Integrand size = 19, antiderivative size = 818 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^{3/2}} \, dx=\frac {e \left (a e^2-c d^2 x^2\right )}{2 a \left (c d^4+a e^4\right ) \sqrt {a+c x^4}}+\frac {c d x \left (d^2+e^2 x^2\right )}{2 a \left (c d^4+a e^4\right ) \sqrt {a+c x^4}}-\frac {\sqrt {c} d e^2 x \sqrt {a+c x^4}}{2 a \left (c d^4+a e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^5 \arctan \left (\frac {\sqrt {-c d^4-a e^4} x}{d e \sqrt {a+c x^4}}\right )}{2 \left (-c d^4-a e^4\right )^{3/2}}-\frac {e^5 \text {arctanh}\left (\frac {a e^2+c d^2 x^2}{\sqrt {c d^4+a e^4} \sqrt {a+c x^4}}\right )}{2 \left (c d^4+a e^4\right )^{3/2}}+\frac {\sqrt [4]{c} d e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} \left (c d^4+a e^4\right ) \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{5/4} \left (c d^4+a e^4\right ) \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} d e^4 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+a e^4\right ) \sqrt {a+c x^4}}-\frac {e^4 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+a e^4\right ) \sqrt {a+c x^4}} \]
-1/2*e^5*arctan(x*(-a*e^4-c*d^4)^(1/2)/d/e/(c*x^4+a)^(1/2))/(-a*e^4-c*d^4) ^(3/2)-1/2*e^5*arctanh((c*d^2*x^2+a*e^2)/(a*e^4+c*d^4)^(1/2)/(c*x^4+a)^(1/ 2))/(a*e^4+c*d^4)^(3/2)+1/2*e*(-c*d^2*x^2+a*e^2)/a/(a*e^4+c*d^4)/(c*x^4+a) ^(1/2)+1/2*c*d*x*(e^2*x^2+d^2)/a/(a*e^4+c*d^4)/(c*x^4+a)^(1/2)-1/2*d*e^2*x *c^(1/2)*(c*x^4+a)^(1/2)/a/(a*e^4+c*d^4)/(a^(1/2)+x^2*c^(1/2))+1/2*c^(1/4) *d*e^2*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a ^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+ x^2*c^(1/2))*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(3/4)/(a*e^4+c*d^ 4)/(c*x^4+a)^(1/2)+1/4*c^(1/4)*d*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2 )/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4 ))),1/2*2^(1/2))*(-e^2*a^(1/2)+d^2*c^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+ a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(5/4)/(a*e^4+c*d^4)/(c*x^4+a)^(1/2)+1/ 2*c^(1/4)*d*e^4*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^ (1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))* (a^(1/2)+x^2*c^(1/2))*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(1/4)/(a *e^4+c*d^4)/(e^2*a^(1/2)+d^2*c^(1/2))/(c*x^4+a)^(1/2)-1/4*e^4*(cos(2*arcta n(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi (sin(2*arctan(c^(1/4)*x/a^(1/4))),1/4*(e^2*a^(1/2)+d^2*c^(1/2))^2/d^2/e^2/ a^(1/2)/c^(1/2),1/2*2^(1/2))*(-e^2*a^(1/2)+d^2*c^(1/2))*(a^(1/2)+x^2*c^(1/ 2))*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(1/4)/c^(1/4)/d/(a*e^4+...
Result contains complex when optimal does not.
