 a^{3} x^{3} \sqrt {b \,x^{4}+a}}{60 b^{7} \left (x^{4}+\frac {a}{b}\right )^{2}}-\frac {157 a^{2} x^{3}}{40 b^{5} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {x^{7} \sqrt {b \,x^{4}+a}}{9 b^{4}}-\frac {34 a \,x^{3} \sqrt {b \,x^{4}+a}}{45 b^{5}}+\frac {1463 i a^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{120 b^{\frac {11}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(422\) |
| risch | \(-\frac {x^{3} \left (-5 d b \,x^{4}+34 a d -9 c b \right ) \sqrt {b \,x^{4}+a}}{45 b^{5}}+\frac {a \left (\frac {i \left (124 a d -54 c 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}}+15 a^{2} \left (5 a d -4 c b \right ) \left (\frac {x^{3} \sqrt {b \,x^{4}+a}}{6 a \,b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {x^{3}}{4 a^{2} \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 )}{4 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )-15 a^{3} \left (a d -c b \right ) \left (\frac {x^{3} \sqrt {b \,x^{4}+a}}{10 a \,b^{3} \left (x^{4}+\frac {a}{b}\right )^{3}}+\frac {7 x^{3} \sqrt {b \,x^{4}+a}}{60 a^{2} b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {7 x^{3}}{40 a^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {7 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 )}{40 a^{\frac {5}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )-30 a \left (5 a d -3 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^{5}}\) | \(633\) |
Input:
int(x^18*(d*x^4+c)/(b*x^4+a)^(7/2),x,method=_RETURNVERBOSE)
Output:
-1/10*a^3*x^3/b^8*(a*d-b*c)*(b*x^4+a)^(1/2)/(x^4+a/b)^3+1/60*a^2*x^3*(43*a *d-33*b*c)/b^7*(b*x^4+a)^(1/2)/(x^4+a/b)^2-1/40/b^5*a*x^3*(157*a*d-87*b*c) /((x^4+a/b)*b)^(1/2)+1/9*d*x^7*(b*x^4+a)^(1/2)/b^4+1/5*(-1/b^4*(3*a*d-b*c) -7/9/b^4*d*a)/b*x^3*(b*x^4+a)^(1/2)+I*(3*a*(2*a*d-b*c)/b^5+1/40/b^5*a*(157 *a*d-87*b*c)-3/5*(-1/b^4*(3*a*d-b*c)-7/9/b^4*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)-Ell ipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I))
Time = 0.10 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.01 \[ \int \frac {x^{18} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=-\frac {231 \, {\left ({\left (9 \, a b^{4} c - 19 \, a^{2} b^{3} d\right )} x^{13} + 3 \, {\left (9 \, a^{2} b^{3} c - 19 \, a^{3} b^{2} d\right )} x^{9} + 3 \, {\left (9 \, a^{3} b^{2} c - 19 \, a^{4} b d\right )} x^{5} + {\left (9 \, a^{4} b c - 19 \, a^{5} d\right )} x\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 231 \, {\left ({\left (9 \, a b^{4} c - 19 \, a^{2} b^{3} d\right )} x^{13} + 3 \, {\left (9 \, a^{2} b^{3} c - 19 \, a^{3} b^{2} d\right )} x^{9} + 3 \, {\left (9 \, a^{3} b^{2} c - 19 \, a^{4} b d\right )} x^{5} + {\left (9 \, a^{4} b c - 19 \, a^{5} d\right )} x\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (40 \, b^{5} d x^{20} + 8 \, {\left (9 \, b^{5} c - 19 \, a b^{4} d\right )} x^{16} - 120 \, {\left (9 \, a b^{4} c - 19 \, a^{2} b^{3} d\right )} x^{12} - 517 \, {\left (9 \, a^{2} b^{3} c - 19 \, a^{3} b^{2} d\right )} x^{8} - 2079 \, a^{4} b c + 4389 \, a^{5} d - 616 \, {\left (9 \, a^{3} b^{2} c - 19 \, a^{4} b d\right )} x^{4}\right )} \sqrt {b x^{4} + a}}{360 \, {\left (b^{9} x^{13} + 3 \, a b^{8} x^{9} + 3 \, a^{2} b^{7} x^{5} + a^{3} b^{6} x\right )}} \] Input:
integrate(x^18*(d*x^4+c)/(b*x^4+a)^(7/2),x, algorithm="fricas")
Output:
-1/360*(231*((9*a*b^4*c - 19*a^2*b^3*d)*x^13 + 3*(9*a^2*b^3*c - 19*a^3*b^2 *d)*x^9 + 3*(9*a^3*b^2*c - 19*a^4*b*d)*x^5 + (9*a^4*b*c - 19*a^5*d)*x)*sqr t(b)*(-a/b)^(3/4)*elliptic_e(arcsin((-a/b)^(1/4)/x), -1) - 231*((9*a*b^4*c - 19*a^2*b^3*d)*x^13 + 3*(9*a^2*b^3*c - 19*a^3*b^2*d)*x^9 + 3*(9*a^3*b^2* c - 19*a^4*b*d)*x^5 + (9*a^4*b*c - 19*a^5*d)*x)*sqrt(b)*(-a/b)^(3/4)*ellip tic_f(arcsin((-a/b)^(1/4)/x), -1) - (40*b^5*d*x^20 + 8*(9*b^5*c - 19*a*b^4 *d)*x^16 - 120*(9*a*b^4*c - 19*a^2*b^3*d)*x^12 - 517*(9*a^2*b^3*c - 19*a^3 *b^2*d)*x^8 - 2079*a^4*b*c + 4389*a^5*d - 616*(9*a^3*b^2*c - 19*a^4*b*d)*x ^4)*sqrt(b*x^4 + a))/(b^9*x^13 + 3*a*b^8*x^9 + 3*a^2*b^7*x^5 + a^3*b^6*x)
