\(\int \frac {(a-b x^2)^{3/2}}{x \sqrt {c+d x}} \, dx\) [1482]

Optimal result

Integrand size = 25, antiderivative size = 489 \[ \int \frac {\left (a-b x^2\right )^{3/2}}{x \sqrt {c+d x}} \, dx=\frac {8 b c \sqrt {c+d x} \sqrt {a-b x^2}}{15 d^2}-\frac {2 b x \sqrt {c+d x} \sqrt {a-b x^2}}{5 d}-\frac {2 \sqrt {a} \sqrt {b} \left (8 b c^2-21 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 d^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} c \left (8 b c^2-23 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 d^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 a^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

8/15*b*c*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/d^2-2/5*b*x*(d*x+c)^(1/2)*(-b*x^2+ 
a)^(1/2)/d-2/15*a^(1/2)*b^(1/2)*(-21*a*d^2+8*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))/d^3/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2) 
*d))^(1/2)/(-b*x^2+a)^(1/2)+2/15*a^(1/2)*b^(1/2)*c*(-23*a*d^2+8*b*c^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))/d^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)-2*a^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))/ 
(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.28 (sec) , antiderivative size = 860, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a-b x^2\right )^{3/2}}{x \sqrt {c+d x}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {2 b (4 c-3 d x) (c+d x)}{d^2}-\frac {2 \left (8 b^2 c^5 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}-29 a b c^3 d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}+21 a^2 c d^4 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}-16 b^2 c^4 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)+42 a b c^2 d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)+8 b^2 c^3 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)^2-21 a b c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)^2-i \sqrt {b} c \left (8 b^{3/2} c^3-8 \sqrt {a} b c^2 d-21 a \sqrt {b} c d^2+21 a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \sqrt {a} d \left (8 b^{3/2} c^3-2 \sqrt {a} b c^2 d-21 a \sqrt {b} c d^2+15 a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+15 i a^2 d^4 \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticPi}\left (\frac {\sqrt {b} c}{\sqrt {b} c-\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{c d^4 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{15 \sqrt {c+d x}} \] Input:

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

Output:

(Sqrt[a - b*x^2]*((2*b*(4*c - 3*d*x)*(c + d*x))/d^2 - (2*(8*b^2*c^5*Sqrt[- 
c + (Sqrt[a]*d)/Sqrt[b]] - 29*a*b*c^3*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 
 21*a^2*c*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 16*b^2*c^4*Sqrt[-c + (Sqrt[ 
a]*d)/Sqrt[b]]*(c + d*x) + 42*a*b*c^2*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*( 
c + d*x) + 8*b^2*c^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - 21*a*b*c 
*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - I*Sqrt[b]*c*(8*b^(3/2)*c 
^3 - 8*Sqrt[a]*b*c^2*d - 21*a*Sqrt[b]*c*d^2 + 21*a^(3/2)*d^3)*Sqrt[(d*(Sqr 
t[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 
*(8*b^(3/2)*c^3 - 2*Sqrt[a]*b*c^2*d - 21*a*Sqrt[b]*c*d^2 + 15*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)] 
+ (15*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)/(Sq 
rt[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)]))/(c*d^4*Sqrt[-c + ( 
Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(15*Sqrt[c + d*x])
 

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.

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

\(\Big \downarrow \) 638

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

Input:

Int[(a - b*x^2)^(3/2)/(x*Sqrt[c + d*x]),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. \(852\) vs. \(2(396)=792\).

Time = 1.26 (sec) , antiderivative size = 853, normalized size of antiderivative = 1.74

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 b x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{5 d}+\frac {8 b c \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{15 d^{2}}+\frac {4 a 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 )}{15 d \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (-\frac {7 a b}{5}+\frac {8 b^{2} c^{2}}{15 d^{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}}}\, \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 {2 a^{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 )}{\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}}\)	\(853\)
default	\(\text {Expression too large to display}\)	\(1461\)

Input:

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

Output:

((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(-2/5*b/d*x*(-b* 
d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+8/15*b*c/d^2*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^( 
1/2)+4/15*a*b*c/d*(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)*Ellipti 
cF(((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*(-7/5*a*b+8/15*b^2*c^2/d^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^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*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(-1)]

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

integrate((-b*x^2+a)^(3/2)/x/(d*x+c)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((-b*x^2+a)^(3/2)/x/(d*x+c)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((-b*x^2+a)^(3/2)/x/(d*x+c)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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