Integrand size = 30, antiderivative size = 411 \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=-\frac {c e^3 (e x)^{5/2}}{d (b c-a d) \sqrt {c-d x^2}}-\frac {(5 b c-2 a d) e^5 \sqrt {e x} \sqrt {c-d x^2}}{3 b d^2 (b c-a d)}+\frac {\sqrt [4]{c} \left (5 b^2 c^2-2 a b c d-6 a^2 d^2\right ) e^{11/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{3 b^2 d^{9/4} (b c-a d) \sqrt {c-d x^2}}+\frac {a^2 \sqrt [4]{c} e^{11/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^2 \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {a^2 \sqrt [4]{c} e^{11/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^2 \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}} \] Output:
-c*e^3*(e*x)^(5/2)/d/(-a*d+b*c)/(-d*x^2+c)^(1/2)-1/3*(-2*a*d+5*b*c)*e^5*(e *x)^(1/2)*(-d*x^2+c)^(1/2)/b/d^2/(-a*d+b*c)+1/3*c^(1/4)*(-6*a^2*d^2-2*a*b* c*d+5*b^2*c^2)*e^(11/2)*(1-d*x^2/c)^(1/2)*EllipticF(d^(1/4)*(e*x)^(1/2)/c^ (1/4)/e^(1/2),I)/b^2/d^(9/4)/(-a*d+b*c)/(-d*x^2+c)^(1/2)+a^2*c^(1/4)*e^(11 /2)*(1-d*x^2/c)^(1/2)*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)/b^2/d^(1/4)/(-a*d+b*c)/(-d*x^2+c)^(1/2)+a^2 *c^(1/4)*e^(11/2)*(1-d*x^2/c)^(1/2)*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)/b^2/d^(1/4)/(-a*d+b*c)/(-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 = 199, normalized size of antiderivative = 0.48 \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=-\frac {e^5 \sqrt {e x} \left (5 a \left (2 a d \left (c-d x^2\right )+b c \left (-5 c+2 d x^2\right )\right )+5 a c (5 b c-2 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+\left (-5 b^2 c^2+2 a b c d+6 a^2 d^2\right ) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{15 a b d^2 (-b c+a d) \sqrt {c-d x^2}} \] Input:
Integrate[(e*x)^(11/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x]
Output:
-1/15*(e^5*Sqrt[e*x]*(5*a*(2*a*d*(c - d*x^2) + b*c*(-5*c + 2*d*x^2)) + 5*a *c*(5*b*c - 2*a*d)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/ c, (b*x^2)/a] + (-5*b^2*c^2 + 2*a*b*c*d + 6*a^2*d^2)*x^2*Sqrt[1 - (d*x^2)/ c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(a*b*d^2*(-(b*c) + a *d)*Sqrt[c - d*x^2])
Time = 0.87 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {368, 27, 970, 1052, 25, 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.
