Integrand size = 30, antiderivative size = 413 \[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\frac {e (e x)^{3/2} \sqrt {c-d x^2}}{2 b \left (a-b x^2\right )}-\frac {5 c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 b^2 \sqrt {c-d x^2}}+\frac {5 c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 b^2 \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (3 b c-5 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (3 b c-5 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}} \]
1/2*e*(e*x)^(3/2)*(-d*x^2+c)^(1/2)/b/(-b*x^2+a)-5/2*c^(3/4)*d^(1/4)*e^(5/2 )*EllipticE(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)*(1-d*x^2/c)^(1/2)/b^2/( -d*x^2+c)^(1/2)+5/2*c^(3/4)*d^(1/4)*e^(5/2)*EllipticF(d^(1/4)*(e*x)^(1/2)/ c^(1/4)/e^(1/2),I)*(1-d*x^2/c)^(1/2)/b^2/(-d*x^2+c)^(1/2)+1/4*c^(1/4)*(-5* a*d+3*b*c)*e^(5/2)*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^(5/2)/d^(1/4)/a^(1/2)/(-d* x^2+c)^(1/2)-1/4*c^(1/4)*(-5*a*d+3*b*c)*e^(5/2)*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^(5/2)/d^(1/4)/a^(1/2)/(-d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.14 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.39 \[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\frac {e (e x)^{3/2} \left (-7 a \left (c-d x^2\right )+7 c \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+5 d x^2 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{14 a b \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
(e*(e*x)^(3/2)*(-7*a*(c - d*x^2) + 7*c*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*App ellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 5*d*x^2*(-a + b*x^2)*Sqrt[ 1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(14*a*b *(-a + b*x^2)*Sqrt[c - d*x^2])
Time = 0.74 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {368, 27, 967, 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.
| \(\displaystyle \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^7 x^3 \sqrt {c-d x^2}}{\left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \int \frac {e^3 x^3 \sqrt {c-d x^2}}{\left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\) |
| \(\Big \downarrow \) 967 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{3/2} \sqrt {c-d x^2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\int \frac {x \left (3 c e^2-5 d e^2 x^2\right )}{e \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 b}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{3/2} \sqrt {c-d x^2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\int \frac {e x \left (3 c e^2-5 d e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{3/2} \sqrt {c-d x^2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\int \left (\frac {5 d e x}{b \sqrt {c-d x^2}}+\frac {e \left (3 b c e^2-5 a d e^2\right ) x}{b \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}\right )d\sqrt {e x}}{4 b e^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 e^3 \left (\frac {(e x)^{3/2} \sqrt {c-d x^2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {-\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (3 b c-5 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (3 b c-5 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {5 c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}+\frac {5 c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b \sqrt {c-d x^2}}}{4 b e^2}\right )\) |
2*e^3*(((e*x)^(3/2)*Sqrt[c - d*x^2])/(4*b*(a*e^2 - b*e^2*x^2)) - ((5*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]) - (5*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)*Sqrt[e ])], -1])/(b*Sqrt[c - d*x^2]) - (c^(1/4)*(3*b*c - 5*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)*(3*b*c - 5*a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticP i[(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]))/(4*b*e^2))
3.9.97.3.1 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^(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[(((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. \(825\) vs. \(2(303)=606\).
Time = 3.16 (sec) , antiderivative size = 826, normalized size of antiderivative = 2.00
| method | result | size |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (\frac {e^{2} x \sqrt {-d e \,x^{3}+c e x}}{2 b \left (-b \,x^{2}+a \right )}+\frac {5 e^{3} 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 b^{2} \sqrt {-d e \,x^{3}+c e x}}-\frac {5 e^{3} 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 b^{2} \sqrt {-d e \,x^{3}+c e x}}-\frac {5 e^{3} \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 ) a}{8 b^{3} \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {3 e^{3} \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 b^{2} d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {5 e^{3} \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 ) a}{8 b^{3} \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}+\frac {3 e^{3} \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 b^{2} 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}}\) | \(826\) |
| default | \(\text {Expression too large to display}\) | \(2530\) |
1/e/x*(e*x)^(1/2)/(-d*x^2+c)^(1/2)*((-d*x^2+c)*e*x)^(1/2)*(1/2/b*e^2*x*(-d *e*x^3+c*e*x)^(1/2)/(-b*x^2+a)+5/2*e^3/b^2*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)* EllipticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-5/4*e^3/b ^2*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))-5/8*e^3/b^3*(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+3/8*e^3/b^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-5 /8*e^3/b^3*(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+3/8*e^3/b^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/...
Timed out. \[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\int \frac {\left (e x\right )^{\frac {5}{2}} \sqrt {c - d x^{2}}}{\left (- a + b x^{2}\right )^{2}}\, dx \]
\[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {\sqrt {-d x^{2} + c} \left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \]
\[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {\sqrt {-d x^{2} + c} \left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}\,\sqrt {c-d\,x^2}}{{\left (a-b\,x^2\right )}^2} \,d x \]