Integrand size = 25, antiderivative size = 532 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {b (c-d x) \sqrt {c+d x}}{a \left (b c^2-a d^2\right ) x \sqrt {a-b x^2}}-\frac {\left (2 b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{a^2 c \left (b c^2-a d^2\right ) x}+\frac {\sqrt {b} \left (2 b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a^{3/2} c \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a^{3/2} \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a c \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
b*(-d*x+c)*(d*x+c)^(1/2)/a/(-a*d^2+b*c^2)/x/(-b*x^2+a)^(1/2)-(-a*d^2+2*b*c ^2)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a^2/c/(-a*d^2+b*c^2)/x+b^(1/2)*(-a*d^2+ 2*b*c^2)*(d*x+c)^(1/2)*(1-b*x^2/a)^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2 ))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(3/2)/ c/(-a*d^2+b*c^2)/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^ (1/2)-2*b^(1/2)*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*(1-b*x^2/a)^ (1/2)*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d /(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(3/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)+d*(b ^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticP i(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2,2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+ a^(1/2)*d))^(1/2))/a/c/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.09 (sec) , antiderivative size = 1066, normalized size of antiderivative = 2.00 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
Integrate[1/(x^2*Sqrt[c + d*x]*(a - b*x^2)^(3/2)),x]
Output:
(Sqrt[a - b*x^2]*(((c + d*x)*(-(1/(c*x)) + (b*(a*d - b*c*x))/((b*c^2 - a*d ^2)*(-a + b*x^2))))/a^2 + (2*b^2*c^5*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 3*a* b*c^3*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + a^2*c*d^4*Sqrt[-c + (Sqrt[a]*d) /Sqrt[b]] - 4*b^2*c^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + 2*a*b*c^2 *d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + 2*b^2*c^3*Sqrt[-c + (Sqrt[ a]*d)/Sqrt[b]]*(c + d*x)^2 - a*b*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - I*Sqrt[b]*c*(2*b^(3/2)*c^3 - 2*Sqrt[a]*b*c^2*d - a*Sqrt[b]*c*d^2 + a^(3/2)*d^3)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a] *d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - I*Sqrt[a]*d*(2*b^(3/2)*c^3 - a*Sqrt[b]*c*d^2 - a^(3/2)*d^ 3)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d )/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d) ] + I*a*b*c^2*d^2*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[ a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticPi[(Sqrt[b]*c)/(S qrt[b]*c - Sqrt[a]*d), I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d *x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - I*a^2*d^4*Sqrt[(d *(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticPi[(Sqrt[b]*c)/(Sqrt[b]*c - Sqrt[a]*d),...
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.
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 638 |
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}dx\) |
Input:
Int[1/(x^2*Sqrt[c + d*x]*(a - b*x^2)^(3/2)),x]
Output:
$Aborted
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Unintegrable[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x] /; FreeQ [{a, b, c, d, e, m, n, p}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(970\) vs. \(2(447)=894\).
Time = 6.84 (sec) , antiderivative size = 971, normalized size of antiderivative = 1.83
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 \left (-b d x -b c \right ) \left (-\frac {b c x}{2 \left (a \,d^{2}-b \,c^{2}\right ) a^{2}}+\frac {d}{2 \left (a \,d^{2}-b \,c^{2}\right ) a}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}-\frac {\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{a^{2} c x}+\frac {2 \left (\frac {b}{a^{2}}-\frac {b \,d^{2}}{2 a \left (a \,d^{2}-b \,c^{2}\right )}+\frac {b^{2} c^{2}}{\left (a \,d^{2}-b \,c^{2}\right ) a^{2}}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (\frac {c d \,b^{2}}{2 a^{2} \left (a \,d^{2}-b \,c^{2}\right )}-\frac {d b}{2 a^{2} c}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {d^{2} \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, -\frac {\left (-\frac {c}{d}+\frac {\sqrt {a b}}{b}\right ) d}{c}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{c^{2} a \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(971\) |