| \(\displaystyle \int \frac {(e x)^{11/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^8 x^6}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \int \frac {e^6 x^6}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}\) |
| \(\Big \downarrow \) 970 |
| \(\displaystyle 2 e \left (\frac {e^2 \int \frac {e^2 x^2 \left (5 a c e^2-(5 b c-2 a d) e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 d (b c-a d)}-\frac {c e^2 (e x)^{5/2}}{2 d \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1052 |
| \(\displaystyle 2 e \left (\frac {e^2 \left (-\frac {e^2 \int -\frac {a c (5 b c-2 a d) e^2-\left (5 b^2 c^2-2 a b d c-6 a^2 d^2\right ) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 b d}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (5 b c-2 a d)}{3 b d}\right )}{2 d (b c-a d)}-\frac {c e^2 (e x)^{5/2}}{2 d \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 e \left (\frac {e^2 \left (\frac {e^2 \int \frac {a c (5 b c-2 a d) e^2-\left (5 b^2 c^2-2 a b d c-6 a^2 d^2\right ) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 b d}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (5 b c-2 a d)}{3 b d}\right )}{2 d (b c-a d)}-\frac {c e^2 (e x)^{5/2}}{2 d \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle 2 e \left (\frac {e^2 \left (\frac {e^2 \left (\frac {6 a^3 d^2 e^2 \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}+\frac {\left (-6 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}}{b}\right )}{3 b d}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (5 b c-2 a d)}{3 b d}\right )}{2 d (b c-a d)}-\frac {c e^2 (e x)^{5/2}}{2 d \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle 2 e \left (\frac {e^2 \left (\frac {e^2 \left (\frac {6 a^3 d^2 e^2 \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}+\frac {\sqrt {1-\frac {d x^2}{c}} \left (-6 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{b \sqrt {c-d x^2}}\right )}{3 b d}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (5 b c-2 a d)}{3 b d}\right )}{2 d (b c-a d)}-\frac {c e^2 (e x)^{5/2}}{2 d \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 2 e \left (\frac {e^2 \left (\frac {e^2 \left (\frac {6 a^3 d^2 e^2 \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-6 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{3 b d}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (5 b c-2 a d)}{3 b d}\right )}{2 d (b c-a d)}-\frac {c e^2 (e x)^{5/2}}{2 d \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle 2 e \left (\frac {e^2 \left (\frac {e^2 \left (\frac {6 a^3 d^2 e^2 \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )}{b}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-6 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{3 b d}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (5 b c-2 a d)}{3 b d}\right )}{2 d (b c-a d)}-\frac {c e^2 (e x)^{5/2}}{2 d \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \left (\frac {e^2 \left (\frac {e^2 \left (\frac {6 a^3 d^2 e^2 \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )}{b}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-6 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{3 b d}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (5 b c-2 a d)}{3 b d}\right )}{2 d (b c-a d)}-\frac {c e^2 (e x)^{5/2}}{2 d \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle 2 e \left (\frac {e^2 \left (\frac {e^2 \left (\frac {6 a^3 d^2 e^2 \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )}{b}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-6 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{3 b d}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (5 b c-2 a d)}{3 b d}\right )}{2 d (b c-a d)}-\frac {c e^2 (e x)^{5/2}}{2 d \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle 2 e \left (\frac {e^2 \left (\frac {e^2 \left (\frac {6 a^3 d^2 e^2 \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )}{b}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-6 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{3 b d}-\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (5 b c-2 a d)}{3 b d}\right )}{2 d (b c-a d)}-\frac {c e^2 (e x)^{5/2}}{2 d \sqrt {c-d x^2} (b c-a d)}\right )\) |
