Integrand size = 24, antiderivative size = 857 \[ \int \frac {x^6 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {x^3 \sqrt {c+d x^4}}{5 b}+\frac {(2 b c-5 a d) x \sqrt {c+d x^4}}{5 b^2 \sqrt {d} \left (\sqrt {c}+\sqrt {d} x^2\right )}-\frac {a \sqrt {-\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} \arctan \left (\frac {\sqrt {-\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} x}{\sqrt {c+d x^4}}\right )}{4 b^2}-\frac {a \sqrt {\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} \arctan \left (\frac {\sqrt {\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} x}{\sqrt {c+d x^4}}\right )}{4 b^2}-\frac {\sqrt [4]{c} (2 b c-5 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{5 b^2 d^{3/4} \sqrt {c+d x^4}}+\frac {\sqrt [4]{c} \left (b^2 c^2+a b c d-5 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{5 b^2 d^{3/4} (b c+a d) \sqrt {c+d x^4}}+\frac {a \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 b^{5/2} \sqrt [4]{c} \left (\sqrt {-a} \sqrt {b} \sqrt {c}-a \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^4}}-\frac {a \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {c} \left (\sqrt {b}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {c}}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 b^{5/2} \sqrt [4]{c} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^4}} \] Output:
1/5*x^3*(d*x^4+c)^(1/2)/b+1/5*(-5*a*d+2*b*c)*x*(d*x^4+c)^(1/2)/b^2/d^(1/2) /(c^(1/2)+d^(1/2)*x^2)-1/4*a*(-(-a*d+b*c)/(-a)^(1/2)/b^(1/2))^(1/2)*arctan ((-(-a*d+b*c)/(-a)^(1/2)/b^(1/2))^(1/2)*x/(d*x^4+c)^(1/2))/b^2-1/4*a*((-a* d+b*c)/(-a)^(1/2)/b^(1/2))^(1/2)*arctan(((-a*d+b*c)/(-a)^(1/2)/b^(1/2))^(1 /2)*x/(d*x^4+c)^(1/2))/b^2-1/5*c^(1/4)*(-5*a*d+2*b*c)*(c^(1/2)+d^(1/2)*x^2 )*((d*x^4+c)/(c^(1/2)+d^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(d^(1/4) *x/c^(1/4))),1/2*2^(1/2))/b^2/d^(3/4)/(d*x^4+c)^(1/2)+1/5*c^(1/4)*(-5*a^2* d^2+a*b*c*d+b^2*c^2)*(c^(1/2)+d^(1/2)*x^2)*((d*x^4+c)/(c^(1/2)+d^(1/2)*x^2 )^2)^(1/2)*InverseJacobiAM(2*arctan(d^(1/4)*x/c^(1/4)),1/2*2^(1/2))/b^2/d^ (3/4)/(a*d+b*c)/(d*x^4+c)^(1/2)+1/8*a*(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2)) *(-a*d+b*c)*(c^(1/2)+d^(1/2)*x^2)*((d*x^4+c)/(c^(1/2)+d^(1/2)*x^2)^2)^(1/2 )*EllipticPi(sin(2*arctan(d^(1/4)*x/c^(1/4))),1/4*(b^(1/2)*c^(1/2)+(-a)^(1 /2)*d^(1/2))^2/(-a)^(1/2)/b^(1/2)/c^(1/2)/d^(1/2),1/2*2^(1/2))/b^(5/2)/c^( 1/4)/((-a)^(1/2)*b^(1/2)*c^(1/2)-a*d^(1/2))/d^(1/4)/(d*x^4+c)^(1/2)-1/8*a* (b^(1/2)*c^(1/2)+(-a)^(1/2)*d^(1/2))*(-a*d+b*c)*(c^(1/2)+d^(1/2)*x^2)*((d* x^4+c)/(c^(1/2)+d^(1/2)*x^2)^2)^(1/2)*EllipticPi(sin(2*arctan(d^(1/4)*x/c^ (1/4))),-1/4*c^(1/2)*(b^(1/2)-(-a)^(1/2)*d^(1/2)/c^(1/2))^2/(-a)^(1/2)/b^( 1/2)/d^(1/2),1/2*2^(1/2))/b^(5/2)/c^(1/4)/((-a)^(1/2)*b^(1/2)*c^(1/2)+a*d^ (1/2))/d^(1/4)/(d*x^4+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.10 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.16 \[ \int \frac {x^6 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {7 a x^3 \left (c+d x^4\right )-7 a c x^3 \sqrt {1+\frac {d x^4}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+(2 b c-5 a d) x^7 \sqrt {1+\frac {d x^4}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{35 a b \sqrt {c+d x^4}} \] Input:
Integrate[(x^6*Sqrt[c + d*x^4])/(a + b*x^4),x]
Output:
(7*a*x^3*(c + d*x^4) - 7*a*c*x^3*Sqrt[1 + (d*x^4)/c]*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^4)/c), -((b*x^4)/a)] + (2*b*c - 5*a*d)*x^7*Sqrt[1 + (d*x^4)/c ]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^4)/c), -((b*x^4)/a)])/(35*a*b*Sqrt[c + d*x^4])
Time = 2.65 (sec) , antiderivative size = 1069, normalized size of antiderivative = 1.25, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {978, 1054, 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.
