Integrand size = 21, antiderivative size = 976 \[ \int \frac {1}{\left (a+b x^4\right )^{5/2} \left (c+d x^4\right )} \, dx=\frac {b x}{6 a (b c-a d) \left (a+b x^4\right )^{3/2}}+\frac {b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt {a+b x^4}}-\frac {d^{9/4} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{4 (-c)^{3/4} (b c-a d)^{5/2}}-\frac {d^{9/4} \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{4 (-c)^{3/4} (-b c+a d)^{5/2}}+\frac {b^{3/4} (5 b c-11 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{24 a^{9/4} (b c-a d)^2 \sqrt {a+b x^4}}+\frac {\sqrt [4]{b} \left (\sqrt {b} c-\sqrt {a} \sqrt {-c} \sqrt {d}\right ) d^2 \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 )}{4 \sqrt [4]{a} c (b c-a d)^2 (b c+a d) \sqrt {a+b x^4}}+\frac {\sqrt [4]{b} \left (\sqrt {b} c+\sqrt {a} \sqrt {-c} \sqrt {d}\right ) d^2 \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 )}{4 \sqrt [4]{a} c (b c-a d)^2 (b c+a d) \sqrt {a+b x^4}}+\frac {\left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right )^2 d^2 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} 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]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c (b c-a d)^2 (b c+a d) \sqrt {a+b x^4}}+\frac {\left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right )^2 d^2 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} 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]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c (b c-a d)^2 (b c+a d) \sqrt {a+b x^4}} \]
1/6*b*x/a/(-a*d+b*c)/(b*x^4+a)^(3/2)-1/4*d^(9/4)*arctan(x*(-a*d+b*c)^(1/2) /(-c)^(1/4)/d^(1/4)/(b*x^4+a)^(1/2))/(-c)^(3/4)/(-a*d+b*c)^(5/2)-1/4*d^(9/ 4)*arctan(x*(a*d-b*c)^(1/2)/(-c)^(1/4)/d^(1/4)/(b*x^4+a)^(1/2))/(-c)^(3/4) /(a*d-b*c)^(5/2)+1/12*b*(-11*a*d+5*b*c)*x/a^2/(-a*d+b*c)^2/(b*x^4+a)^(1/2) +1/24*b^(3/4)*(-11*a*d+5*b*c)*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/c os(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))) ,1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1 /2)/a^(9/4)/(-a*d+b*c)^2/(b*x^4+a)^(1/2)+1/8*d^2*(cos(2*arctan(b^(1/4)*x/a ^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan (b^(1/4)*x/a^(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))*(a^(1/2)+x^2*b^(1/2))*(b^(1/2)*(-c)^ (1/2)-a^(1/2)*d^(1/2))^2*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^(1/4) /b^(1/4)/c/(-a*d+b*c)^2/(a*d+b*c)/(b*x^4+a)^(1/2)+1/8*d^2*(cos(2*arctan(b^ (1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticPi(sin (2*arctan(b^(1/4)*x/a^(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))*(a^(1/2)+x^2*b^(1/2))*(b^( 1/2)*(-c)^(1/2)+a^(1/2)*d^(1/2))^2*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/ 2)/a^(1/4)/b^(1/4)/c/(-a*d+b*c)^2/(a*d+b*c)/(b*x^4+a)^(1/2)+1/4*b^(1/4)*d^ 2*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4 )))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^...