Input:
Int[(e*x)^(11/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x]
Output:
2*e*(-1/2*(c*e^2*(e*x)^(5/2))/(d*(b*c - a*d)*Sqrt[c - d*x^2]) + (e^2*(-1/3 *((5*b*c - 2*a*d)*e^2*Sqrt[e*x]*Sqrt[c - d*x^2])/(b*d) + (e^2*((c^(1/4)*(5 *b^2*c^2 - 2*a*b*c*d - 6*a^2*d^2)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticF[Ar cSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(b*d^(1/4)*Sqrt[c - d*x^ 2]) + (6*a^3*d^2*e^2*((c^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*S qrt[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] *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])))/b))/(3* b*d)))/(2*d*(b*c - a*d)))
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[(-a)*e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n) ^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Simp[e^(2*n) /(b*n*(b*c - a*d)*(p + 1)) Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d *x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m - n + n*q + 1) + b*c*n*(p + 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] && GtQ[m - n + 1, n] && 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[((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]
Time = 2.73 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.56
| method | result | size |
| risch | \(-\frac {2 \sqrt {-x^{2} d +c}\, x \,e^{6}}{3 b \,d^{2} \sqrt {e x}}+\frac {\left (\frac {\left (3 a d +4 b c \right ) \sqrt {c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{b d \sqrt {-d e \,x^{3}+c e x}}-\frac {3 b \,c^{3} \left (-\frac {x}{c \sqrt {-\left (x^{2}-\frac {c}{d}\right ) d e x}}-\frac {\sqrt {c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{2 c d \sqrt {-d e \,x^{3}+c e x}}\right )}{a d -b c}+\frac {3 a^{3} d^{2} \left (\frac {\sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {\sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{b \left (a d -b c \right )}\right ) e^{6} \sqrt {\left (-x^{2} d +c \right ) e x}}{3 b \,d^{2} \sqrt {e x}\, \sqrt {-x^{2} d +c}}\) | \(643\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-x^{2} d +c \right ) e x}\, \left (\frac {e^{6} x \,c^{2}}{d^{2} \left (a d -b c \right ) \sqrt {-\left (x^{2}-\frac {c}{d}\right ) d e x}}-\frac {2 e^{5} \sqrt {-d e \,x^{3}+c e x}}{3 b \,d^{2}}+\frac {\sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) e^{6} a}{d^{2} \sqrt {-d e \,x^{3}+c e x}\, b^{2}}+\frac {4 \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) e^{6} c}{3 d^{3} \sqrt {-d e \,x^{3}+c e x}\, b}+\frac {\sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) c^{2} e^{6}}{2 d^{3} \sqrt {-d e \,x^{3}+c e x}\, \left (a d -b c \right )}+\frac {e^{6} a^{3} \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \left (a d -b c \right ) b^{2} \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {e^{6} a^{3} \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \left (a d -b c \right ) b^{2} \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-x^{2} d +c}}\) | \(717\) |
| default | \(-\frac {\left (6 \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a^{3} d^{3} \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}\, \sqrt {c d}-4 \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b c \,d^{2} \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}\, \sqrt {c d}-7 \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} c^{2} d \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}\, \sqrt {c d}+5 \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b^{3} c^{3} \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}\, \sqrt {c d}+3 \sqrt {2}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {a b}\, d +\sqrt {c d}\, b}, \frac {\sqrt {2}}{2}\right ) a^{3} b c \,d^{3} \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}-3 \sqrt {2}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {a b}\, d +\sqrt {c d}\, b}, \frac {\sqrt {2}}{2}\right ) a^{3} d^{3} \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}\, \sqrt {c d}-3 \sqrt {2}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) a^{3} b c \,d^{3} \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}-3 \sqrt {2}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) a^{3} d^{3} \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}\, \sqrt {c d}+4 a^{2} b \,d^{4} x^{3} \sqrt {a b}-8 a \,b^{2} c \,d^{3} x^{3} \sqrt {a b}+4 b^{3} c^{2} d^{2} x^{3} \sqrt {a b}-4 a^{2} b c \,d^{3} x \sqrt {a b}+14 a \,b^{2} c^{2} d^{2} x \sqrt {a b}-10 b^{3} c^{3} d x \sqrt {a b}\right ) e^{5} \sqrt {e x}}{6 b \,d^{2} x \sqrt {-x^{2} d +c}\, \left (\sqrt {c d}\, b -\sqrt {a b}\, d \right ) \left (\sqrt {a b}\, d +\sqrt {c d}\, b \right ) \sqrt {a b}\, \left (a d -b c \right )}\) | \(990\) |