| \(\displaystyle \int \frac {x^6 \sqrt {c+d x^4}}{a+b x^4} \, dx\) |
| \(\Big \downarrow \) 978 |
| \(\displaystyle \frac {x^3 \sqrt {c+d x^4}}{5 b}-\frac {\int \frac {x^2 \left (3 a c-(2 b c-5 a d) x^4\right )}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{5 b}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {x^3 \sqrt {c+d x^4}}{5 b}-\frac {\int \left (-\frac {(2 b c-5 a d) x^2}{b \sqrt {d x^4+c}}-\frac {5 \left (a^2 d-a b c\right ) x^2}{b \left (b x^4+a\right ) \sqrt {d x^4+c}}\right )dx}{5 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^3 \sqrt {d x^4+c}}{5 b}-\frac {\frac {5 \sqrt {-a} (b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right ) \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{8 b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^4+c}}-\frac {5 (-a)^{3/4} \sqrt {b c-a d} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^4+c}}\right )}{4 b^{5/4}}+\frac {5 (-a)^{3/4} \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^4+c}}\right )}{4 b^{5/4}}+\frac {\sqrt [4]{c} (2 b c-5 a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{b d^{3/4} \sqrt {d x^4+c}}-\frac {\sqrt [4]{c} (2 b c-5 a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 b d^{3/4} \sqrt {d x^4+c}}-\frac {5 a \left (\sqrt {c}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 b \sqrt [4]{c} (b c+a d) \sqrt {d x^4+c}}-\frac {5 a \left (\sqrt {c}+\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 b \sqrt [4]{c} (b c+a d) \sqrt {d x^4+c}}-\frac {5 \sqrt {-a} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 (b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {c} \left (\sqrt {b}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {c}}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^4+c}}-\frac {(2 b c-5 a d) x \sqrt {d x^4+c}}{b \sqrt {d} \left (\sqrt {d} x^2+\sqrt {c}\right )}}{5 b}\) |
Input:
Int[(x^6*Sqrt[c + d*x^4])/(a + b*x^4),x]
Output:
(x^3*Sqrt[c + d*x^4])/(5*b) - (-(((2*b*c - 5*a*d)*x*Sqrt[c + d*x^4])/(b*Sq rt[d]*(Sqrt[c] + Sqrt[d]*x^2))) - (5*(-a)^(3/4)*Sqrt[b*c - a*d]*ArcTan[(Sq rt[b*c - a*d]*x)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^4])])/(4*b^(5/4)) + (5*( -a)^(3/4)*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b*c - a*d]*x)/((-a)^(1/4)*b^(1/4)* Sqrt[c + d*x^4])])/(4*b^(5/4)) + (c^(1/4)*(2*b*c - 5*a*d)*(Sqrt[c] + Sqrt[ d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticE[2*ArcTan[(d^ (1/4)*x)/c^(1/4)], 1/2])/(b*d^(3/4)*Sqrt[c + d*x^4]) - (c^(1/4)*(2*b*c - 5 *a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]* EllipticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(2*b*d^(3/4)*Sqrt[c + d*x^4 ]) - (5*a*(Sqrt[c] - (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(1/4)*(b*c - a*d)*(Sqrt [c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[2 *ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(4*b*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^ 4]) - (5*a*(Sqrt[c] + (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(1/4)*(b*c - a*d)*(Sqr t[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[ 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(4*b*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x ^4]) + (5*Sqrt[-a]*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2*(b*c - a*d)*(Sqr t[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticPi [(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt[d ]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*b^(3/2)*c^(1/4)*d^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4]) - (5*Sqrt[-a]*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d...
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* ((c + d*x^n)^q/(b*(m + n*(p + q) + 1))), x] - Simp[e^n/(b*(m + n*(p + q) + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[a*c*(m - n + 1) + (a*d*(m - n + 1) - n*q*(b*c - a*d))*x^n, x], x], x] /; FreeQ[{a, b, c , d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Result contains complex when optimal does not.