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.82 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\left (a+b x^4\right )^{5/2} \left (c+d x^4\right )} \, dx=-\frac {x \left (\frac {b d (-5 b c+11 a d) x^4 \sqrt {1+\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{c}+\frac {5 \left (5 a c \left (12 a^3 d^2+5 b^3 c x^4 \left (2 c+d x^4\right )-a^2 b d \left (24 c+d x^4\right )+a b^2 \left (12 c^2-15 c d x^4-11 d^2 x^8\right )\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+2 b x^4 \left (c+d x^4\right ) \left (13 a^2 d-5 b^2 c x^4+a b \left (-7 c+11 d x^4\right )\right ) \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}{\left (a+b x^4\right ) \left (c+d x^4\right ) \left (-5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}\right )}{60 a^2 (b c-a d)^2 \sqrt {a+b x^4}} \]
-1/60*(x*((b*d*(-5*b*c + 11*a*d)*x^4*Sqrt[1 + (b*x^4)/a]*AppellF1[5/4, 1/2 , 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/c + (5*(5*a*c*(12*a^3*d^2 + 5*b^3*c *x^4*(2*c + d*x^4) - a^2*b*d*(24*c + d*x^4) + a*b^2*(12*c^2 - 15*c*d*x^4 - 11*d^2*x^8))*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + 2*b *x^4*(c + d*x^4)*(13*a^2*d - 5*b^2*c*x^4 + a*b*(-7*c + 11*d*x^4))*(2*a*d*A ppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*AppellF1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))/((a + b*x^4)*(c + d*x^4)*(-5*a *c*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + 2*x^4*(2*a*d*A ppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*AppellF1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))))/(a^2*(b*c - a*d)^2*Sqrt[a + b*x^4])
Time = 1.64 (sec) , antiderivative size = 1071, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {931, 25, 1024, 25, 1021, 761, 925, 1541, 27, 761, 2221, 2223}
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.
| \(\displaystyle \int \frac {1}{\left (a+b x^4\right )^{5/2} \left (c+d x^4\right )} \, dx\) |
| \(\Big \downarrow \) 931 |
| \(\displaystyle \frac {b x}{6 a \left (a+b x^4\right )^{3/2} (b c-a d)}-\frac {\int -\frac {5 b d x^4+5 b c-6 a d}{\left (b x^4+a\right )^{3/2} \left (d x^4+c\right )}dx}{6 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {5 b d x^4+5 b c-6 a d}{\left (b x^4+a\right )^{3/2} \left (d x^4+c\right )}dx}{6 a (b c-a d)}+\frac {b x}{6 a \left (a+b x^4\right )^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle \frac {\frac {b x (5 b c-11 a d)}{2 a \sqrt {a+b x^4} (b c-a d)}-\frac {\int -\frac {b d (5 b c-11 a d) x^4+5 b^2 c^2+12 a^2 d^2-11 a b c d}{\sqrt {b x^4+a} \left (d x^4+c\right )}dx}{2 a (b c-a d)}}{6 a (b c-a d)}+\frac {b x}{6 a \left (a+b x^4\right )^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {b d (5 b c-11 a d) x^4+5 b^2 c^2+12 a^2 d^2-11 a b c d}{\sqrt {b x^4+a} \left (d x^4+c\right )}dx}{2 a (b c-a d)}+\frac {b x (5 b c-11 a d)}{2 a \sqrt {a+b x^4} (b c-a d)}}{6 a (b c-a d)}+\frac {b x}{6 a \left (a+b x^4\right )^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle \frac {\frac {12 a^2 d^2 \int \frac {1}{\sqrt {b x^4+a} \left (d x^4+c\right )}dx+b (5 b c-11 a d) \int \frac {1}{\sqrt {b x^4+a}}dx}{2 a (b c-a d)}+\frac {b x (5 b c-11 a d)}{2 a \sqrt {a+b x^4} (b c-a d)}}{6 a (b c-a d)}+\frac {b x}{6 a \left (a+b x^4\right )^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {12 a^2 d^2 \int \frac {1}{\sqrt {b x^4+a} \left (d x^4+c\right )}dx+\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (5 b c-11 a d) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4}}}{2 a (b c-a d)}+\frac {b x (5 b c-11 a d)}{2 a \sqrt {a+b x^4} (b c-a d)}}{6 a (b c-a d)}+\frac {b x}{6 a \left (a+b x^4\right )^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle \frac {\frac {12 a^2 d^2 \left (\frac {\int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {b x^4+a}}dx}{2 c}+\frac {\int \frac {1}{\left (\frac {\sqrt {d} x^2}{\sqrt {-c}}+1\right ) \sqrt {b x^4+a}}dx}{2 c}\right )+\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (5 b c-11 a d) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4}}}{2 a (b c-a d)}+\frac {b x (5 b c-11 a d)}{2 a \sqrt {a+b x^4} (b c-a d)}}{6 a (b c-a d)}+\frac {b x}{6 a \left (a+b x^4\right )^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 1541 |