Time = 10.65 (sec) , antiderivative size = 455, normalized size of antiderivative = 0.56 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^{3/2}} \, dx=-\frac {-\sqrt {a} c^{3/4} d^2 e^2 \sqrt {-c d^4-a e^4} \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+c^{3/4} d^2 \left (-i \sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {-c d^4-a e^4} \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (\sqrt [4]{c} d \left (\sqrt {-c d^4-a e^4} \left (a e^3+c d x \left (d^2-d e x+e^2 x^2\right )\right )+2 a e^5 \sqrt {a+c x^4} \arctan \left (\frac {\sqrt {c} \left (d^2-e^2 x^2\right )+e^2 \sqrt {a+c x^4}}{\sqrt {-c d^4-a e^4}}\right )\right )-2 \sqrt [4]{-1} a^{5/4} e^4 \sqrt {-c d^4-a e^4} \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (\frac {i \sqrt {a} e^2}{\sqrt {c} d^2},\arcsin \left (\frac {(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \sqrt [4]{c} d \left (-c d^4-a e^4\right )^{3/2} \sqrt {a+c x^4}} \]
-1/2*(-(Sqrt[a]*c^(3/4)*d^2*e^2*Sqrt[-(c*d^4) - a*e^4]*Sqrt[1 + (c*x^4)/a] *EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1]) + c^(3/4)*d^2*((-I )*Sqrt[c]*d^2 + Sqrt[a]*e^2)*Sqrt[-(c*d^4) - a*e^4]*Sqrt[1 + (c*x^4)/a]*El lipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + Sqrt[(I*Sqrt[c])/Sqr t[a]]*(c^(1/4)*d*(Sqrt[-(c*d^4) - a*e^4]*(a*e^3 + c*d*x*(d^2 - d*e*x + e^2 *x^2)) + 2*a*e^5*Sqrt[a + c*x^4]*ArcTan[(Sqrt[c]*(d^2 - e^2*x^2) + e^2*Sqr t[a + c*x^4])/Sqrt[-(c*d^4) - a*e^4]]) - 2*(-1)^(1/4)*a^(5/4)*e^4*Sqrt[-(c *d^4) - a*e^4]*Sqrt[1 + (c*x^4)/a]*EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2 ), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1]))/(a*Sqrt[(I*Sqrt[c])/Sqrt[ a]]*c^(1/4)*d*(-(c*d^4) - a*e^4)^(3/2)*Sqrt[a + c*x^4])
Time = 1.53 (sec) , antiderivative size = 727, normalized size of antiderivative = 0.89, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.895, Rules used = {2267, 1548, 27, 1577, 496, 25, 27, 488, 219, 2223, 2397, 25, 27, 1512, 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.
| \(\displaystyle \int \frac {1}{\left (a+c x^4\right )^{3/2} (d+e x)} \, dx\) |
| \(\Big \downarrow \) 2267 |
| \(\displaystyle d \int \frac {1}{\left (d^2-e^2 x^2\right ) \left (c x^4+a\right )^{3/2}}dx-e \int \frac {x}{\left (d^2-e^2 x^2\right ) \left (c x^4+a\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 1548 |
| \(\displaystyle d \left (\frac {\int \frac {\frac {c^{3/2} e^4 x^4}{\sqrt {a}}+c e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) x^2+\sqrt {c} \left (\frac {c d^4}{\sqrt {a}}+\sqrt {c} e^2 d^2+\sqrt {a} e^4\right )}{\left (c x^4+a\right )^{3/2}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}\right )-e \int \frac {x}{\left (d^2-e^2 x^2\right ) \left (c x^4+a\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {\int \frac {\frac {c^{3/2} e^4 x^4}{\sqrt {a}}+c e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) x^2+\sqrt {c} \left (\frac {c d^4}{\sqrt {a}}+\sqrt {c} e^2 d^2+\sqrt {a} e^4\right )}{\left (c x^4+a\right )^{3/2}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}\right )-e \int \frac {x}{\left (d^2-e^2 x^2\right ) \left (c x^4+a\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 1577 |
| \(\displaystyle d \left (\frac {\int \frac {\frac {c^{3/2} e^4 x^4}{\sqrt {a}}+c e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) x^2+\sqrt {c} \left (\frac {c d^4}{\sqrt {a}}+\sqrt {c} e^2 d^2+\sqrt {a} e^4\right )}{\left (c x^4+a\right )^{3/2}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}\right )-\frac {1}{2} e \int \frac {1}{\left (d^2-e^2 x^2\right ) \left (c x^4+a\right )^{3/2}}dx^2\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle d \left (\frac {\int \frac {\frac {c^{3/2} e^4 x^4}{\sqrt {a}}+c e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) x^2+\sqrt {c} \left (\frac {c d^4}{\sqrt {a}}+\sqrt {c} e^2 d^2+\sqrt {a} e^4\right )}{\left (c x^4+a\right )^{3/2}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}\right )-\frac {1}{2} e \left (-\frac {\int -\frac {a e^4}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{a \left (a e^4+c d^4\right )}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d \left (\frac {\int \frac {\frac {c^{3/2} e^4 x^4}{\sqrt {a}}+c e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) x^2+\sqrt {c} \left (\frac {c d^4}{\sqrt {a}}+\sqrt {c} e^2 d^2+\sqrt {a} e^4\right )}{\left (c x^4+a\right )^{3/2}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}\right )-\frac {1}{2} e \left (\frac {\int \frac {a e^4}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{a \left (a e^4+c d^4\right )}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {\int \frac {\frac {c^{3/2} e^4 x^4}{\sqrt {a}}+c e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) x^2+\sqrt {c} \left (\frac {c d^4}{\sqrt {a}}+\sqrt {c} e^2 d^2+\sqrt {a} e^4\right )}{\left (c x^4+a\right )^{3/2}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}\right )-\frac {1}{2} e \left (\frac {e^4 \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{a e^4+c d^4}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle d \left (\frac {\int \frac {\frac {c^{3/2} e^4 x^4}{\sqrt {a}}+c e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) x^2+\sqrt {c} \left (\frac {c d^4}{\sqrt {a}}+\sqrt {c} e^2 d^2+\sqrt {a} e^4\right )}{\left (c x^4+a\right )^{3/2}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}\right )-\frac {1}{2} e \left (-\frac {e^4 \int \frac {1}{c d^4+a e^4-x^4}d\frac {-a e^2-c d^2 x^2}{\sqrt {c x^4+a}}}{a e^4+c d^4}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle d \left (\frac {\int \frac {\frac {c^{3/2} e^4 x^4}{\sqrt {a}}+c e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) x^2+\sqrt {c} \left (\frac {c d^4}{\sqrt {a}}+\sqrt {c} e^2 d^2+\sqrt {a} e^4\right )}{\left (c x^4+a\right )^{3/2}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}\right )-\frac {1}{2} e \left (-\frac {e^4 \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 2223 |
| \(\displaystyle d \left (\frac {\int \frac {\frac {c^{3/2} e^4 x^4}{\sqrt {a}}+c e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) x^2+\sqrt {c} \left (\frac {c d^4}{\sqrt {a}}+\sqrt {c} e^2 d^2+\sqrt {a} e^4\right )}{\left (c x^4+a\right )^{3/2}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}+\frac {e^6 \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+c d^4}}{d e \sqrt {a+c x^4}}\right )}{2 d e \sqrt {a e^4+c d^4}}\right )}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}\right )-\frac {1}{2} e \left (-\frac {e^4 \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle d \left (\frac {\frac {c x \left (d^2+e^2 x^2\right ) \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}{2 a \sqrt {a+c x^4}}-\frac {\int -\frac {c^{3/2} \left (c d^4+\sqrt {a} \sqrt {c} e^2 d^2+2 a e^4-\sqrt {a} \sqrt {c} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) x^2\right )}{\sqrt {a} \sqrt {c x^4+a}}dx}{2 a c}}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}+\frac {e^6 \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+c d^4}}{d e \sqrt {a+c x^4}}\right )}{2 d e \sqrt {a e^4+c d^4}}\right )}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}\right )-\frac {1}{2} e \left (-\frac {e^4 \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d \left (\frac {\frac {\int \frac {c^{3/2} \left (c d^4+\sqrt {a} \sqrt {c} e^2 d^2+2 a e^4-\sqrt {c} e^2 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) x^2\right )}{\sqrt {a} \sqrt {c x^4+a}}dx}{2 a c}+\frac {c x \left (d^2+e^2 x^2\right ) \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}{2 a \sqrt {a+c x^4}}}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}+\frac {e^6 \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+c d^4}}{d e \sqrt {a+c x^4}}\right )}{2 d e \sqrt {a e^4+c d^4}}\right )}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}\right )-\frac {1}{2} e \left (-\frac {e^4 \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {\frac {\sqrt {c} \int \frac {c d^4+\sqrt {a} \sqrt {c} e^2 d^2+2 a e^4-\sqrt {c} e^2 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) x^2}{\sqrt {c x^4+a}}dx}{2 a^{3/2}}+\frac {c x \left (d^2+e^2 x^2\right ) \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}{2 a \sqrt {a+c x^4}}}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}+\frac {e^6 \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+c d^4}}{d e \sqrt {a+c x^4}}\right )}{2 d e \sqrt {a e^4+c d^4}}\right )}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}\right )-\frac {1}{2} e \left (-\frac {e^4 \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle d \left (\frac {\frac {\sqrt {c} \left (\left (a e^4+c