Integrand size = 30, antiderivative size = 392 \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=-\frac {2 \sqrt {c-d x^2}}{a e \sqrt {e x}}-\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a e^{3/2} \sqrt {c-d x^2}}+\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{3/2} \sqrt {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{3/2} \sqrt {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}} \] Output:
-2*(-d*x^2+c)^(1/2)/a/e/(e*x)^(1/2)-2*c^(3/4)*d^(1/4)*(1-d*x^2/c)^(1/2)*El lipticE(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/a/e^(3/2)/(-d*x^2+c)^(1/2)+ 2*c^(3/4)*d^(1/4)*(1-d*x^2/c)^(1/2)*EllipticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/ e^(1/2),I)/a/e^(3/2)/(-d*x^2+c)^(1/2)-c^(1/4)*(-a*d+b*c)*(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)/a^(3/2)/b^(1/2)/d^(1/4)/e^(3/2)/(-d*x^2+c)^(1/2)+c^(1/4)*(-a*d+b *c)*(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)/a^(3/2)/b^(1/2)/d^(1/4)/e^(3/2)/(-d*x^2+c)^( 1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.09 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.36 \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\frac {x \left (-42 a \left (c-d x^2\right )+14 (b c-2 a d) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+6 b d x^4 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{21 a^2 (e x)^{3/2} \sqrt {c-d x^2}} \] Input:
Integrate[Sqrt[c - d*x^2]/((e*x)^(3/2)*(a - b*x^2)),x]
Output:
(x*(-42*a*(c - d*x^2) + 14*(b*c - 2*a*d)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[ 3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 6*b*d*x^4*Sqrt[1 - (d*x^2)/c]*Ap pellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(21*a^2*(e*x)^(3/2)*Sqrt [c - d*x^2])
Time = 0.71 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {368, 27, 975, 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.
| \(\displaystyle \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e \sqrt {c-d x^2}}{x \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \int \frac {\sqrt {c-d x^2}}{e x \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}\) |
| \(\Big \downarrow \) 975 |
| \(\displaystyle 2 e \left (\frac {\int \frac {x \left (b d x^2 e^2+(b c-2 a d) e^2\right )}{e \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a e^2}-\frac {\sqrt {c-d x^2}}{a e^2 \sqrt {e x}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \left (\frac {\int \frac {e x \left (b d x^2 e^2+(b c-2 a d) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a e^4}-\frac {\sqrt {c-d x^2}}{a e^2 \sqrt {e x}}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 e \left (\frac {\int \left (\frac {e \left (b c e^2-a d e^2\right ) x}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}-\frac {d e x}{\sqrt {c-d x^2}}\right )d\sqrt {e x}}{a e^4}-\frac {\sqrt {c-d x^2}}{a e^2 \sqrt {e x}}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 e \left (\frac {-\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}-\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {c-d x^2}}}{a e^4}-\frac {\sqrt {c-d x^2}}{a e^2 \sqrt {e x}}\right )\) |
Input:
Int[Sqrt[c - d*x^2]/((e*x)^(3/2)*(a - b*x^2)),x]
Output:
2*e*(-(Sqrt[c - d*x^2]/(a*e^2*Sqrt[e*x])) + (-((c^(3/4)*d^(1/4)*e^(3/2)*Sq rt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/Sqrt[c - d*x^2]) + (c^(3/4)*d^(1/4)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*Elli pticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/Sqrt[c - d*x^2] - (c^(1/4)*(b*c - a*d)*e^(3/2)*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*Sqrt[a]*Sqrt[b]*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[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*Sqrt[a]*Sqrt[b]*d^ (1/4)*Sqrt[c - d*x^2]))/(a*e^4))
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*(m + 1))), x] - Simp[1/(a*e^n*(m + 1)) Int[(e*x)^(m + n)*(a + b*x^n) ^p*(c + d*x^n)^(q - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + b*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomi alQ[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. \(818\) vs. \(2(292)=584\).
Time = 0.93 (sec) , antiderivative size = 819, normalized size of antiderivative = 2.09
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-x^{2} d +c \right ) e x}\, \left (-\frac {2 \left (-d e \,x^{2}+c e \right )}{e^{2} a \sqrt {x \left (-d e \,x^{2}+c e \right )}}+\frac {2 c \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 {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{e a \sqrt {-d e \,x^{3}+c e x}}-\frac {c \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 a \sqrt {-d e \,x^{3}+c e x}}+\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 e b \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 ) c}{2 a e 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 e b \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 ) c}{2 a e d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{\sqrt {e x}\, \sqrt {-x^{2} d +c}}\) | \(819\) |
| default | \(\text {Expression too large to display}\) | \(1263\) |
Input:
int((-d*x^2+c)^(1/2)/(e*x)^(3/2)/(-b*x^2+a),x,method=_RETURNVERBOSE)
Output:
((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)*(-2*(-d*e*x^2+c*e)/e^2 /a/(x*(-d*e*x^2+c*e))^(1/2)+2/e/a*c*(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)*EllipticE( ((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/e/a*c*(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))+1/2/e/b*(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/e/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))*El lipticPi(((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+1/2/e/b*(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/e/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*...
Timed out. \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((-d*x^2+c)^(1/2)/(e*x)^(3/2)/(-b*x^2+a),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=- \int \frac {\sqrt {c - d x^{2}}}{- a \left (e x\right )^{\frac {3}{2}} + b x^{2} \left (e x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((-d*x**2+c)**(1/2)/(e*x)**(3/2)/(-b*x**2+a),x)
Output:
-Integral(sqrt(c - d*x**2)/(-a*(e*x)**(3/2) + b*x**2*(e*x)**(3/2)), x)
\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\int { -\frac {\sqrt {-d x^{2} + c}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((-d*x^2+c)^(1/2)/(e*x)^(3/2)/(-b*x^2+a),x, algorithm="maxima")
Output:
-integrate(sqrt(-d*x^2 + c)/((b*x^2 - a)*(e*x)^(3/2)), x)
\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\int { -\frac {\sqrt {-d x^{2} + c}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((-d*x^2+c)^(1/2)/(e*x)^(3/2)/(-b*x^2+a),x, algorithm="giac")
Output:
integrate(-sqrt(-d*x^2 + c)/((b*x^2 - a)*(e*x)^(3/2)), x)
Timed out. \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\int \frac {\sqrt {c-d\,x^2}}{{\left (e\,x\right )}^{3/2}\,\left (a-b\,x^2\right )} \,d x \] Input:
int((c - d*x^2)^(1/2)/((e*x)^(3/2)*(a - b*x^2)),x)
Output:
int((c - d*x^2)^(1/2)/((e*x)^(3/2)*(a - b*x^2)), x)
\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {-d \,x^{2}+c}}{\sqrt {x}\, a x -\sqrt {x}\, b \,x^{3}}d x \right )}{e^{2}} \] Input:
int((-d*x^2+c)^(1/2)/(e*x)^(3/2)/(-b*x^2+a),x)
Output:
(sqrt(e)*int(sqrt(c - d*x**2)/(sqrt(x)*a*x - sqrt(x)*b*x**3),x))/e**2