| \(\displaystyle \frac {\frac {12 a^2 d^2 \left (\frac {\frac {\sqrt {b} \left (\sqrt {a} \sqrt {-c} \sqrt {d}+\sqrt {b} c\right ) \int \frac {1}{\sqrt {b x^4+a}}dx}{a d+b c}-\frac {\sqrt {a} \sqrt {d} \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \left (1-\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {b x^4+a}}dx}{a d+b c}}{2 c}+\frac {\frac {\sqrt {b} c \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}+\sqrt {b}\right ) \int \frac {1}{\sqrt {b x^4+a}}dx}{a d+b c}+\frac {\sqrt {a} \sqrt {d} \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {-c}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \left (\frac {\sqrt {d} x^2}{\sqrt {-c}}+1\right ) \sqrt {b x^4+a}}dx}{a d+b c}}{2 c}\right )+\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (5 b c-11 a d) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4}}}{2 a (b c-a d)}+\frac {b x (5 b c-11 a d)}{2 a \sqrt {a+b x^4} (b c-a d)}}{6 a (b c-a d)}+\frac {b x}{6 a \left (a+b x^4\right )^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {12 a^2 d^2 \left (\frac {\frac {\sqrt {b} \left (\sqrt {a} \sqrt {-c} \sqrt {d}+\sqrt {b} c\right ) \int \frac {1}{\sqrt {b x^4+a}}dx}{a d+b c}-\frac {\sqrt {d} \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {b x^4+a}}dx}{a d+b c}}{2 c}+\frac {\frac {\sqrt {b} c \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}+\sqrt {b}\right ) \int \frac {1}{\sqrt {b x^4+a}}dx}{a d+b c}+\frac {\sqrt {d} \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {-c}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (\frac {\sqrt {d} x^2}{\sqrt {-c}}+1\right ) \sqrt {b x^4+a}}dx}{a d+b c}}{2 c}\right )+\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (5 b c-11 a d) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4}}}{2 a (b c-a d)}+\frac {b x (5 b c-11 a d)}{2 a \sqrt {a+b x^4} (b c-a d)}}{6 a (b c-a d)}+\frac {b x}{6 a \left (a+b x^4\right )^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {12 a^2 d^2 \left (\frac {\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\sqrt {a} \sqrt {-c} \sqrt {d}+\sqrt {b} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4} (a d+b c)}-\frac {\sqrt {d} \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {b x^4+a}}dx}{a d+b c}}{2 c}+\frac {\frac {\sqrt {d} \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {-c}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (\frac {\sqrt {d} x^2}{\sqrt {-c}}+1\right ) \sqrt {b x^4+a}}dx}{a d+b c}+\frac {\sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}+\sqrt {b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4} (a d+b c)}}{2 c}\right )+\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (5 b c-11 a d) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4}}}{2 a (b c-a d)}+\frac {b x (5 b c-11 a d)}{2 a \sqrt {a+b x^4} (b c-a d)}}{6 a (b c-a d)}+\frac {b x}{6 a \left (a+b x^4\right )^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 2221 |