d^4\right ) \int \frac {1}{\sqrt {c x^4+a}}dx+\sqrt {a} e^2 \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx\right )}{2 a^{3/2}}+\frac {c x \left (d^2+e^2 x^2\right ) \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}{2 a \sqrt {a+c x^4}}}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}+\frac {e^6 \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+c d^4}}{d e \sqrt {a+c x^4}}\right )}{2 d e \sqrt {a e^4+c d^4}}\right )}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}\right )-\frac {1}{2} e \left (-\frac {e^4 \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {\frac {\sqrt {c} \left (\left (a e^4+c d^4\right ) \int \frac {1}{\sqrt {c x^4+a}}dx+e^2 \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx\right )}{2 a^{3/2}}+\frac {c x \left (d^2+e^2 x^2\right ) \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}{2 a \sqrt {a+c x^4}}}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}+\frac {e^6 \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+c d^4}}{d e \sqrt {a+c x^4}}\right )}{2 d e \sqrt {a e^4+c d^4}}\right )}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}\right )-\frac {1}{2} e \left (-\frac {e^4 \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle d \left (\frac {\frac {\sqrt {c} \left (e^2 \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx+\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (a e^4+c d^4\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}\right )}{2 a^{3/2}}+\frac {c x \left (d^2+e^2 x^2\right ) \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}{2 a \sqrt {a+c x^4}}}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}+\frac {e^6 \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+c d^4}}{d e \sqrt {a+c x^4}}\right )}{2 d e \sqrt {a e^4+c d^4}}\right )}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}\right )-\frac {1}{2} e \left (-\frac {e^4 \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle d \left (\frac {\frac {\sqrt {c} \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (a e^4+c d^4\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+e^2 \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )\right )}{2 a^{3/2}}+\frac {c x \left (d^2+e^2 x^2\right ) \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}{2 a \sqrt {a+c x^4}}}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}+\frac {e^6 \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+c d^4}}{d e \sqrt {a+c x^4}}\right )}{2 d e \sqrt {a e^4+c d^4}}\right )}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+c d^4\right )}\right )-\frac {1}{2} e \left (-\frac {e^4 \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (a e^4+c d^4\right )}\right )\) |
-1/2*(e*(-((a*e^2 - c*d^2*x^2)/(a*(c*d^4 + a*e^4)*Sqrt[a + c*x^4])) - (e^4 *ArcTanh[(-(a*e^2) - c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])])/(c *d^4 + a*e^4)^(3/2))) + d*(((c*((Sqrt[c]*d^2)/Sqrt[a] + e^2)*x*(d^2 + e^2* x^2))/(2*a*Sqrt[a + c*x^4]) + (Sqrt[c]*(e^2*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*(- ((x*Sqrt[a + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c ]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^( 1/4)*x)/a^(1/4)], 1/2])/(c^(1/4)*Sqrt[a + c*x^4])) + ((c*d^4 + a*e^4)*(Sqr t[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[ 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*c^(1/4)*Sqrt[a + c*x^4]))) /(2*a^(3/2)))/(((Sqrt[c]*d^2)/Sqrt[a] + e^2)*(c*d^4 + a*e^4)) + (e^6*(((Sq rt[c]*d^2 + Sqrt[a]*e^2)*ArcTanh[(Sqrt[c*d^4 + a*e^4]*x)/(d*e*Sqrt[a + c*x ^4])])/(2*d*e*Sqrt[c*d^4 + a*e^4]) + ((Sqrt[a]/d^2 - Sqrt[c]/e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[(Sq rt[c]*d^2 + Sqrt[a]*e^2)^2/(4*Sqrt[a]*Sqrt[c]*d^2*e^2), 2*ArcTan[(c^(1/4)* x)/a^(1/4)], 1/2])/(4*a^(1/4)*c^(1/4)*Sqrt[a + c*x^4])))/(Sqrt[a]*((Sqrt[c ]*d^2)/Sqrt[a] + e^2)*(c*d^4 + a*e^4)))
3.3.23.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 *p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad raticQ[a, 0, b, c, d, n, p, 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[((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]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/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] /; NeQ[e + d*q, 0]] /; FreeQ[{a, c , d, e}, x] && PosQ[c/a]
