Integrand size = 30, antiderivative size = 362 \[ \int \frac {(e x)^{7/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\frac {7 e^3 \sqrt {e x} \sqrt {c-d x^2}}{6 b^2}+\frac {e (e x)^{5/2} \sqrt {c-d x^2}}{2 b \left (a-b x^2\right )}+\frac {\sqrt [4]{c} (8 b c-21 a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 b^3 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-7 a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^3 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-7 a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^3 \sqrt [4]{d} \sqrt {c-d x^2}} \] Output:
7/6*e^3*(e*x)^(1/2)*(-d*x^2+c)^(1/2)/b^2+1/2*e*(e*x)^(5/2)*(-d*x^2+c)^(1/2 )/b/(-b*x^2+a)+1/6*c^(1/4)*(-21*a*d+8*b*c)*e^(7/2)*(1-d*x^2/c)^(1/2)*Ellip ticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/b^3/d^(1/4)/(-d*x^2+c)^(1/2)-1 /4*c^(1/4)*(-7*a*d+5*b*c)*e^(7/2)*(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^3/d^(1/4)/( -d*x^2+c)^(1/2)-1/4*c^(1/4)*(-7*a*d+5*b*c)*e^(7/2)*(1-d*x^2/c)^(1/2)*Ellip ticPi(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^3/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 = 11.15 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.51 \[ \int \frac {(e x)^{7/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\frac {e^3 \sqrt {e x} \left (5 a \left (7 a-4 b x^2\right ) \left (-c+d x^2\right )+35 a c \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )-(-8 b c+21 a d) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{30 a b^2 \left (-a+b x^2\right ) \sqrt {c-d x^2}} \] Input:
Integrate[((e*x)^(7/2)*Sqrt[c - d*x^2])/(a - b*x^2)^2,x]
Output:
(e^3*Sqrt[e*x]*(5*a*(7*a - 4*b*x^2)*(-c + d*x^2) + 35*a*c*(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] - (-8*b*c + 21*a*d)*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]))/(30*a*b^2*(-a + b*x^2)*Sqrt[c - d*x^2])
Time = 0.82 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.09, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {368, 27, 967, 27, 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.
| \(\displaystyle \int \frac {(e x)^{7/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^8 x^4 \sqrt {c-d x^2}}{\left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \int \frac {e^4 x^4 \sqrt {c-d x^2}}{\left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\) |
| \(\Big \downarrow \) 967 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \sqrt {c-d x^2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\int \frac {x^2 \left (5 c e^2-7 d e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 b}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \sqrt {c-d x^2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\int \frac {e^2 x^2 \left (5 c e^2-7 d e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 1052 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \sqrt {c-d x^2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {-\frac {e^2 \int -\frac {d \left ((8 b c-21 a d) x^2 e^2+7 a c e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 b d}-\frac {7 e^2 \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \sqrt {c-d x^2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \int \frac {d \left ((8 b c-21 a d) x^2 e^2+7 a c e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 b d}-\frac {7 e^2 \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \sqrt {c-d x^2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \int \frac {(8 b c-21 a d) x^2 e^2+7 a c e^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 b}-\frac {7 e^2 \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \sqrt {c-d x^2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {3 a e^2 (5 b c-7 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}-\frac {(8 b c-21 a d) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}}{b}\right )}{3 b}-\frac {7 e^2 \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \sqrt {c-d x^2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {3 a e^2 (5 b c-7 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}-\frac {\sqrt {1-\frac {d x^2}{c}} (8 b c-21 a d) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{b \sqrt {c-d x^2}}\right )}{3 b}-\frac {7 e^2 \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \sqrt {c-d x^2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {3 a e^2 (5 b c-7 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (8 