Time = 5.67 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.39
| method | result | size |
| risch | \(\frac {x^{3} \sqrt {d \,x^{4}+c}}{5 b}-\frac {\frac {i \left (5 a d -2 c b \right ) \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{b \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}\, \sqrt {d}}-\frac {5 a \left (a d -c b \right ) \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}}{\underline {\hspace {1.25 ex}}\alpha }\right )}{8 b^{2}}}{5 b}\) | \(332\) |
| elliptic | \(\frac {x^{3} \sqrt {d \,x^{4}+c}}{5 b}+\frac {i \left (-\frac {a d -c b}{b^{2}}-\frac {3 c}{5 b}\right ) \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}\, \sqrt {d}}+\frac {a \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (a d -c b \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{8 b^{3}}\) | \(334\) |
| default | \(\frac {\frac {x^{3} \sqrt {d \,x^{4}+c}}{5}+\frac {2 i c^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}\, \sqrt {d}}}{b}-\frac {a \left (\frac {i \sqrt {d}\, \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{b \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (a d -c b \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\underline {\hspace {1.25 ex}}\alpha }}{8 b^{2}}\right )}{b}\) | \(421\) |
Input:
int(x^6*(d*x^4+c)^(1/2)/(b*x^4+a),x,method=_RETURNVERBOSE)
Output:
1/5*x^3*(d*x^4+c)^(1/2)/b-1/5/b*(I*(5*a*d-2*b*c)/b*c^(1/2)/(I/c^(1/2)*d^(1 /2))^(1/2)*(1-I/c^(1/2)*d^(1/2)*x^2)^(1/2)*(1+I/c^(1/2)*d^(1/2)*x^2)^(1/2) /(d*x^4+c)^(1/2)/d^(1/2)*(EllipticF(x*(I/c^(1/2)*d^(1/2))^(1/2),I)-Ellipti cE(x*(I/c^(1/2)*d^(1/2))^(1/2),I))-5/8*a*(a*d-b*c)/b^2*sum(1/_alpha*(-1/(( -a*d+b*c)/b)^(1/2)*arctanh(1/2*(2*_alpha^2*d*x^2+2*c)/((-a*d+b*c)/b)^(1/2) /(d*x^4+c)^(1/2))+2/(I/c^(1/2)*d^(1/2))^(1/2)*_alpha^3*b/a*(1-I/c^(1/2)*d^ (1/2)*x^2)^(1/2)*(1+I/c^(1/2)*d^(1/2)*x^2)^(1/2)/(d*x^4+c)^(1/2)*EllipticP i(x*(I/c^(1/2)*d^(1/2))^(1/2),I*c^(1/2)/d^(1/2)*_alpha^2/a*b,(-I/c^(1/2)*d ^(1/2))^(1/2)/(I/c^(1/2)*d^(1/2))^(1/2))),_alpha=RootOf(_Z^4*b+a)))
Timed out. \[ \int \frac {x^6 \sqrt {c+d x^4}}{a+b x^4} \, dx=\text {Timed out} \] Input:
integrate(x^6*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x^6 \sqrt {c+d x^4}}{a+b x^4} \, dx=\int \frac {x^{6} \sqrt {c + d x^{4}}}{a + b x^{4}}\, dx \] Input:
integrate(x**6*(d*x**4+c)**(1/2)/(b*x**4+a),x)
Output:
Integral(x**6*sqrt(c + d*x**4)/(a + b*x**4), x)
\[ \int \frac {x^6 \sqrt {c+d x^4}}{a+b x^4} \, dx=\int { \frac {\sqrt {d x^{4} + c} x^{6}}{b x^{4} + a} \,d x } \] Input:
integrate(x^6*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="maxima")
Output:
integrate(sqrt(d*x^4 + c)*x^6/(b*x^4 + a), x)
\[ \int \frac {x^6 \sqrt {c+d x^4}}{a+b x^4} \, dx=\int { \frac {\sqrt {d x^{4} + c} x^{6}}{b x^{4} + a} \,d x } \] Input:
integrate(x^6*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="giac")
Output:
integrate(sqrt(d*x^4 + c)*x^6/(b*x^4 + a), x)
Timed out. \[ \int \frac {x^6 \sqrt {c+d x^4}}{a+b x^4} \, dx=\int \frac {x^6\,\sqrt {d\,x^4+c}}{b\,x^4+a} \,d x \] Input:
int((x^6*(c + d*x^4)^(1/2))/(a + b*x^4),x)
Output:
int((x^6*(c + d*x^4)^(1/2))/(a + b*x^4), x)
\[ \int \frac {x^6 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {\sqrt {d \,x^{4}+c}\, x^{3}-5 \left (\int \frac {\sqrt {d \,x^{4}+c}\, x^{6}}{b d \,x^{8}+a d \,x^{4}+b c \,x^{4}+a c}d x \right ) a d +2 \left (\int \frac {\sqrt {d \,x^{4}+c}\, x^{6}}{b d \,x^{8}+a d \,x^{4}+b c \,x^{4}+a c}d x \right ) b c -3 \left (\int \frac {\sqrt {d \,x^{4}+c}\, x^{2}}{b d \,x^{8}+a d \,x^{4}+b c \,x^{4}+a c}d x \right ) a c}{5 b} \] Input:
int(x^6*(d*x^4+c)^(1/2)/(b*x^4+a),x)
Output:
(sqrt(c + d*x**4)*x**3 - 5*int((sqrt(c + d*x**4)*x**6)/(a*c + a*d*x**4 + b *c*x**4 + b*d*x**8),x)*a*d + 2*int((sqrt(c + d*x**4)*x**6)/(a*c + a*d*x**4 + b*c*x**4 + b*d*x**8),x)*b*c - 3*int((sqrt(c + d*x**4)*x**2)/(a*c + a*d* x**4 + b*c*x**4 + b*d*x**8),x)*a*c)/(5*b)