Int[((a_) + (c_.)*(x_)^4)^(p_)/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Simp[-( c*d^2 + a*e^2)^(p + 1/2)/(e^(2*p)*(Rt[c/a, 2]*d - e)) Int[(1 + Rt[c/a, 2] *x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] + Simp[(c*d^2 + a*e^2)^(p + 1/2 )/(Rt[c/a, 2]*d - e) Int[(a + c*x^4)^p*ExpandToSum[((Rt[c/a, 2]*d - e)*(c *d^2 + a*e^2)^(-p - 1/2) + ((1 + Rt[c/a, 2]*x^2)*(a + c*x^4)^(-p - 1/2))/e^ (2*p))/(d + e*x^2), x], x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e ^2, 0] && ILtQ[p + 1/2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, c, d, e, p, q}, x]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) , x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTanh[Rt[(-c)* (d/e) - a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[(-c)*(d/e) - a*(e/d), 2] )), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^ 2)]/(4*d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*Arc Tan[q*x], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0 ] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[c*(d/e) + a*(e /d)]
Int[((a_) + (c_.)*(x_)^4)^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[d Int[(a + c*x^4)^p/(d^2 - e^2*x^2), x], x] - Simp[e Int[x*((a + c*x^4)^p/( d^2 - e^2*x^2)), x], x] /; FreeQ[{a, c, d, e}, x] && IntegerQ[p + 1/2]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x]}, S imp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[(a + b*x^n)^(p + 1)* ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]
Result contains complex when optimal does not.
Time = 0.97 (sec) , antiderivative size = 496, normalized size of antiderivative = 0.61
| method | result | size |
| default | \(-\frac {2 c \left (-\frac {d \,e^{2} x^{3}}{4 a \left (e^{4} a +d^{4} c \right )}+\frac {d^{2} e \,x^{2}}{4 a \left (e^{4} a +d^{4} c \right )}-\frac {d^{3} x}{4 a \left (e^{4} a +d^{4} c \right )}-\frac {e^{3}}{4 \left (e^{4} a +d^{4} c \right ) c}\right )}{\sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}+\frac {d^{3} c \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \left (e^{4} a +d^{4} c \right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {i \sqrt {c}\, d \,e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \left (e^{4} a +d^{4} c \right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {e^{3} \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{e^{4} a +d^{4} c}\) | \(496\) |
| elliptic | \(-\frac {2 c \left (-\frac {d \,e^{2} x^{3}}{4 a \left (e^{4} a +d^{4} c \right )}+\frac {d^{2} e \,x^{2}}{4 a \left (e^{4} a +d^{4} c \right )}-\frac {d^{3} x}{4 a \left (e^{4} a +d^{4} c \right )}-\frac {e^{3}}{4 \left (e^{4} a +d^{4} c \right ) c}\right )}{\sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}+\frac {d^{3} c \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \left (e^{4} a +d^{4} c \right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {i \sqrt {c}\, d \,e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \left (e^{4} a +d^{4} c \right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {e^{3} \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{e^{4} a +d^{4} c}\) | \(496\) |
-2*c*(-1/4/a*d*e^2/(a*e^4+c*d^4)*x^3+1/4/a*d^2*e/(a*e^4+c*d^4)*x^2-1/4*d^3 /a/(a*e^4+c*d^4)*x-1/4*e^3/(a*e^4+c*d^4)/c)/((x^4+a/c)*c)^(1/2)+1/2*d^3/a* c/(a*e^4+c*d^4)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)* (1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^( 1/2))^(1/2),I)-1/2*I*c^(1/2)*d/a^(1/2)*e^2/(a*e^4+c*d^4)/(I/a^(1/2)*c^(1/2 ))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/( c*x^4+a)^(1/2)*(EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-EllipticE(x*(I/a^ (1/2)*c^(1/2))^(1/2),I))+e^3/(a*e^4+c*d^4)*(-1/2/(c/e^4*d^4+a)^(1/2)*arcta nh(1/2*(2*c*x^2/e^2*d^2+2*a)/(c/e^4*d^4+a)^(1/2)/(c*x^4+a)^(1/2))+1/(I/a^( 1/2)*c^(1/2))^(1/2)*e/d*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/ 2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticPi(x*(I/a^(1/2)*c^(1/2))^(1/2),-I*a^ (1/2)/c^(1/2)*e^2/d^2,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2) ))
Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (a + c x^{4}\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
\[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}} \,d x } \]
\[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x^4+a\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]