Integrand size = 30, antiderivative size = 447 \[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {5 d e \sqrt {e x}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (14 b c+a d) e \sqrt {e x}}{6 c (b c-a d)^3 \sqrt {c-d x^2}}+\frac {d^{3/4} (14 b c+a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 c^{3/4} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {b \sqrt [4]{c} (b c+9 a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {b \sqrt [4]{c} (b c+9 a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}} \]
5/6*d*e*(e*x)^(1/2)/(-a*d+b*c)^2/(-d*x^2+c)^(3/2)+1/2*e*(e*x)^(1/2)/(-a*d+ b*c)/(-b*x^2+a)/(-d*x^2+c)^(3/2)+1/6*d*(a*d+14*b*c)*e*(e*x)^(1/2)/c/(-a*d+ b*c)^3/(-d*x^2+c)^(1/2)+1/6*d^(3/4)*(a*d+14*b*c)*e^(3/2)*EllipticF(d^(1/4) *(e*x)^(1/2)/c^(1/4)/e^(1/2),I)*(1-d*x^2/c)^(1/2)/c^(3/4)/(-a*d+b*c)^3/(-d *x^2+c)^(1/2)-1/4*b*c^(1/4)*(9*a*d+b*c)*e^(3/2)*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) /a/d^(1/4)/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)-1/4*b*c^(1/4)*(9*a*d+b*c)*e^(3/2) *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)/a/d^(1/4)/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.34 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.62 \[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {e \sqrt {e x} \left (5 a \left (a^2 d^2 \left (c+d x^2\right )+b^2 c \left (-3 c^2+19 c d x^2-14 d^2 x^4\right )-a b d \left (13 c^2-10 c d x^2+d^2 x^4\right )\right )-5 \left (-3 b^2 c^2-13 a b c d+a^2 d^2\right ) \left (a-b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+b d (14 b c+a d) x^2 \left (a-b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{30 a c (b c-a d)^3 \left (-a+b x^2\right ) \left (c-d x^2\right )^{3/2}} \]
(e*Sqrt[e*x]*(5*a*(a^2*d^2*(c + d*x^2) + b^2*c*(-3*c^2 + 19*c*d*x^2 - 14*d ^2*x^4) - a*b*d*(13*c^2 - 10*c*d*x^2 + d^2*x^4)) - 5*(-3*b^2*c^2 - 13*a*b* c*d + a^2*d^2)*(a - b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1 /2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + b*d*(14*b*c + a*d)*x^2*(a - b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2) /a]))/(30*a*c*(b*c - a*d)^3*(-a + b*x^2)*(c - d*x^2)^(3/2))
Time = 0.93 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.05, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {368, 27, 971, 27, 1024, 27, 1024, 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.
| \(\displaystyle \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^6 x^2}{\left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \int \frac {e^2 x^2}{\left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\) |
| \(\Big \downarrow \) 971 |
| \(\displaystyle 2 e^3 \left (\frac {\sqrt {e x}}{4 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\int \frac {9 d x^2 e^2+c e^2}{e^2 \left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\sqrt {e x}}{4 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\int \frac {9 d x^2 e^2+c e^2}{\left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 e^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle 2 e^3 \left (\frac {\sqrt {e x}}{4 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {-\frac {\int -\frac {2 c \left (25 b d x^2 e^2+(3 b c+2 a d) e^2\right )}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{6 c (b c-a d)}-\frac {5 d \sqrt {e x}}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 e^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\sqrt {e x}}{4 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\int \frac {25 b d x^2 e^2+(3 b c+2 a d) e^2}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 (b c-a d)}-\frac {5 d \sqrt {e x}}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 e^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle 2 e^3 \left (\frac {\sqrt {e x}}{4 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {-\frac {\int -\frac {2 \left (b d (14 b c+a d) x^2 e^2+\left (3 b^2 c^2+13 a b d c-a^2 d^2\right ) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}-\frac {d \sqrt {e x} (a d+14 b c)}{c \sqrt {c-d x^2} (b c-a d)}}{3 (b c-a d)}-\frac {5 d \sqrt {e x}}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 e^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\sqrt {e x}}{4 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {\int \frac {b d (14 b c+a d) x^2 e^2+\left (3 b^2 c^2+13 a b d c-a^2 d^2\right ) e^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{c (b c-a d)}-\frac {d \sqrt {e x} (a d+14 b c)}{c \sqrt {c-d x^2} (b c-a d)}}{3 (b c-a d)}-\frac {5 d \sqrt {e x}}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 e^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle 2 e^3 \left (\frac {\sqrt {e x}}{4 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {3 b c e^2 (9 a d+b c) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}-d (a d+14 b c) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}}{c (b c-a d)}-\frac {d \sqrt {e x} (a d+14 b c)}{c \sqrt {c-d x^2} (b c-a d)}}{3 (b c-a d)}-\frac {5 d \sqrt {e x}}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 e^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle 2 e^3 \left (\frac {\sqrt {e x}}{4 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {3 b c e^2 (9 a d+b c) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}-\frac {d \sqrt {1-\frac {d x^2}{c}} (a d+14 b c) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{\sqrt {c-d x^2}}}{c (b c-a d)}-\frac {d \sqrt {e x} (a d+14 b c)}{c \sqrt {c-d x^2} (b c-a d)}}{3 (b c-a d)}-\frac {5 d \sqrt {e x}}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 e^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 2 e^3 \left (\frac {\sqrt {e x}}{4 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {3 b c e^2 (9 a d+b c) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}-\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (a d+14 b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{c (b c-a d)}-\frac {d \sqrt {e x} (a d+14 b c)}{c \sqrt {c-d x^2} (b c-a d)}}{3 (b c-a d)}-\frac {5 d \sqrt {e x}}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 e^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle 2 e^3 \left (\frac {\sqrt {e x}}{4 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {3 b c e^2 (9 a d+b c) \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )-\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (a d+14 b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{c (b c-a d)}-\frac {d \sqrt {e x} (a d+14 b c)}{c \sqrt {c-d x^2} (b c-a d)}}{3 (b c-a d)}-\frac {5 d \sqrt {e x}}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 e^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\sqrt {e x}}{4 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {3 b c e^2 (9 a d+b c) \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )-\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (a d+14 b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{c (b c-a d)}-\frac {d \sqrt {e x} (a d+14 b c)}{c \sqrt {c-d x^2} (b c-a d)}}{3 (b c-a d)}-\frac {5 d \sqrt {e x}}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 e^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle 2 e^3 \left (\frac {\sqrt {e x}}{4 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {3 b c e^2 (9 a d+b c) \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )-\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (a d+14 b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{c (b c-a d)}-\frac {d \sqrt {e x} (a d+14 b c)}{c \sqrt {c-d x^2} (b c-a d)}}{3 (b c-a d)}-\frac {5 d \sqrt {e x}}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 e^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle 2 e^3 \left (\frac {\sqrt {e x}}{4 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {3 b c e^2 (9 a d+b c) \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )-\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (a d+14 b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{c (b c-a d)}-\frac {d \sqrt {e x} (a d+14 b c)}{c \sqrt {c-d x^2} (b c-a d)}}{3 (b c-a d)}-\frac {5 d \sqrt {e x}}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 e^2 (b c-a d)}\right )\) |
2*e^3*(Sqrt[e*x]/(4*(b*c - a*d)*(c - d*x^2)^(3/2)*(a*e^2 - b*e^2*x^2)) - ( (-5*d*Sqrt[e*x])/(3*(b*c - a*d)*(c - d*x^2)^(3/2)) + (-((d*(14*b*c + a*d)* Sqrt[e*x])/(c*(b*c - a*d)*Sqrt[c - d*x^2])) + (-((c^(1/4)*d^(3/4)*(14*b*c + a*d)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c ^(1/4)*Sqrt[e])], -1])/Sqrt[c - d*x^2]) + 3*b*c*(b*c + 9*a*d)*e^2*((c^(1/4 )*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), A rcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*d^(1/4)*e^(3/2)*Sq rt[c - d*x^2]) + (c^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(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])))/(c*(b*c - a*d)))/(3*(b*c - a*d)))/(4 *(b*c - a*d)*e^2))
3.10.29.3.1 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 + 1)/(n*(b*c - a*d)*(p + 1))), x] - Simp[e^n/(n*(b*c - a*d) *(p + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e , q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[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[((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, 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. \(1191\) vs. \(2(353)=706\).
Time = 3.29 (sec) , antiderivative size = 1192, normalized size of antiderivative = 2.67
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1192\) |
| default | \(\text {Expression too large to display}\) | \(4391\) |
1/e/x*(e*x)^(1/2)/(-d*x^2+c)^(1/2)*((-d*x^2+c)*e*x)^(1/2)*(1/2*b^2*d*e/(a^ 2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)*(-d*e*x^3+c*e*x)^(1/2)/(b*d*x^2-a*d)+1/ 3*e/(a*d-b*c)^2/d*(-d*e*x^3+c*e*x)^(1/2)/(x^2-c/d)^2-1/6*d*e^2*x/c*(a*d+11 *b*c)/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/(-(x^2-c/d)*d*e*x)^(1/2)-7/6*( 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))*b*e^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c) -1/12*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)*EllipticF(((x+1/d*(c*d)^( 1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))/c*e^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/ (a*d-b*c)*a-9/8*e^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)*b/(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))*a-1/8*e^2/(a^2*d^2-2*a*b*c*d+b ^2*c^2)/(a*d-b*c)*b^2/(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))*c+9/8*e^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)*b/(a*b)^(1/2)*(...
Timed out. \[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int \frac {\left (e x\right )^{\frac {3}{2}}}{\left (- a + b x^{2}\right )^{2} \left (c - d x^{2}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int \frac {{\left (e\,x\right )}^{3/2}}{{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{5/2}} \,d x \]