| \(\displaystyle \frac {b x}{6 a (b c-a d) \left (b x^4+a\right )^{3/2}}+\frac {\frac {b (5 b c-11 a d) x}{2 a (b c-a d) \sqrt {b x^4+a}}+\frac {12 a^2 \left (\frac {\frac {\sqrt [4]{b} \left (\sqrt {b} c+\sqrt {a} \sqrt {-c} \sqrt {d}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} (b c+a d) \sqrt {b x^4+a}}-\frac {\left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) \sqrt {d} \left (\frac {\sqrt [4]{-c} \left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right ) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {b x^4+a}}\right )}{2 \sqrt [4]{d} \sqrt {b c-a d}}+\frac {\left (\sqrt {a}-\frac {\sqrt {b} \sqrt {-c}}{\sqrt {d}}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\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]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x^4+a}}\right )}{b c+a d}}{2 c}+\frac {\frac {\sqrt [4]{b} c \left (\sqrt {b}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} (b c+a d) \sqrt {b x^4+a}}+\frac {\left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right ) \sqrt {d} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (\frac {\sqrt {d} x^2}{\sqrt {-c}}+1\right ) \sqrt {b x^4+a}}dx}{b c+a d}}{2 c}\right ) d^2+\frac {b^{3/4} (5 b c-11 a d) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {b x^4+a}}}{2 a (b c-a d)}}{6 a (b c-a d)}\) |
| \(\Big \downarrow \) 2223 |
| \(\displaystyle \frac {b x}{6 a (b c-a d) \left (b x^4+a\right )^{3/2}}+\frac {\frac {b (5 b c-11 a d) x}{2 a (b c-a d) \sqrt {b x^4+a}}+\frac {12 a^2 \left (\frac {\frac {\sqrt [4]{b} c \left (\sqrt {b}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} (b c+a d) \sqrt {b x^4+a}}+\frac {\left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right ) \sqrt {d} \left (\frac {\left (\sqrt {a}+\frac {\sqrt {b} \sqrt {-c}}{\sqrt {d}}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\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]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x^4+a}}-\frac {(-c)^{3/4} \left (\sqrt {b}-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {b x^4+a}}\right )}{2 \sqrt [4]{d} \sqrt {b c-a d}}\right )}{b c+a d}}{2 c}+\frac {\frac {\sqrt [4]{b} \left (\sqrt {b} c+\sqrt {a} \sqrt {-c} \sqrt {d}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} (b c+a d) \sqrt {b x^4+a}}-\frac {\left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) \sqrt {d} \left (\frac {\sqrt [4]{-c} \left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right ) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {b x^4+a}}\right )}{2 \sqrt [4]{d} \sqrt {b c-a d}}+\frac {\left (\sqrt {a}-\frac {\sqrt {b} \sqrt {-c}}{\sqrt {d}}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\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]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x^4+a}}\right )}{b c+a d}}{2 c}\right ) d^2+\frac {b^{3/4} (5 b c-11 a d) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {b x^4+a}}}{2 a (b c-a d)}}{6 a (b c-a d)}\) |
(b*x)/(6*a*(b*c - a*d)*(a + b*x^4)^(3/2)) + ((b*(5*b*c - 11*a*d)*x)/(2*a*( b*c - a*d)*Sqrt[a + b*x^4]) + ((b^(3/4)*(5*b*c - 11*a*d)*(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*a^(1/4)*Sqrt[a + b*x^4]) + 12*a^2*d^2*(((b^(1/4 )*c*(Sqrt[b] + (Sqrt[a]*Sqrt[d])/Sqrt[-c])*(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*a^(1/4)*(b*c + a*d)*Sqrt[a + b*x^4]) + ((Sqrt[b]*Sqrt[-c] + S qrt[a]*Sqrt[d])*Sqrt[d]*(-1/2*((-c)^(3/4)*(Sqrt[b] - (Sqrt[a]*Sqrt[d])/Sqr t[-c])*ArcTanh[(Sqrt[b*c - a*d]*x)/((-c)^(1/4)*d^(1/4)*Sqrt[a + b*x^4])])/ (d^(1/4)*Sqrt[b*c - a*d]) + ((Sqrt[a] + (Sqrt[b]*Sqrt[-c])/Sqrt[d])*(Sqrt[ a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[- 1/4*(Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])^2/(Sqrt[a]*Sqrt[b]*Sqrt[-c]*Sqrt[ d]), 2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*b^(1/4)*Sqrt[a + b*x^ 4])))/(b*c + a*d))/(2*c) + ((b^(1/4)*(Sqrt[b]*c + Sqrt[a]*Sqrt[-c]*Sqrt[d] )*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Elli pticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(b*c + a*d)*Sqrt[a + b*x^4]) - ((Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])*Sqrt[d]*(((-c)^(1/4)*(Sqr t[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d])*ArcTan[(Sqrt[b*c - a*d]*x)/((-c)^(1/4)*d^ (1/4)*Sqrt[a + b*x^4])])/(2*d^(1/4)*Sqrt[b*c - a*d]) + ((Sqrt[a] - (Sqrt[b ]*Sqrt[-c])/Sqrt[d])*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] ...
