Integrand size = 32, antiderivative size = 330 \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {(B c-A d) x \sqrt {a-c x^4}}{3 c d^2}-\frac {B x^3 \sqrt {a-c x^4}}{5 c d}+\frac {a^{3/4} \left (5 B c^3-5 A c^2 d+3 a B d^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} d^3 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (5 \sqrt {c} (B c-A d) \left (3 c^3+a d^2\right )+3 \sqrt {a} d \left (5 B c^3-5 A c^2 d+3 a B d^2\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{15 c^{7/4} d^4 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} c^{7/4} (B c-A d) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{d^4 \sqrt {a-c x^4}} \] Output:
1/3*(-A*d+B*c)*x*(-c*x^4+a)^(1/2)/c/d^2-1/5*B*x^3*(-c*x^4+a)^(1/2)/c/d+1/5 *a^(3/4)*(-5*A*c^2*d+3*B*a*d^2+5*B*c^3)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4 )*x/a^(1/4),I)/c^(7/4)/d^3/(-c*x^4+a)^(1/2)-1/15*a^(1/4)*(5*c^(1/2)*(-A*d+ B*c)*(a*d^2+3*c^3)+3*a^(1/2)*d*(-5*A*c^2*d+3*B*a*d^2+5*B*c^3))*(1-c*x^4/a) ^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/c^(7/4)/d^4/(-c*x^4+a)^(1/2)+a^(1/4) *c^(7/4)*(-A*d+B*c)*(1-c*x^4/a)^(1/2)*EllipticPi(c^(1/4)*x/a^(1/4),-a^(1/2 )*d/c^(3/2),I)/d^4/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.48 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.07 \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {-3 i \sqrt {a} d \left (5 B c^3-5 A c^2 d+3 a B d^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+i \left (-5 A \sqrt {c} d \left (3 c^3+3 \sqrt {a} c^{3/2} d+a d^2\right )+B \left (15 c^{9/2}+15 \sqrt {a} c^3 d+5 a c^{3/2} d^2+9 a^{3/2} d^3\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\sqrt {c} \left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d^2 x \left (5 B c-5 A d-3 B d x^2\right ) \left (a-c x^4\right )-15 i c^3 (B c-A d) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{15 \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^{3/2} d^4 \sqrt {a-c x^4}} \] Input:
Integrate[(x^6*(A + B*x^2))/((c + d*x^2)*Sqrt[a - c*x^4]),x]
Output:
((-3*I)*Sqrt[a]*d*(5*B*c^3 - 5*A*c^2*d + 3*a*B*d^2)*Sqrt[1 - (c*x^4)/a]*El lipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + I*(-5*A*Sqrt[c]*d*(3* c^3 + 3*Sqrt[a]*c^(3/2)*d + a*d^2) + B*(15*c^(9/2) + 15*Sqrt[a]*c^3*d + 5* a*c^(3/2)*d^2 + 9*a^(3/2)*d^3))*Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcSinh[Sq rt[-(Sqrt[c]/Sqrt[a])]*x], -1] + Sqrt[c]*(Sqrt[-(Sqrt[c]/Sqrt[a])]*d^2*x*( 5*B*c - 5*A*d - 3*B*d*x^2)*(a - c*x^4) - (15*I)*c^3*(B*c - A*d)*Sqrt[1 - ( c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt [a])]*x], -1]))/(15*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^(3/2)*d^4*Sqrt[a - c*x^4])
Time = 1.25 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {2237, 25, 2237, 2235, 25, 27, 1513, 27, 765, 762, 1390, 1389, 327, 1543, 1542}
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.