b c-21 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{3 b}-\frac {7 e^2 \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \sqrt {c-d x^2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {3 a e^2 (5 b c-7 a d) \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )}{b}-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (8 b c-21 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{3 b}-\frac {7 e^2 \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \sqrt {c-d x^2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {3 a e^2 (5 b c-7 a d) \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )}{b}-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (8 b c-21 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{3 b}-\frac {7 e^2 \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \sqrt {c-d x^2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {3 a e^2 (5 b c-7 a d) \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )}{b}-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (8 b c-21 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{3 b}-\frac {7 e^2 \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{5/2} \sqrt {c-d x^2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {3 a e^2 (5 b c-7 a d) \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )}{b}-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (8 b c-21 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{3 b}-\frac {7 e^2 \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\) |
Input:
Int[((e*x)^(7/2)*Sqrt[c - d*x^2])/(a - b*x^2)^2,x]
Output:
2*e^3*(((e*x)^(5/2)*Sqrt[c - d*x^2])/(4*b*(a*e^2 - b*e^2*x^2)) - ((-7*e^2* Sqrt[e*x]*Sqrt[c - d*x^2])/(3*b) + (e^2*(-((c^(1/4)*(8*b*c - 21*a*d)*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])) + (3*a*(5*b*c - 7*a*d)*e^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)*S qrt[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))/(4*b*e^2))
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^(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. \(886\) vs. \(2(274)=548\).
Time = 2.74 (sec) , antiderivative size = 887, normalized size of antiderivative = 2.45
| method | result | size |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-x^{2} d +c \right ) e x}\, \left (\frac {a \,e^{3} \sqrt {-d e \,x^{3}+c e x}}{2 b^{2} \left (-b \,x^{2}+a \right )}+\frac {2 e^{3} \sqrt {-d e \,x^{3}+c e x}}{3 b^{2}}-\frac {7 \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 ) a \,e^{4}}{4 \sqrt {-d e \,x^{3}+c e x}\, b^{3}}+\frac {2 \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^{4} c}{3 d \sqrt {-d e \,x^{3}+c e x}\, b^{2}}-\frac {7 a^{2} e^{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 {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 )}{8 b^{3} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {5 a \,e^{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 {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 ) c}{8 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 {7 a^{2} e^{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 {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 )}{8 b^{3} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}-\frac {5 a \,e^{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 {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 ) c}{8 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}}\) | \(887\) |
| risch | \(\text {Expression too large to display}\) | \(1271\) |
| default | \(\text {Expression too large to display}\) | \(2549\) |
Input:
int((e*x)^(7/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
1/e/x*(e*x)^(1/2)/(-d*x^2+c)^(1/2)*((-d*x^2+c)*e*x)^(1/2)*(1/2*a/b^2*e^3*( -d*e*x^3+c*e*x)^(1/2)/(-b*x^2+a)+2/3/b^2*e^3*(-d*e*x^3+c*e*x)^(1/2)-7/4*(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)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c* d)^(1/2))^(1/2),1/2*2^(1/2))*a*e^4/b^3+2/3/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)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2) )/b^2*e^4*c-7/8*a^2*e^4/b^3/(a*b)^(1/2)*(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))+5/8*a*e^4/b^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))*c+7/8*a^2*e^4/b^3/(a*b)^(1/2)*(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))-5/8*a*e^4/b^2/(a*b)^(1/2)/d*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^...