3.2.83.3.1 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[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[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 1/(2*c) Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[1/(2 *c) Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Simp[1/(a*n*(p + 1)*(b*c - a*d)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, n, p, q, x]
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x _)^(n_)]), x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^n], x], x] + Simp[(b* e - a*f)/b Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f _.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( p + 1)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b *c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[c/a, 2]}, Simp[(c*d + a*e*q)/(c*d^2 - a*e^2) Int[1/Sqrt[a + c*x^4 ], x], x] - Simp[(a*e*(e + d*q))/(c*d^2 - a*e^2) Int[(1 + q*x^2)/((d + e* x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e ^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
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))*(ArcTan[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*ArcTan[q*x ], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && Po sQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[c*(d/e) + a*(e/d)]
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)]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.16 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.38
| method | result | size |
| default | \(-\frac {x \sqrt {b \,x^{4}+a}}{6 a b \left (a d -b c \right ) \left (x^{4}+\frac {a}{b}\right )^{2}}-\frac {b x \left (11 a d -5 b c \right )}{12 a^{2} \left (a d -b c \right )^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {b \left (11 a d -5 b c \right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{12 a^{2} \left (a d -b c \right )^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {d \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, c \sqrt {b \,x^{4}+a}}}{\left (a d -b c \right )^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8}\) | \(371\) |
| elliptic | \(-\frac {x \sqrt {b \,x^{4}+a}}{6 a b \left (a d -b c \right ) \left (x^{4}+\frac {a}{b}\right )^{2}}-\frac {b x \left (11 a d -5 b c \right )}{12 a^{2} \left (a d -b c \right )^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {b \left (11 a d -5 b c \right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{12 a^{2} \left (a d -b c \right )^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {d \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, c \sqrt {b \,x^{4}+a}}}{\left (a d -b c \right )^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8}\) | \(371\) |
-1/6*x/a/b/(a*d-b*c)*(b*x^4+a)^(1/2)/(x^4+a/b)^2-1/12*b*x/a^2*(11*a*d-5*b* c)/(a*d-b*c)^2/((x^4+a/b)*b)^(1/2)-1/12*b/a^2*(11*a*d-5*b*c)/(a*d-b*c)^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)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)+1/ 8*d*sum(1/(a*d-b*c)^2/_alpha^3*(-1/(1/d*(a*d-b*c))^(1/2)*arctanh(1/2*(2*_a lpha^2*b*x^2+2*a)/(1/d*(a*d-b*c))^(1/2)/(b*x^4+a)^(1/2))+2/(I/a^(1/2)*b^(1 /2))^(1/2)*_alpha^3*d/c*(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)*EllipticPi(x*(I/a^(1/2)*b^(1/2))^(1/2),I*a^( 1/2)/b^(1/2)*_alpha^2/c*d,(-I/a^(1/2)*b^(1/2))^(1/2)/(I/a^(1/2)*b^(1/2))^( 1/2))),_alpha=RootOf(_Z^4*d+c))
Timed out. \[ \int \frac {1}{\left (a+b x^4\right )^{5/2} \left (c+d x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (a+b x^4\right )^{5/2} \left (c+d x^4\right )} \, dx=\int \frac {1}{\left (a + b x^{4}\right )^{\frac {5}{2}} \left (c + d x^{4}\right )}\, dx \]
\[ \int \frac {1}{\left (a+b x^4\right )^{5/2} \left (c+d x^4\right )} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {5}{2}} {\left (d x^{4} + c\right )}} \,d x } \]
\[ \int \frac {1}{\left (a+b x^4\right )^{5/2} \left (c+d x^4\right )} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {5}{2}} {\left (d x^{4} + c\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b x^4\right )^{5/2} \left (c+d x^4\right )} \, dx=\int \frac {1}{{\left (b\,x^4+a\right )}^{5/2}\,\left (d\,x^4+c\right )} \,d x \]