\(\int \frac {1}{\sqrt {c+d x} (a-b x^2)} \, dx\) [588]

Optimal result

Integrand size = 20, antiderivative size = 134 \[ \int \frac {1}{\sqrt {c+d x} \left (a-b x^2\right )} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{\sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{\sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {a} d}} \] Output:

-arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/a^(1/2)/b^(1/4 
)/(b^(1/2)*c-a^(1/2)*d)^(1/2)+arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+a^( 
1/2)*d)^(1/2))/a^(1/2)/b^(1/4)/(b^(1/2)*c+a^(1/2)*d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\sqrt {c+d x} \left (a-b x^2\right )} \, dx=\frac {\frac {\arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {-b c-\sqrt {a} \sqrt {b} d}}-\frac {\arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {-b c+\sqrt {a} \sqrt {b} d}}}{\sqrt {a}} \] Input:

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

Output:

(ArcTan[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt 
[a]*d)]/Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d] - ArcTan[(Sqrt[-(b*c) + Sqrt[a]*S 
qrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqrt[a]*d)]/Sqrt[-(b*c) + Sqrt[a]*Sq 
rt[b]*d])/Sqrt[a]
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {484, 1406, 221}

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.

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

\(\Big \downarrow \) 484

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

\(\Big \downarrow \) 1406

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

\(\Big \downarrow \) 221

\(\displaystyle 2 d \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} \sqrt [4]{b} d \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} d \sqrt {\sqrt {b} c-\sqrt {a} d}}\right )\)

Input:

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

Output:

2*d*(-1/2*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]]/(Sq 
rt[a]*b^(1/4)*d*Sqrt[Sqrt[b]*c - Sqrt[a]*d]) + ArcTanh[(b^(1/4)*Sqrt[c + d 
*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]]/(2*Sqrt[a]*b^(1/4)*d*Sqrt[Sqrt[b]*c + Sq 
rt[a]*d]))
 

Defintions of rubi rules used

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 484

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[2* 
d   Subst[Int[1/(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4), x], x, Sqrt[c + d*x]], 
 x] /; FreeQ[{a, b, c, d}, x]
 

rule 1406

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 
2 - 4*a*c, 2]}, Simp[c/q   Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q   I 
nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c 
, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.75

method	result	size
pseudoelliptic	\(\frac {b d \left (\frac {\operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\sqrt {a b \,d^{2}}}\)	\(101\)
derivativedivides	\(-2 d b \left (-\frac {\operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )\)	\(112\)
default	\(-2 d b \left (-\frac {\operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )\)	\(112\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 949 vs. \(2 (94) = 188\).

Time = 0.10 (sec) , antiderivative size = 949, normalized size of antiderivative = 7.08 \[ \int \frac {1}{\sqrt {c+d x} \left (a-b x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*sqrt(((a*b*c^2 - a^2*d^2)*sqrt(d^2/(a*b^3*c^4 - 2*a^2*b^2*c^2*d^2 + a^ 
3*b*d^4)) + c)/(a*b*c^2 - a^2*d^2))*log(sqrt(d*x + c)*d + (a*d^2 - (a*b^2* 
c^3 - a^2*b*c*d^2)*sqrt(d^2/(a*b^3*c^4 - 2*a^2*b^2*c^2*d^2 + a^3*b*d^4)))* 
sqrt(((a*b*c^2 - a^2*d^2)*sqrt(d^2/(a*b^3*c^4 - 2*a^2*b^2*c^2*d^2 + a^3*b* 
d^4)) + c)/(a*b*c^2 - a^2*d^2))) - 1/2*sqrt(((a*b*c^2 - a^2*d^2)*sqrt(d^2/ 
(a*b^3*c^4 - 2*a^2*b^2*c^2*d^2 + a^3*b*d^4)) + c)/(a*b*c^2 - a^2*d^2))*log 
(sqrt(d*x + c)*d - (a*d^2 - (a*b^2*c^3 - a^2*b*c*d^2)*sqrt(d^2/(a*b^3*c^4 
- 2*a^2*b^2*c^2*d^2 + a^3*b*d^4)))*sqrt(((a*b*c^2 - a^2*d^2)*sqrt(d^2/(a*b 
^3*c^4 - 2*a^2*b^2*c^2*d^2 + a^3*b*d^4)) + c)/(a*b*c^2 - a^2*d^2))) + 1/2* 
sqrt(-((a*b*c^2 - a^2*d^2)*sqrt(d^2/(a*b^3*c^4 - 2*a^2*b^2*c^2*d^2 + a^3*b 
*d^4)) - c)/(a*b*c^2 - a^2*d^2))*log(sqrt(d*x + c)*d + (a*d^2 + (a*b^2*c^3 
 - a^2*b*c*d^2)*sqrt(d^2/(a*b^3*c^4 - 2*a^2*b^2*c^2*d^2 + a^3*b*d^4)))*sqr 
t(-((a*b*c^2 - a^2*d^2)*sqrt(d^2/(a*b^3*c^4 - 2*a^2*b^2*c^2*d^2 + a^3*b*d^ 
4)) - c)/(a*b*c^2 - a^2*d^2))) - 1/2*sqrt(-((a*b*c^2 - a^2*d^2)*sqrt(d^2/( 
a*b^3*c^4 - 2*a^2*b^2*c^2*d^2 + a^3*b*d^4)) - c)/(a*b*c^2 - a^2*d^2))*log( 
sqrt(d*x + c)*d - (a*d^2 + (a*b^2*c^3 - a^2*b*c*d^2)*sqrt(d^2/(a*b^3*c^4 - 
 2*a^2*b^2*c^2*d^2 + a^3*b*d^4)))*sqrt(-((a*b*c^2 - a^2*d^2)*sqrt(d^2/(a*b 
^3*c^4 - 2*a^2*b^2*c^2*d^2 + a^3*b*d^4)) - c)/(a*b*c^2 - a^2*d^2)))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (94) = 188\).