| \(\displaystyle \int \frac {x^6 \left (A+B x^2\right )}{\sqrt {a-c x^4} \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2237 |
| \(\displaystyle -\frac {\int -\frac {5 c d \left (B x^2+A\right ) x^6+B \left (d x^2+c\right ) \left (3 a-5 c x^4\right ) x^2}{\left (d x^2+c\right ) \sqrt {a-c x^4}}dx}{5 c d}-\frac {B x^3 \sqrt {a-c x^4}}{5 c d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {5 c d \left (B x^2+A\right ) x^6+B \left (d x^2+c\right ) \left (3 a-5 c x^4\right ) x^2}{\left (d x^2+c\right ) \sqrt {a-c x^4}}dx}{5 c d}-\frac {B x^3 \sqrt {a-c x^4}}{5 c d}\) |
| \(\Big \downarrow \) 2237 |
| \(\displaystyle \frac {\frac {5 x \sqrt {a-c x^4} (B c-A d)}{3 d}-\frac {\int \frac {5 c (B c-A d) \left (d x^2+c\right ) \left (a-3 c x^4\right )-3 c d \left (5 c d \left (B x^2+A\right ) x^6+B \left (d x^2+c\right ) \left (3 a-5 c x^4\right ) x^2\right )}{\left (d x^2+c\right ) \sqrt {a-c x^4}}dx}{3 c d}}{5 c d}-\frac {B x^3 \sqrt {a-c x^4}}{5 c d}\) |
| \(\Big \downarrow \) 2235 |
| \(\displaystyle \frac {\frac {5 x \sqrt {a-c x^4} (B c-A d)}{3 d}-\frac {-\frac {15 c^5 (B c-A d) \int \frac {1}{\left (d x^2+c\right ) \sqrt {a-c x^4}}dx}{d^2}-\frac {\int -\frac {c \left (5 (B c-A d) \left (3 c^3+a d^2\right )-3 d \left (5 B c^3-5 A d c^2+3 a B d^2\right ) x^2\right )}{\sqrt {a-c x^4}}dx}{d^2}}{3 c d}}{5 c d}-\frac {B x^3 \sqrt {a-c x^4}}{5 c d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {5 x \sqrt {a-c x^4} (B c-A d)}{3 d}-\frac {\frac {\int \frac {c \left (5 (B c-A d) \left (3 c^3+a d^2\right )-3 d \left (5 B c^3-5 A d c^2+3 a B d^2\right ) x^2\right )}{\sqrt {a-c x^4}}dx}{d^2}-\frac {15 c^5 (B c-A d) \int \frac {1}{\left (d x^2+c\right ) \sqrt {a-c x^4}}dx}{d^2}}{3 c d}}{5 c d}-\frac {B x^3 \sqrt {a-c x^4}}{5 c d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5 x \sqrt {a-c x^4} (B c-A d)}{3 d}-\frac {\frac {c \int \frac {5 (B c-A d) \left (3 c^3+a d^2\right )-3 d \left (5 B c^3-5 A d c^2+3 a B d^2\right ) x^2}{\sqrt {a-c x^4}}dx}{d^2}-\frac {15 c^5 (B c-A d) \int \frac {1}{\left (d x^2+c\right ) \sqrt {a-c x^4}}dx}{d^2}}{3 c d}}{5 c d}-\frac {B x^3 \sqrt {a-c x^4}}{5 c d}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {\frac {5 x \sqrt {a-c x^4} (B c-A d)}{3 d}-\frac {\frac {c \left (\left (5 \left (a d^2+3 c^3\right ) (B c-A d)+\frac {3 \sqrt {a} d \left (3 a B d^2-5 A c^2 d+5 B c^3\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a-c x^4}}dx-\frac {3 \sqrt {a} d \left (3 a B d^2-5 A c^2 d+5 B c^3\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-c x^4}}dx}{\sqrt {c}}\right )}{d^2}-\frac {15 c^5 (B c-A d) \int \frac {1}{\left (d x^2+c\right ) \sqrt {a-c x^4}}dx}{d^2}}{3 c d}}{5 c d}-\frac {B x^3 \sqrt {a-c x^4}}{5 c d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5 x \sqrt {a-c x^4} (B c-A d)}{3 d}-\frac {\frac {c \left (\left (5 \left (a d^2+3 c^3\right ) (B c-A d)+\frac {3 \sqrt {a} d \left (3 a B d^2-5 A c^2 d+5 B c^3\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a-c x^4}}dx-\frac {3 d \left (3 a B d^2-5 A c^2 d+5 B c^3\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\right )}{d^2}-\frac {15 c^5 (B c-A d) \int \frac {1}{\left (d x^2+c\right ) \sqrt {a-c x^4}}dx}{d^2}}{3 c d}}{5 c d}-\frac {B x^3 \sqrt {a-c x^4}}{5 c d}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\frac {5 x \sqrt {a-c x^4} (B c-A d)}{3 d}-\frac {\frac {c \left (\frac {\sqrt {1-\frac {c x^4}{a}} \left (5 \left (a d^2+3 c^3\right ) (B c-A d)+\frac {3 \sqrt {a} d \left (3 a B d^2-5 A c^2 d+5 B c^3\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}-\frac {3 d \left (3 a B d^2-5 A c^2 d+5 B