Timed out. \[ \int \frac {(e x)^{7/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x)^(7/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a)^2,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {(e x)^{7/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x)**(7/2)*(-d*x**2+c)**(1/2)/(-b*x**2+a)**2,x)
Output:
Timed out
\[ \int \frac {(e x)^{7/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {\sqrt {-d x^{2} + c} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \] Input:
integrate((e*x)^(7/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a)^2,x, algorithm="maxima")
Output:
integrate(sqrt(-d*x^2 + c)*(e*x)^(7/2)/(b*x^2 - a)^2, x)
\[ \int \frac {(e x)^{7/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {\sqrt {-d x^{2} + c} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \] Input:
integrate((e*x)^(7/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a)^2,x, algorithm="giac")
Output:
integrate(sqrt(-d*x^2 + c)*(e*x)^(7/2)/(b*x^2 - a)^2, x)
Timed out. \[ \int \frac {(e x)^{7/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}\,\sqrt {c-d\,x^2}}{{\left (a-b\,x^2\right )}^2} \,d x \] Input:
int(((e*x)^(7/2)*(c - d*x^2)^(1/2))/(a - b*x^2)^2,x)
Output:
int(((e*x)^(7/2)*(c - d*x^2)^(1/2))/(a - b*x^2)^2, x)
\[ \int \frac {(e x)^{7/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\text {too large to display} \] Input:
int((e*x)^(7/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a)^2,x)
Output:
(sqrt(e)*e**3*( - 10*sqrt(x)*sqrt(c - d*x**2)*a*c - 6*sqrt(x)*sqrt(c - d*x **2)*a*d*x**2 + 6*sqrt(x)*sqrt(c - d*x**2)*b*c*x**2 + 5*int(sqrt(c - d*x** 2)/(sqrt(x)*a**3*c*d - sqrt(x)*a**3*d**2*x**2 - sqrt(x)*a**2*b*c**2 - sqrt (x)*a**2*b*c*d*x**2 + 2*sqrt(x)*a**2*b*d**2*x**4 + 2*sqrt(x)*a*b**2*c**2*x **2 - sqrt(x)*a*b**2*c*d*x**4 - sqrt(x)*a*b**2*d**2*x**6 - sqrt(x)*b**3*c* *2*x**4 + sqrt(x)*b**3*c*d*x**6),x)*a**4*c**2*d - 5*int(sqrt(c - d*x**2)/( sqrt(x)*a**3*c*d - sqrt(x)*a**3*d**2*x**2 - sqrt(x)*a**2*b*c**2 - sqrt(x)* a**2*b*c*d*x**2 + 2*sqrt(x)*a**2*b*d**2*x**4 + 2*sqrt(x)*a*b**2*c**2*x**2 - sqrt(x)*a*b**2*c*d*x**4 - sqrt(x)*a*b**2*d**2*x**6 - sqrt(x)*b**3*c**2*x **4 + sqrt(x)*b**3*c*d*x**6),x)*a**3*b*c**3 - 5*int(sqrt(c - d*x**2)/(sqrt (x)*a**3*c*d - sqrt(x)*a**3*d**2*x**2 - sqrt(x)*a**2*b*c**2 - sqrt(x)*a**2 *b*c*d*x**2 + 2*sqrt(x)*a**2*b*d**2*x**4 + 2*sqrt(x)*a*b**2*c**2*x**2 - sq rt(x)*a*b**2*c*d*x**4 - sqrt(x)*a*b**2*d**2*x**6 - sqrt(x)*b**3*c**2*x**4 + sqrt(x)*b**3*c*d*x**6),x)*a**3*b*c**2*d*x**2 + 5*int(sqrt(c - d*x**2)/(s qrt(x)*a**3*c*d - sqrt(x)*a**3*d**2*x**2 - sqrt(x)*a**2*b*c**2 - sqrt(x)*a **2*b*c*d*x**2 + 2*sqrt(x)*a**2*b*d**2*x**4 + 2*sqrt(x)*a*b**2*c**2*x**2 - sqrt(x)*a*b**2*c*d*x**4 - sqrt(x)*a*b**2*d**2*x**6 - sqrt(x)*b**3*c**2*x* *4 + sqrt(x)*b**3*c*d*x**6),x)*a**2*b**2*c**3*x**2 - 21*int((sqrt(x)*sqrt( c - d*x**2)*x**3)/(a**3*c*d - a**3*d**2*x**2 - a**2*b*c**2 - a**2*b*c*d*x* *2 + 2*a**2*b*d**2*x**4 + 2*a*b**2*c**2*x**2 - a*b**2*c*d*x**4 - a*b**2...