Timed out. \[ \int \frac {x^{18} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\text {Timed out} \] Input:
integrate(x**18*(d*x**4+c)/(b*x**4+a)**(7/2),x)
Output:
Timed out
\[ \int \frac {x^{18} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{18}}{{\left (b x^{4} + a\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(x^18*(d*x^4+c)/(b*x^4+a)^(7/2),x, algorithm="maxima")
Output:
integrate((d*x^4 + c)*x^18/(b*x^4 + a)^(7/2), x)
\[ \int \frac {x^{18} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{18}}{{\left (b x^{4} + a\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(x^18*(d*x^4+c)/(b*x^4+a)^(7/2),x, algorithm="giac")
Output:
integrate((d*x^4 + c)*x^18/(b*x^4 + a)^(7/2), x)
Timed out. \[ \int \frac {x^{18} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\int \frac {x^{18}\,\left (d\,x^4+c\right )}{{\left (b\,x^4+a\right )}^{7/2}} \,d x \] Input:
int((x^18*(c + d*x^4))/(a + b*x^4)^(7/2),x)
Output:
int((x^18*(c + d*x^4))/(a + b*x^4)^(7/2), x)
\[ \int \frac {x^{18} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\frac {1045 \sqrt {b \,x^{4}+a}\, a^{4} d \,x^{3}-495 \sqrt {b \,x^{4}+a}\, a^{3} b c \,x^{3}+1045 \sqrt {b \,x^{4}+a}\, a^{3} b d \,x^{7}-495 \sqrt {b \,x^{4}+a}\, a^{2} b^{2} c \,x^{7}+285 \sqrt {b \,x^{4}+a}\, a^{2} b^{2} d \,x^{11}-135 \sqrt {b \,x^{4}+a}\, a \,b^{3} c \,x^{11}-19 \sqrt {b \,x^{4}+a}\, a \,b^{3} d \,x^{15}+9 \sqrt {b \,x^{4}+a}\, b^{4} c \,x^{15}+5 \sqrt {b \,x^{4}+a}\, b^{4} d \,x^{19}-3135 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{8} d +1485 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{7} b c -9405 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{7} b d \,x^{4}+4455 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{6} b^{2} c \,x^{4}-9405 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{6} b^{2} d \,x^{8}+4455 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{5} b^{3} c \,x^{8}-3135 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{5} b^{3} d \,x^{12}+1485 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{4} b^{4} c \,x^{12}}{45 b^{5} \left (b^{3} x^{12}+3 a \,b^{2} x^{8}+3 a^{2} b \,x^{4}+a^{3}\right )} \] Input:
int(x^18*(d*x^4+c)/(b*x^4+a)^(7/2),x)
Output:
(1045*sqrt(a + b*x**4)*a**4*d*x**3 - 495*sqrt(a + b*x**4)*a**3*b*c*x**3 + 1045*sqrt(a + b*x**4)*a**3*b*d*x**7 - 495*sqrt(a + b*x**4)*a**2*b**2*c*x** 7 + 285*sqrt(a + b*x**4)*a**2*b**2*d*x**11 - 135*sqrt(a + b*x**4)*a*b**3*c *x**11 - 19*sqrt(a + b*x**4)*a*b**3*d*x**15 + 9*sqrt(a + b*x**4)*b**4*c*x* *15 + 5*sqrt(a + b*x**4)*b**4*d*x**19 - 3135*int((sqrt(a + b*x**4)*x**2)/( a**4 + 4*a**3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b**4*x**16),x)* a**8*d + 1485*int((sqrt(a + b*x**4)*x**2)/(a**4 + 4*a**3*b*x**4 + 6*a**2*b **2*x**8 + 4*a*b**3*x**12 + b**4*x**16),x)*a**7*b*c - 9405*int((sqrt(a + b *x**4)*x**2)/(a**4 + 4*a**3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b **4*x**16),x)*a**7*b*d*x**4 + 4455*int((sqrt(a + b*x**4)*x**2)/(a**4 + 4*a **3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b**4*x**16),x)*a**6*b**2* c*x**4 - 9405*int((sqrt(a + b*x**4)*x**2)/(a**4 + 4*a**3*b*x**4 + 6*a**2*b **2*x**8 + 4*a*b**3*x**12 + b**4*x**16),x)*a**6*b**2*d*x**8 + 4455*int((sq rt(a + b*x**4)*x**2)/(a**4 + 4*a**3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x **12 + b**4*x**16),x)*a**5*b**3*c*x**8 - 3135*int((sqrt(a + b*x**4)*x**2)/ (a**4 + 4*a**3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b**4*x**16),x) *a**5*b**3*d*x**12 + 1485*int((sqrt(a + b*x**4)*x**2)/(a**4 + 4*a**3*b*x** 4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b**4*x**16),x)*a**4*b**4*c*x**12)/ (45*b**5*(a**3 + 3*a**2*b*x**4 + 3*a*b**2*x**8 + b**3*x**12))