c^3\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\right )}{d^2}-\frac {15 c^5 (B c-A d) \int \frac {1}{\left (d x^2+c\right ) \sqrt {a-c x^4}}dx}{d^2}}{3 c d}}{5 c d}-\frac {B x^3 \sqrt {a-c x^4}}{5 c d}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\frac {5 x \sqrt {a-c x^4} (B c-A d)}{3 d}-\frac {\frac {c \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (5 \left (a d^2+3 c^3\right ) (B c-A d)+\frac {3 \sqrt {a} d \left (3 a B d^2-5 A c^2 d+5 B c^3\right )}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {3 d \left (3 a B d^2-5 A c^2 d+5 B c^3\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\right )}{d^2}-\frac {15 c^5 (B c-A d) \int \frac {1}{\left (d x^2+c\right ) \sqrt {a-c x^4}}dx}{d^2}}{3 c d}}{5 c d}-\frac {B x^3 \sqrt {a-c x^4}}{5 c d}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {\frac {5 x \sqrt {a-c x^4} (B c-A d)}{3 d}-\frac {\frac {c \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (5 \left (a d^2+3 c^3\right ) (B c-A d)+\frac {3 \sqrt {a} d \left (3 a B d^2-5 A c^2 d+5 B c^3\right )}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {3 d \sqrt {1-\frac {c x^4}{a}} \left (3 a B d^2-5 A c^2 d+5 B c^3\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}\right )}{d^2}-\frac {15 c^5 (B c-A d) \int \frac {1}{\left (d x^2+c\right ) \sqrt {a-c x^4}}dx}{d^2}}{3 c d}}{5 c d}-\frac {B x^3 \sqrt {a-c x^4}}{5 c d}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {\frac {5 x \sqrt {a-c x^4} (B c-A d)}{3 d}-\frac {\frac {c \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (5 \left (a d^2+3 c^3\right ) (B c-A d)+\frac {3 \sqrt {a} d \left (3 a B d^2-5 A c^2 d+5 B c^3\right )}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {3 \sqrt {a} d \sqrt {1-\frac {c x^4}{a}} \left (3 a B d^2-5 A c^2 d+5 B c^3\right ) \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {c} \sqrt {a-c x^4}}\right )}{d^2}-\frac {15 c^5 (B c-A d) \int \frac {1}{\left (d x^2+c\right ) \sqrt {a-c x^4}}dx}{d^2}}{3 c d}}{5 c d}-\frac {B x^3 \sqrt {a-c x^4}}{5 c d}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {5 x \sqrt {a-c x^4} (B c-A d)}{3 d}-\frac {\frac {c \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (5 \left (a d^2+3 c^3\right ) (B c-A d)+\frac {3 \sqrt {a} d \left (3 a B d^2-5 A c^2 d+5 B c^3\right )}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {3 a^{3/4} d \sqrt {1-\frac {c x^4}{a}} \left (3 a B d^2-5 A c^2 d+5 B c^3\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}\right )}{d^2}-\frac {15 c^5 (B c-A d) \int \frac {1}{\left (d x^2+c\right ) \sqrt {a-c x^4}}dx}{d^2}}{3 c d}}{5 c d}-\frac {B x^3 \sqrt {a-c x^4}}{5 c d}\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle \frac {\frac {5 x \sqrt {a-c x^4} (B c-A d)}{3 d}-\frac {\frac {c \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (5 \left (a d^2+3 c^3\right ) (B c-A d)+\frac {3 \sqrt {a} d \left (3 a B d^2-5 A c^2 d+5 B c^3\right )}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {3 a^{3/4} d \sqrt {1-\frac {c x^4}{a}} \left (3 a B d^2-5 A c^2 d+5 B c^3\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}\right )}{d^2}-\frac {15 c^5 \sqrt {1-\frac {c x^4}{a}} (B c-A d) \int \frac {1}{\left (d x^2+c\right ) \sqrt {1-\frac {c x^4}{a}}}dx}{d^2 \sqrt {a-c x^4}}}{3 c d}}{5 c d}-\frac {B x^3 \sqrt {a-c x^4}}{5 c d}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {\frac {5 x \sqrt {a-c x^4} (B c-A d)}{3 d}-\frac {\frac {c \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (5 \left (a d^2+3 c^3\right ) (B c-A d)+\frac {3 \sqrt {a} d \left (3 a B d^2-5 A c^2 d+5 B c^3\right )}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {3 a^{3/4} d \sqrt {1-\frac {c x^4}{a}} \left (3 a B d^2-5 A c^2 d+5 B c^3\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}\right )}{d^2}-\frac {15 \sqrt [4]{a} c^{15/4} \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{d^2 \sqrt {a-c x^4}}}{3 c d}}{5 c d}-\frac {B x^3 \sqrt {a-c x^4}}{5 c d}\) |
