\(\int \frac {1}{x \sqrt {c+d x} (a-b x^2)^{3/2}} \, dx\) [1559]

Optimal result

Integrand size = 25, antiderivative size = 321 \[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {b (c-d x) \sqrt {c+d x}}{a \left (b c^2-a d^2\right ) \sqrt {a-b x^2}}-\frac {\sqrt {b} d \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

b*(-d*x+c)*(d*x+c)^(1/2)/a/(-a*d^2+b*c^2)/(-b*x^2+a)^(1/2)-b^(1/2)*d*(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^(1/2)/(-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) 
*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticPi(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/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.18 (sec) , antiderivative size = 872, normalized size of antiderivative = 2.72 \[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {b c^3 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}-a c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}-2 b c^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)+b c \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)^2+b c \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (c^2-d^2 x^2\right )-i \sqrt {b} c \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \left (b c^2+\sqrt {a} \sqrt {b} c d-2 a d^2\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+2 i b c^2 \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticPi}\left (\frac {\sqrt {b} c}{\sqrt {b} c-\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-2 i a d^2 \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticPi}\left (\frac {\sqrt {b} c}{\sqrt {b} c-\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{a c \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Input:

Integrate[1/(x*Sqrt[c + d*x]*(a - b*x^2)^(3/2)),x]
 

Output:

(b*c^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - a*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt 
[b]] - 2*b*c^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + b*c*Sqrt[-c + (S 
qrt[a]*d)/Sqrt[b]]*(c + d*x)^2 + b*c*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c^2 - 
 d^2*x^2) - I*Sqrt[b]*c*(Sqrt[b]*c - Sqrt[a]*d)*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]], (S 
qrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - I*(b*c^2 + Sqrt[a]*Sqrt[b 
]*c*d - 2*a*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)*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)] + (2*I)*b*c^2*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sq 
rt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticPi[(S 
qrt[b]*c)/(Sqrt[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)] - (2*I) 
*a*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)/(Sqrt[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)])/(a*c*Sqrt[-c + (Sqrt[a]*d)/Sq 
rt[b]]*(b*c^2 - a*d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2])
 

Rubi [F]

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.

\(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 638

\(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}dx\)

Input:

Int[1/(x*Sqrt[c + d*x]*(a - b*x^2)^(3/2)),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 638

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(886\) vs. \(2(267)=534\).

Time = 4.72 (sec) , antiderivative size = 887, normalized size of antiderivative = 2.76

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 {d x}{2 \left (a \,d^{2}-b \,c^{2}\right ) a}-\frac {c}{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 {d b c \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 )}{a \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {b \,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}}}\, \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 )}{a \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {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}}}\, d \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 )}{a \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}\, c}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(887\)
default	\(\text {Expression too large to display}\)	\(1287\)

Input:

int(1/x/(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*d/(a*d^2-b*c^2)/a*x-1/2*c/(a*d^2-b*c^2)/a)/((x^2-a/b)*(-b*d*x-b*c))^ 
(1/2)-d*b*c/a/(a*d^2-b*c^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))-1/a*b*d^2/(a*d^2-b*c^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)*((-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)*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/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)/c*d*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)))
 

Fricas [F]

\[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x + c} x} \,d x } \] Input:

integrate(1/x/(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^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), x)
 

Sympy [F]

\[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {1}{x \left (a - b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x}}\, dx \] Input:

integrate(1/x/(d*x+c)**(1/2)/(-b*x**2+a)**(3/2),x)
 

Output:

Integral(1/(x*(a - b*x**2)**(3/2)*sqrt(c + d*x)), x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x + c} x} \,d x } \] Input:

integrate(1/x/(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), x)
 

Giac [F]

\[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x + c} x} \,d x } \] Input:

integrate(1/x/(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), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {1}{x\,{\left (a-b\,x^2\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \] Input:

int(1/(x*(a - b*x^2)^(3/2)*(c + d*x)^(1/2)),x)
 

Output:

int(1/(x*(a - b*x^2)^(3/2)*(c + d*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{b^{2} d \,x^{6}+b^{2} c \,x^{5}-2 a b d \,x^{4}-2 a b c \,x^{3}+a^{2} d \,x^{2}+a^{2} c x}d x \] Input:

int(1/x/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x)
 

Output:

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)