Integrand size = 30, antiderivative size = 417 \[ \int \frac {\sqrt {e x} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\frac {(e x)^{3/2} \sqrt {c-d x^2}}{2 a e \left (a-b x^2\right )}-\frac {c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a b \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a b \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c+a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c+a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}} \]
1/2*(e*x)^(3/2)*(-d*x^2+c)^(1/2)/a/e/(-b*x^2+a)-1/2*c^(3/4)*d^(1/4)*Ellipt icE(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)*e^(1/2)*(1-d*x^2/c)^(1/2)/a/b/( -d*x^2+c)^(1/2)+1/2*c^(3/4)*d^(1/4)*EllipticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/ e^(1/2),I)*e^(1/2)*(1-d*x^2/c)^(1/2)/a/b/(-d*x^2+c)^(1/2)-1/4*c^(1/4)*(a*d +b*c)*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)*e^(1/2)*(1-d*x^2/c)^(1/2)/a^(3/2)/b^(3/2)/d^(1/4)/(-d*x^2+c )^(1/2)+1/4*c^(1/4)*(a*d+b*c)*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)*e^(1/2)*(1-d*x^2/c)^(1/2)/a^(3/2)/b^ (3/2)/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 = 11.12 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {e x} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\frac {\sqrt {e x} \left (21 a x \left (-c+d x^2\right )+7 c x \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 d x^3 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a^2 \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
(Sqrt[e*x]*(21*a*x*(-c + d*x^2) + 7*c*x*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*A ppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*d*x^3*(-a + b*x^2)*Sqr t[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(42*a ^2*(-a + b*x^2)*Sqrt[c - d*x^2])
Time = 0.75 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {368, 27, 969, 25, 27, 1054, 2009}
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.
| \(\displaystyle \int \frac {\sqrt {e x} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^5 x \sqrt {c-d x^2}}{\left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \int \frac {e x \sqrt {c-d x^2}}{\left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\) |
| \(\Big \downarrow \) 969 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{3/2} \sqrt {c-d x^2}}{4 a e^2 \left (a e^2-b e^2 x^2\right )}-\frac {\int -\frac {x \left (d x^2 e^2+c e^2\right )}{e \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 e^3 \left (\frac {\int \frac {x \left (d x^2 e^2+c e^2\right )}{e \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^2}+\frac {(e x)^{3/2} \sqrt {c-d x^2}}{4 a e^2 \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\int \frac {e x \left (d x^2 e^2+c e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^4}+\frac {(e x)^{3/2} \sqrt {c-d x^2}}{4 a e^2 \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 e^3 \left (\frac {\int \left (\frac {e \left (b c e^2+a d e^2\right ) x}{b \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}-\frac {d e x}{b \sqrt {c-d x^2}}\right )d\sqrt {e x}}{4 a e^4}+\frac {(e x)^{3/2} \sqrt {c-d x^2}}{4 a e^2 \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 e^3 \left (\frac {-\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (a d+b c) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (a d+b c) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}-\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b \sqrt {c-d x^2}}}{4 a e^4}+\frac {(e x)^{3/2} \sqrt {c-d x^2}}{4 a e^2 \left (a e^2-b e^2 x^2\right )}\right )\) |
2*e^3*(((e*x)^(3/2)*Sqrt[c - d*x^2])/(4*a*e^2*(a*e^2 - b*e^2*x^2)) + (-((c ^(3/4)*d^(1/4)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[ e*x])/(c^(1/4)*Sqrt[e])], -1])/(b*Sqrt[c - d*x^2])) + (c^(3/4)*d^(1/4)*e^( 3/2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqr t[e])], -1])/(b*Sqrt[c - d*x^2]) - (c^(1/4)*(b*c + a*d)*e^(3/2)*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*Sqrt[a]*b^(3/2)*d^(1/4)*Sqrt[c - d*x^2]) + (c^(1/4)*(b*c + a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sq rt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt [e])], -1])/(2*Sqrt[a]*b^(3/2)*d^(1/4)*Sqrt[c - d*x^2]))/(4*a*e^4))
3.9.99.3.1 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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[(-(e*x)^(m + 1))*(a + b*x^n)^(p + 1)*((c + d*x^n )^q/(a*e*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) Int[(e*x)^m*(a + b*x^n)^( p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m + n*(p + 1) + 1) + d*(m + n*(p + q + 1 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(820\) vs. \(2(307)=614\).
Time = 3.06 (sec) , antiderivative size = 821, normalized size of antiderivative = 1.97
| method | result | size |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (\frac {x \sqrt {-d e \,x^{3}+c e x}}{2 a \left (-b \,x^{2}+a \right )}+\frac {e c \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{2 a b \sqrt {-d e \,x^{3}+c e x}}-\frac {e c \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{4 a b \sqrt {-d e \,x^{3}+c e x}}-\frac {e \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \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^{2} \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {e \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \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 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 \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \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^{2} \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}-\frac {e \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \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 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 {-d \,x^{2}+c}}\) | \(821\) |
| default | \(\text {Expression too large to display}\) | \(2522\) |
1/e/x*(e*x)^(1/2)/(-d*x^2+c)^(1/2)*((-d*x^2+c)*e*x)^(1/2)*(1/2*x/a*(-d*e*x ^3+c*e*x)^(1/2)/(-b*x^2+a)+1/2*e/a/b*c*(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)*Ellipt icE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/4*e/a/b*c*(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))-1/8*e/b^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))-1 /8*e/a/b/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-1/8*e/b^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))*Ell ipticPi(((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))-1/8*e/a/b/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...
Timed out. \[ \int \frac {\sqrt {e x} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {e x} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\int \frac {\sqrt {e x} \sqrt {c - d x^{2}}}{\left (- a + b x^{2}\right )^{2}}\, dx \]
\[ \int \frac {\sqrt {e x} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {\sqrt {-d x^{2} + c} \sqrt {e x}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \]
\[ \int \frac {\sqrt {e x} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {\sqrt {-d x^{2} + c} \sqrt {e x}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {e x} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\int \frac {\sqrt {e\,x}\,\sqrt {c-d\,x^2}}{{\left (a-b\,x^2\right )}^2} \,d x \]