Input:
Int[(x^6*(A + B*x^2))/((c + d*x^2)*Sqrt[a - c*x^4]),x]
Output:
-1/5*(B*x^3*Sqrt[a - c*x^4])/(c*d) + ((5*(B*c - A*d)*x*Sqrt[a - c*x^4])/(3 *d) - ((c*((-3*a^(3/4)*d*(5*B*c^3 - 5*A*c^2*d + 3*a*B*d^2)*Sqrt[1 - (c*x^4 )/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3/4)*Sqrt[a - c*x^4]) + (a^(1/4)*(5*(B*c - A*d)*(3*c^3 + a*d^2) + (3*Sqrt[a]*d*(5*B*c^3 - 5*A*c ^2*d + 3*a*B*d^2))/Sqrt[c])*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)* x)/a^(1/4)], -1])/(c^(1/4)*Sqrt[a - c*x^4])))/d^2 - (15*a^(1/4)*c^(15/4)*( B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), ArcSin[( c^(1/4)*x)/a^(1/4)], -1])/(d^2*Sqrt[a - c*x^4]))/(3*c*d))/(5*c*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-c/a, 2]}, Simp[(d*q - e)/q Int[1/Sqrt[a + c*x^4], x], x] + Simp[e/q Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && Neg Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) ]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && !GtQ[a, 0]
Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Si mp[-(e^2)^(-1) Int[(C*d - B*e - C*e*x^2)/Sqrt[a + c*x^4], x], x] + Simp[( C*d^2 - B*d*e + A*e^2)/e^2 Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x], x]] / ; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 - a*e^2, 0]
Int[(Px_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> W ith[{q = Expon[Px, x]}, Simp[Coeff[Px, x, q]*x^(q - 5)*(Sqrt[a + c*x^4]/(c* e*(q - 3))), x] + Simp[1/(c*e*(q - 3)) Int[(c*e*(q - 3)*Px - Coeff[Px, x, q]*x^(q - 6)*(d + e*x^2)*(a*(q - 5) + c*(q - 3)*x^4))/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; GtQ[q, 4]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x]
Time = 5.10 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.08
| method | result | size |
| risch | \(-\frac {x \left (3 B \,x^{2} d +5 A d -5 B c \right ) \sqrt {-c \,x^{4}+a}}{15 c \,d^{2}}-\frac {-\frac {5 \left (A a \,d^{3}+3 A \,c^{3} d -B a c \,d^{2}-3 B \,c^{4}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 \left (5 A \,c^{2} d -3 B a \,d^{2}-5 B \,c^{3}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}+\frac {15 c^{3} \left (A d -B c \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{15 c \,d^{2}}\) | \(357\) |
| default | \(\frac {d^{2} \left (A d -B c \right ) \left (-\frac {x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{3 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+\frac {\sqrt {c}\, d \left (A d -B c \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {A \,c^{2} d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+B \,d^{3} \left (-\frac {x^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {3 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 c^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )-\frac {B \,c^{3} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{d^{4}}-\frac {c^{2} \left (A d -B c \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d^{4} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(551\) |
| elliptic | \(\text {Expression too large to display}\) | \(970\) |
Input:
int(x^6*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/15*x*(3*B*d*x^2+5*A*d-5*B*c)*(-c*x^4+a)^(1/2)/c/d^2-1/15/c/d^2*(-5*(A*a *d^3+3*A*c^3*d-B*a*c*d^2-3*B*c^4)/d^2/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x ^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticF (x*(c^(1/2)/a^(1/2))^(1/2),I)-3/d*(5*A*c^2*d-3*B*a*d^2-5*B*c^3)*a^(1/2)/(c ^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2) )^(1/2)/(-c*x^4+a)^(1/2)/c^(1/2)*(EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-E llipticE(x*(c^(1/2)/a^(1/2))^(1/2),I))+15*c^3*(A*d-B*c)/d^2/(c^(1/2)/a^(1/ 2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c* x^4+a)^(1/2)*EllipticPi(x*(c^(1/2)/a^(1/2))^(1/2),-a^(1/2)*d/c^(3/2),(-c^( 1/2)/a^(1/2))^(1/2)/(c^(1/2)/a^(1/2))^(1/2)))
\[ \int \frac {x^6 \left (A+B x^2\right )}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{6}}{\sqrt {-c x^{4} + a} {\left (d x^{2} + c\right )}} \,d x } \] Input:
integrate(x^6*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(1/2),x, algorithm="fricas")
Output:
integral(-(B*x^8 + A*x^6)*sqrt(-c*x^4 + a)/(c*d*x^6 + c^2*x^4 - a*d*x^2 - a*c), x)
\[ \int \frac {x^6 \left (A+B x^2\right )}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int \frac {x^{6} \left (A + B x^{2}\right )}{\sqrt {a - c x^{4}} \left (c + d x^{2}\right )}\, dx \] Input:
integrate(x**6*(B*x**2+A)/(d*x**2+c)/(-c*x**4+a)**(1/2),x)
Output:
Integral(x**6*(A + B*x**2)/(sqrt(a - c*x**4)*(c + d*x**2)), x)
\[ \int \frac {x^6 \left (A+B x^2\right )}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{6}}{\sqrt {-c x^{4} + a} {\left (d x^{2} + c\right )}} \,d x } \] Input:
integrate(x^6*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate((B*x^2 + A)*x^6/(sqrt(-c*x^4 + a)*(d*x^2 + c)), x)
\[ \int \frac {x^6 \left (A+B x^2\right )}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{6}}{\sqrt {-c x^{4} + a} {\left (d x^{2} + c\right )}} \,d x } \] Input:
integrate(x^6*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((B*x^2 + A)*x^6/(sqrt(-c*x^4 + a)*(d*x^2 + c)), x)
Timed out. \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int \frac {x^6\,\left (B\,x^2+A\right )}{\sqrt {a-c\,x^4}\,\left (d\,x^2+c\right )} \,d x \] Input:
int((x^6*(A + B*x^2))/((a - c*x^4)^(1/2)*(c + d*x^2)),x)
Output:
int((x^6*(A + B*x^2))/((a - c*x^4)^(1/2)*(c + d*x^2)), x)
\[ \int \frac {x^6 \left (A+B x^2\right )}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {-5 \sqrt {-c \,x^{4}+a}\, a d x +5 \sqrt {-c \,x^{4}+a}\, b c x -3 \sqrt {-c \,x^{4}+a}\, b d \,x^{3}+5 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a^{2} c d -5 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a b \,c^{2}+9 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a b \,d^{2}-15 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a \,c^{2} d +15 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) b \,c^{3}+5 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a^{2} d^{2}+4 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a b c d}{15 c \,d^{2}} \] Input:
int(x^6*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(1/2),x)
Output:
( - 5*sqrt(a - c*x**4)*a*d*x + 5*sqrt(a - c*x**4)*b*c*x - 3*sqrt(a - c*x** 4)*b*d*x**3 + 5*int(sqrt(a - c*x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x** 6),x)*a**2*c*d - 5*int(sqrt(a - c*x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d* x**6),x)*a*b*c**2 + 9*int((sqrt(a - c*x**4)*x**4)/(a*c + a*d*x**2 - c**2*x **4 - c*d*x**6),x)*a*b*d**2 - 15*int((sqrt(a - c*x**4)*x**4)/(a*c + a*d*x* *2 - c**2*x**4 - c*d*x**6),x)*a*c**2*d + 15*int((sqrt(a - c*x**4)*x**4)/(a *c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*b*c**3 + 5*int((sqrt(a - c*x**4)* x**2)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a**2*d**2 + 4*int((sqrt(a - c*x**4)*x**2)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a*b*c*d)/(15*c *d**2)