\(\int \frac {1}{\sqrt {c+d x} (a-b x^2)^{3/2}} \, dx\) [1558]

Optimal result

Integrand size = 22, antiderivative size = 328 \[ \int \frac {1}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=-\frac {(a d-b c x) \sqrt {c+d x}}{a \left (b c^2-a d^2\right ) \sqrt {a-b x^2}}+\frac {\sqrt {b} c \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.94 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {i \left (-i d \left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} x \sqrt {c+d x}+\sqrt {b} c \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^2 E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+\sqrt {a} d \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^2 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{a d \left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x) \sqrt {a-b x^2}} \] Input:

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

Output:

(I*((-I)*d*(Sqrt[b]*c + Sqrt[a]*d)*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*x*Sqrt[c 
 + d*x] + Sqrt[b]*c*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqr 
t[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^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)] + 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)^2*EllipticF[I*ArcSinh[ 
Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sq 
rt[b]*c - Sqrt[a]*d)]))/(a*d*(Sqrt[b]*c + Sqrt[a]*d)*Sqrt[-c + (Sqrt[a]*d) 
/Sqrt[b]]*(c + d*x)*Sqrt[a - b*x^2])
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {496, 27, 600, 509, 508, 327, 512, 511, 321}

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.

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

\(\Big \downarrow \) 496

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

\(\displaystyle -\frac {d \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} (a d-b c x)}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {d \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} (a d-b c x)}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\)

Input:

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

Output:

-(((a*d - b*c*x)*Sqrt[c + d*x])/(a*(b*c^2 - a*d^2)*Sqrt[a - b*x^2])) - (d* 
((-2*Sqrt[a]*Sqrt[b]*c*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[ 
Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/ 
(d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2 
*Sqrt[a]*(b*c^2 - a*d^2)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)] 
*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2 
]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b* 
x^2])))/(2*a*(b*c^2 - a*d^2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 496

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 
 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^n*(a 
 + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 
*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad 
raticQ[a, 0, b, c, d, n, p, x]
 

rule 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 509

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq 
rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], 
x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 512

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 
2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(667\) vs. \(2(270)=540\).

Time = 3.36 (sec) , antiderivative size = 668, normalized size of antiderivative = 2.04

method	result	size
default	\(\frac {\left (-\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) \sqrt {a b}\, a \,d^{3}+\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) \sqrt {a b}\, b \,c^{2} d +\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) a b c \,d^{2}-\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b^{2} c^{3}-b^{2} c \,d^{2} x^{2}+a b \,d^{3} x -b^{2} c^{2} d x +a b c \,d^{2}\right ) \sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}{d b a \left (a \,d^{2}-b \,c^{2}\right ) \left (-b d \,x^{3}-b c \,x^{2}+a d x +a c \right )}\)	\(668\)
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 \left (-b d x -b c \right ) \left (-\frac {c x}{2 \left (a \,d^{2}-b \,c^{2}\right ) a}+\frac {d}{2 \left (a \,d^{2}-b \,c^{2}\right ) b}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {2 \left (\frac {1}{a}-\frac {d^{2}}{2 \left (a \,d^{2}-b \,c^{2}\right )}+\frac {b \,c^{2}}{\left (a \,d^{2}-b \,c^{2}\right ) a}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {d b c \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{a \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(678\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=-\frac {{\left (a b c^{2} - 3 \, a^{2} d^{2} - {\left (b^{2} c^{2} - 3 \, a b d^{2}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) - 3 \, {\left (b^{2} c d x^{2} - a b c d\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (b^{2} c d x - a b d^{2}\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{2} c^{2} d - a^{3} b d^{3} - {\left (a b^{3} c^{2} d - a^{2} b^{2} d^{3}\right )} x^{2}\right )}} \] Input:

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

Output:

-1/3*((a*b*c^2 - 3*a^2*d^2 - (b^2*c^2 - 3*a*b*d^2)*x^2)*sqrt(-b*d)*weierst 
rassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d 
^3), 1/3*(3*d*x + c)/d) - 3*(b^2*c*d*x^2 - a*b*c*d)*sqrt(-b*d)*weierstrass 
Zeta(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), wei 
erstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/ 
(b*d^3), 1/3*(3*d*x + c)/d)) - 3*(b^2*c*d*x - a*b*d^2)*sqrt(-b*x^2 + a)*sq 
rt(d*x + c))/(a^2*b^2*c^2*d - a^3*b*d^3 - (a*b^3*c^2*d - a^2*b^2*d^3)*x^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(1/(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)
 

Giac [F]

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

integrate(1/(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)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int(1/(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 + 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)