Input:
int((e*x)^(11/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/3*(-d*x^2+c)^(1/2)*x/b/d^2*e^6/(e*x)^(1/2)+1/3/b/d^2*((3*a*d+4*b*c)/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)*(-x*d/(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))-3*b*c^3/( a*d-b*c)*(-x/c/(-(x^2-c/d)*d*e*x)^(1/2)-1/2/c/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)*( -x*d/(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)))+3/b*a^3*d^2/(a*d-b*c)*(1/2/(a*b)^(1 /2)/d*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(- x*d/(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))-1/2/(a*b)^(1/2)/d*(c*d) ^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(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))*Elli pticPi(((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))))*e^6*((-d*x^2+c)*e*x)^(1/2)/(e*x) ^(1/2)/(-d*x^2+c)^(1/2)
Timed out. \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x)^(11/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x)**(11/2)/(-b*x**2+a)/(-d*x**2+c)**(3/2),x)
Output:
Timed out
\[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int { -\frac {\left (e x\right )^{\frac {11}{2}}}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x)^(11/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x, algorithm="maxima")
Output:
-integrate((e*x)^(11/2)/((b*x^2 - a)*(-d*x^2 + c)^(3/2)), x)
\[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int { -\frac {\left (e x\right )^{\frac {11}{2}}}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x)^(11/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x, algorithm="giac")
Output:
integrate(-(e*x)^(11/2)/((b*x^2 - a)*(-d*x^2 + c)^(3/2)), x)
Timed out. \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{11/2}}{\left (a-b\,x^2\right )\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \] Input:
int((e*x)^(11/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x)
Output:
int((e*x)^(11/2)/((a - b*x^2)*(c - d*x^2)^(3/2)), x)
\[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int((e*x)^(11/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x)
Output:
(sqrt(e)*e**5*( - 10*sqrt(x)*sqrt(c - d*x**2)*a*c + 2*sqrt(x)*sqrt(c - d*x **2)*a*d*x**2 - 2*sqrt(x)*sqrt(c - d*x**2)*b*c*x**2 + 5*int(sqrt(c - d*x** 2)/(sqrt(x)*a**2*c**2*d - 2*sqrt(x)*a**2*c*d**2*x**2 + sqrt(x)*a**2*d**3*x **4 - sqrt(x)*a*b*c**3 + sqrt(x)*a*b*c**2*d*x**2 + sqrt(x)*a*b*c*d**2*x**4 - sqrt(x)*a*b*d**3*x**6 + sqrt(x)*b**2*c**3*x**2 - 2*sqrt(x)*b**2*c**2*d* x**4 + sqrt(x)*b**2*c*d**2*x**6),x)*a**3*c**3*d - 5*int(sqrt(c - d*x**2)/( sqrt(x)*a**2*c**2*d - 2*sqrt(x)*a**2*c*d**2*x**2 + sqrt(x)*a**2*d**3*x**4 - sqrt(x)*a*b*c**3 + sqrt(x)*a*b*c**2*d*x**2 + sqrt(x)*a*b*c*d**2*x**4 - s qrt(x)*a*b*d**3*x**6 + sqrt(x)*b**2*c**3*x**2 - 2*sqrt(x)*b**2*c**2*d*x**4 + sqrt(x)*b**2*c*d**2*x**6),x)*a**3*c**2*d**2*x**2 - 5*int(sqrt(c - d*x** 2)/(sqrt(x)*a**2*c**2*d - 2*sqrt(x)*a**2*c*d**2*x**2 + sqrt(x)*a**2*d**3*x **4 - sqrt(x)*a*b*c**3 + sqrt(x)*a*b*c**2*d*x**2 + sqrt(x)*a*b*c*d**2*x**4 - sqrt(x)*a*b*d**3*x**6 + sqrt(x)*b**2*c**3*x**2 - 2*sqrt(x)*b**2*c**2*d* x**4 + sqrt(x)*b**2*c*d**2*x**6),x)*a**2*b*c**4 + 5*int(sqrt(c - d*x**2)/( sqrt(x)*a**2*c**2*d - 2*sqrt(x)*a**2*c*d**2*x**2 + sqrt(x)*a**2*d**3*x**4 - sqrt(x)*a*b*c**3 + sqrt(x)*a*b*c**2*d*x**2 + sqrt(x)*a*b*c*d**2*x**4 - s qrt(x)*a*b*d**3*x**6 + sqrt(x)*b**2*c**3*x**2 - 2*sqrt(x)*b**2*c**2*d*x**4 + sqrt(x)*b**2*c*d**2*x**6),x)*a**2*b*c**3*d*x**2 + 3*int((sqrt(x)*sqrt(c - d*x**2)*x**3)/(a**2*c**2*d - 2*a**2*c*d**2*x**2 + a**2*d**3*x**4 - a*b* c**3 + a*b*c**2*d*x**2 + a*b*c*d**2*x**4 - a*b*d**3*x**6 + b**2*c**3*x*...