Integrand size = 30, antiderivative size = 429 \[ \int \frac {(e x)^{7/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {(57 b c-77 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{42 b^3}-\frac {11 d e (e x)^{5/2} \sqrt {c-d x^2}}{14 b^2}+\frac {e (e x)^{5/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}+\frac {\sqrt [4]{c} \left (48 b^2 c^2-259 a b c d+231 a^2 d^2\right ) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{42 b^4 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-11 a d) (b c-a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^4 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-11 a d) (b c-a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^4 \sqrt [4]{d} \sqrt {c-d x^2}} \]
1/2*e*(e*x)^(5/2)*(-d*x^2+c)^(3/2)/b/(-b*x^2+a)-11/14*d*e*(e*x)^(5/2)*(-d* x^2+c)^(1/2)/b^2+1/42*(-77*a*d+57*b*c)*e^3*(e*x)^(1/2)*(-d*x^2+c)^(1/2)/b^ 3+1/42*c^(1/4)*(231*a^2*d^2-259*a*b*c*d+48*b^2*c^2)*e^(7/2)*EllipticF(d^(1 /4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)*(1-d*x^2/c)^(1/2)/b^4/d^(1/4)/(-d*x^2+c )^(1/2)-1/4*c^(1/4)*(-11*a*d+5*b*c)*(-a*d+b*c)*e^(7/2)*EllipticPi(d^(1/4)* (e*x)^(1/2)/c^(1/4)/e^(1/2),-b^(1/2)*c^(1/2)/a^(1/2)/d^(1/2),I)*(1-d*x^2/c )^(1/2)/b^4/d^(1/4)/(-d*x^2+c)^(1/2)-1/4*c^(1/4)*(-11*a*d+5*b*c)*(-a*d+b*c )*e^(7/2)*EllipticPi(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),b^(1/2)*c^(1/2)/a ^(1/2)/d^(1/2),I)*(1-d*x^2/c)^(1/2)/b^4/d^(1/4)/(-d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.22 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.54 \[ \int \frac {(e x)^{7/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {e^3 \sqrt {e x} \left (5 a \left (c-d x^2\right ) \left (77 a^2 d-12 b^2 x^2 \left (-3 c+d x^2\right )-a b \left (57 c+44 d x^2\right )\right )-5 a c (-57 b c+77 a d) \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+\left (48 b^2 c^2-259 a b c d+231 a^2 d^2\right ) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{210 a b^3 \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
(e^3*Sqrt[e*x]*(5*a*(c - d*x^2)*(77*a^2*d - 12*b^2*x^2*(-3*c + d*x^2) - a* b*(57*c + 44*d*x^2)) - 5*a*c*(-57*b*c + 77*a*d)*(a - b*x^2)*Sqrt[1 - (d*x^ 2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + (48*b^2*c^2 - 259 *a*b*c*d + 231*a^2*d^2)*x^2*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(210*a*b^3*(-a + b*x^2)*Sqrt[c - d*x^ 2])
Time = 1.03 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.07, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {368, 27, 967, 27, 1051, 25, 1052, 25, 27, 1021, 765, 762, 925, 27, 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.
| \(\displaystyle \int \frac {(e x)^{7/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^8 x^4 \left (c-d x^2\right )^{3/2}}{\left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \int \frac {e^4 x^4 \left (c-d x^2\right )^{3/2}}{\left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\) |
| \(\Big \downarrow \) 967 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\int \frac {x^2 \sqrt {c-d x^2} \left (5 c e^2-11 d e^2 x^2\right )}{a e^2-b e^2 x^2}d\sqrt {e x}}{4 b}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\int \frac {e^2 x^2 \sqrt {c-d x^2} \left (5 c e^2-11 d e^2 x^2\right )}{a e^2-b e^2 x^2}d\sqrt {e x}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 1051 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {11 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b}-\frac {\int -\frac {e^2 x^2 \left (5 c (7 b c-11 a d) e^2-d (57 b c-77 a d) e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{7 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\int \frac {e^2 x^2 \left (5 c (7 b c-11 a d) e^2-d (57 b c-77 a d) e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{7 b}+\frac {11 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 1052 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {-\frac {e^2 \int -\frac {d \left (\left (48 b^2 c^2-259 a b d c+231 a^2 d^2\right ) x^2 e^2+a c (57 b c-77 a d) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 b d}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (57 b c-77 a d)}{3 b}}{7 b}+\frac {11 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {e^2 \int \frac {d \left (\left (48 b^2 c^2-259 a b d c+231 a^2 d^2\right ) x^2 e^2+a c (57 b c-77 a d) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 b d}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (57 b c-77 a d)}{3 b}}{7 b}+\frac {11 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {e^2 \int \frac {\left (48 b^2 c^2-259 a b d c+231 a^2 d^2\right ) x^2 e^2+a c (57 b c-77 a d) e^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 b}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (57 b c-77 a d)}{3 b}}{7 b}+\frac {11 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {e^2 \left (\frac {21 a e^2 (5 b c-11 a d) (b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}-\frac {\left (231 a^2 d^2-259 a b c d+48 b^2 c^2\right ) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}}{b}\right )}{3 b}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (57 b c-77 a d)}{3 b}}{7 b}+\frac {11 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {e^2 \left (\frac {21 a e^2 (5 b c-11 a d) (b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}-\frac {\sqrt {1-\frac {d x^2}{c}} \left (231 a^2 d^2-259 a b c d+48 b^2 c^2\right ) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{b \sqrt {c-d x^2}}\right )}{3 b}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (57 b c-77 a d)}{3 b}}{7 b}+\frac {11 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {e^2 \left (\frac {21 a e^2 (5 b c-11 a d) (b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (231 a^2 d^2-259 a b c d+48 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{3 b}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (57 b c-77 a d)}{3 b}}{7 b}+\frac {11 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {e^2 \left (\frac {21 a e^2 (5 b c-11 a d) (b c-a d) \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )}{b}-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (231 a^2 d^2-259 a b c d+48 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{3 b}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (57 b c-77 a d)}{3 b}}{7 b}+\frac {11 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {e^2 \left (\frac {21 a e^2 (5 b c-11 a d) (b c-a d) \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )}{b}-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (231 a^2 d^2-259 a b c d+48 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{3 b}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (57 b c-77 a d)}{3 b}}{7 b}+\frac {11 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {e^2 \left (\frac {21 a e^2 (5 b c-11 a d) (b c-a d) \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )}{b}-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (231 a^2 d^2-259 a b c d+48 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{3 b}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (57 b c-77 a d)}{3 b}}{7 b}+\frac {11 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {e^2 \left (\frac {21 a e^2 (5 b c-11 a d) (b c-a d) \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )}{b}-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (231 a^2 d^2-259 a b c d+48 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{3 b}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (57 b c-77 a d)}{3 b}}{7 b}+\frac {11 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b}}{4 b e^2}\right )\) |
2*e^3*(((e*x)^(5/2)*(c - d*x^2)^(3/2))/(4*b*(a*e^2 - b*e^2*x^2)) - ((11*d* (e*x)^(5/2)*Sqrt[c - d*x^2])/(7*b) + (-1/3*((57*b*c - 77*a*d)*e^2*Sqrt[e*x ]*Sqrt[c - d*x^2])/b + (e^2*(-((c^(1/4)*(48*b^2*c^2 - 259*a*b*c*d + 231*a^ 2*d^2)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c ^(1/4)*Sqrt[e])], -1])/(b*d^(1/4)*Sqrt[c - d*x^2])) + (21*a*(5*b*c - 11*a* d)*(b*c - a*d)*e^2*((c^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqr t[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], - 1])/(2*a*d^(1/4)*e^(3/2)*Sqrt[c - d*x^2]) + (c^(1/4)*Sqrt[1 - (d*x^2)/c]*E llipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/ (c^(1/4)*Sqrt[e])], -1])/(2*a*d^(1/4)*e^(3/2)*Sqrt[c - d*x^2])))/b))/(3*b) )/(7*b))/(4*b*e^2))
3.10.3.3.1 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[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
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[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[((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*n*(p + 1))), x] - Simp[e^n/(b*n*(p + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m - n + 1) + d*(m + n*( q - 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && GtQ[m - n + 1, 0] && IntBino mialQ[a, b, c, d, e, m, 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[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*g*(m + n*(p + q + 1) + 1))), x] + Simp[1/( b*(m + n*(p + q + 1) + 1)) Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 1)* Simp[c*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*n*q*(b*c - a*d) + b*e*d*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c , d, e, f, g, m, p}, x] && IGtQ[n, 0] && GtQ[q, 0] && !(EqQ[q, 1] && Simpl erQ[e + f*x^n, c + d*x^n])
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q + 1) + 1))), x] - Simp[g^n/(b*d*(m + n*(p + q + 1) + 1)) Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*( f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 1]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1325\) vs. \(2(335)=670\).
Time = 5.86 (sec) , antiderivative size = 1326, normalized size of antiderivative = 3.09
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1326\) |
| elliptic | \(\text {Expression too large to display}\) | \(1357\) |
| default | \(\text {Expression too large to display}\) | \(3778\) |
-2/21*(3*b*d*x^2+14*a*d-9*b*c)*(-d*x^2+c)^(1/2)*x/b^3*e^4/(e*x)^(1/2)+1/21 /b^3*((63*a^2*d^2-70*a*b*c*d+12*b^2*c^2)/b/d*(c*d)^(1/2)*((x+1/d*(c*d)^(1/ 2))*d/(c*d)^(1/2))^(1/2)*(-2*(x-1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2)*(-d* x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d)^(1/2)) *d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+21*a^2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b*(1 /2*b/(a*d-b*c)/a/e*(-d*e*x^3+c*e*x)^(1/2)/(b*x^2-a)-1/4/(a*d-b*c)/a*(c*d)^ (1/2)*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^( 1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^ (1/2))^(1/2),1/2*2^(1/2))-5/8/(a*d-b*c)/(a*b)^(1/2)*(c*d)^(1/2)*(d*x/(c*d) ^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d* e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*( c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a *b)^(1/2)),1/2*2^(1/2))+3/8/(a*d-b*c)/a/(a*b)^(1/2)/d*(c*d)^(1/2)*(d*x/(c* d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(- d*e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))*EllipticPi(((x+1/d *(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)-1/b* (a*b)^(1/2)),1/2*2^(1/2))*b*c+5/8/(a*d-b*c)/(a*b)^(1/2)*(c*d)^(1/2)*(d*x/( c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/ (-d*e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2))*EllipticPi(((x+1 /d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)...
Timed out. \[ \int \frac {(e x)^{7/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e x)^{7/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {(e x)^{7/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \]
\[ \int \frac {(e x)^{7/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(e x)^{7/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}\,{\left (c-d\,x^2\right )}^{3/2}}{{\left (a-b\,x^2\right )}^2} \,d x \]