| risch | \(\text {Expression too large to display}\) | \(1022\) |
| default | \(\text {Expression too large to display}\) | \(1838\) |
Input:
int(1/x^2/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(-2*(-b*d*x-b*c) *(-1/2/(a*d^2-b*c^2)*b*c/a^2*x+1/2*d/(a*d^2-b*c^2)/a)/((x^2-a/b)*(-b*d*x-b *c))^(1/2)-1/a^2/c/x*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(b/a^2-1/2/a*b*d ^2/(a*d^2-b*c^2)+b^2*c^2/(a*d^2-b*c^2)/a^2)*(c/d-1/b*(a*b)^(1/2))*((x+c/d) /(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2))) ^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^ 2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+ 1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+2*(1/2*c*d*b^2/a^2/(a*d^2- b*c^2)-1/2/a^2*d*b/c)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)) )^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^( 1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((- c/d-1/b*(a*b)^(1/2))*EllipticE(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/ d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+1/b*(a*b)^(1/2)*Elliptic F(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b* (a*b)^(1/2)))^(1/2)))+1/c^2*d^2/a*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b* (a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x +1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a* c)^(1/2)*EllipticPi(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),-(-c/d+1/b*(a*b) ^(1/2))/c*d,((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)))
\[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x + c} x^{2}} \,d x } \] Input:
integrate(1/x^2/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(-b*x^2 + a)*sqrt(d*x + c)/(b^2*d*x^7 + b^2*c*x^6 - 2*a*b*d*x ^5 - 2*a*b*c*x^4 + a^2*d*x^3 + a^2*c*x^2), x)
\[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (a - b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x}}\, dx \] Input:
integrate(1/x**2/(d*x+c)**(1/2)/(-b*x**2+a)**(3/2),x)
Output:
Integral(1/(x**2*(a - b*x**2)**(3/2)*sqrt(c + d*x)), x)
\[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x + c} x^{2}} \,d x } \] Input:
integrate(1/x^2/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((-b*x^2 + a)^(3/2)*sqrt(d*x + c)*x^2), x)
\[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x + c} x^{2}} \,d x } \] Input:
integrate(1/x^2/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate(1/((-b*x^2 + a)^(3/2)*sqrt(d*x + c)*x^2), x)
Timed out. \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (a-b\,x^2\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \] Input:
int(1/(x^2*(a - b*x^2)^(3/2)*(c + d*x)^(1/2)),x)
Output:
int(1/(x^2*(a - b*x^2)^(3/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {-2 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}+2 \left (\int \frac {\sqrt {d x +c}}{\sqrt {-b \,x^{2}+a}\, a \,c^{2}-\sqrt {-b \,x^{2}+a}\, a \,d^{2} x^{2}-\sqrt {-b \,x^{2}+a}\, b \,c^{2} x^{2}+\sqrt {-b \,x^{2}+a}\, b \,d^{2} x^{4}}d x \right ) a b \,c^{2} x +\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{3}}{b^{2} d \,x^{5}+b^{2} c \,x^{4}-2 a b d \,x^{3}-2 a b c \,x^{2}+a^{2} d x +a^{2} c}d x \right ) b^{2} d x -\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{b^{2} d \,x^{6}+b^{2} c \,x^{5}-2 a b d \,x^{4}-2 a b c \,x^{3}+a^{2} d \,x^{2}+a^{2} c x}d x \right ) a^{2} d x -2 \left (\int \frac {\sqrt {d x +c}\, x}{\sqrt {-b \,x^{2}+a}\, a \,c^{2}-\sqrt {-b \,x^{2}+a}\, a \,d^{2} x^{2}-\sqrt {-b \,x^{2}+a}\, b \,c^{2} x^{2}+\sqrt {-b \,x^{2}+a}\, b \,d^{2} x^{4}}d x \right ) a b c d x}{2 a^{2} c x} \] Input:
int(1/x^2/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x)
Output:
( - 2*sqrt(c + d*x)*sqrt(a - b*x**2) + 2*int(sqrt(c + d*x)/(sqrt(a - b*x** 2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a*b*c**2*x + int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c*x* *4 + b**2*d*x**5),x)*b**2*d*x - int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a**2 *c*x + a**2*d*x**2 - 2*a*b*c*x**3 - 2*a*b*d*x**4 + b**2*c*x**5 + b**2*d*x* *6),x)*a**2*d*x - 2*int((sqrt(c + d*x)*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt( a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)* b*d**2*x**4),x)*a*b*c*d*x)/(2*a**2*c*x)