Time = 0.13 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.54 \[ \int \frac {1}{\sqrt {c+d x} \left (a-b x^2\right )} \, dx=\frac {{\left (a d {\left | b \right |} {\left | d \right |} - \sqrt {a b} c d {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b c + \sqrt {b^{2} c^{2} - {\left (b c^{2} - a d^{2}\right )} b}}{b}}}\right )}{{\left (a b c - \sqrt {a b} a d\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | d \right |}} + \frac {{\left (a d {\left | b \right |} {\left | d \right |} + \sqrt {a b} c d {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b c - \sqrt {b^{2} c^{2} - {\left (b c^{2} - a d^{2}\right )} b}}{b}}}\right )}{{\left (a b c + \sqrt {a b} a d\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | d \right |}} \] Input:

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

Output:

(a*d*abs(b)*abs(d) - sqrt(a*b)*c*d*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(b*c 
 + sqrt(b^2*c^2 - (b*c^2 - a*d^2)*b))/b))/((a*b*c - sqrt(a*b)*a*d)*sqrt(-b 
^2*c - sqrt(a*b)*b*d)*abs(d)) + (a*d*abs(b)*abs(d) + sqrt(a*b)*c*d*abs(b)) 
*arctan(sqrt(d*x + c)/sqrt(-(b*c - sqrt(b^2*c^2 - (b*c^2 - a*d^2)*b))/b))/ 
((a*b*c + sqrt(a*b)*a*d)*sqrt(-b^2*c + sqrt(a*b)*b*d)*abs(d))
 

Mupad [B] (verification not implemented)

Time = 8.72 (sec) , antiderivative size = 1366, normalized size of antiderivative = 10.19 \[ \int \frac {1}{\sqrt {c+d x} \left (a-b x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

2*atanh((32*a^2*b^5*c^2*d^2*(- (d*(a^3*b)^(1/2))/(4*(a^3*b*d^2 - a^2*b^2*c 
^2)) - (a*b*c)/(4*(a^3*b*d^2 - a^2*b^2*c^2)))^(1/2)*(c + d*x)^(1/2))/((16* 
a^4*b^6*c^3*d^3)/(a^3*b*d^2 - a^2*b^2*c^2) - (16*a^4*b^4*d^6*(a^3*b)^(1/2) 
)/(a^3*b*d^2 - a^2*b^2*c^2) - (16*a^5*b^5*c*d^5)/(a^3*b*d^2 - a^2*b^2*c^2) 
 + (16*a^3*b^5*c^2*d^4*(a^3*b)^(1/2))/(a^3*b*d^2 - a^2*b^2*c^2)) - (32*b^3 
*d^2*(- (d*(a^3*b)^(1/2))/(4*(a^3*b*d^2 - a^2*b^2*c^2)) - (a*b*c)/(4*(a^3* 
b*d^2 - a^2*b^2*c^2)))^(1/2)*(c + d*x)^(1/2))/((16*a^2*b^4*c*d^3)/(a^3*b*d 
^2 - a^2*b^2*c^2) + (16*a*b^3*d^4*(a^3*b)^(1/2))/(a^3*b*d^2 - a^2*b^2*c^2) 
) + (32*a*b^4*c*d^3*(a^3*b)^(1/2)*(- (d*(a^3*b)^(1/2))/(4*(a^3*b*d^2 - a^2 
*b^2*c^2)) - (a*b*c)/(4*(a^3*b*d^2 - a^2*b^2*c^2)))^(1/2)*(c + d*x)^(1/2)) 
/((16*a^4*b^6*c^3*d^3)/(a^3*b*d^2 - a^2*b^2*c^2) - (16*a^4*b^4*d^6*(a^3*b) 
^(1/2))/(a^3*b*d^2 - a^2*b^2*c^2) - (16*a^5*b^5*c*d^5)/(a^3*b*d^2 - a^2*b^ 
2*c^2) + (16*a^3*b^5*c^2*d^4*(a^3*b)^(1/2))/(a^3*b*d^2 - a^2*b^2*c^2)))*(- 
(d*(a^3*b)^(1/2) + a*b*c)/(4*(a^3*b*d^2 - a^2*b^2*c^2)))^(1/2) - 2*atanh(( 
32*b^3*d^2*((d*(a^3*b)^(1/2))/(4*(a^3*b*d^2 - a^2*b^2*c^2)) - (a*b*c)/(4*( 
a^3*b*d^2 - a^2*b^2*c^2)))^(1/2)*(c + d*x)^(1/2))/((16*a^2*b^4*c*d^3)/(a^3 
*b*d^2 - a^2*b^2*c^2) - (16*a*b^3*d^4*(a^3*b)^(1/2))/(a^3*b*d^2 - a^2*b^2* 
c^2)) - (32*a^2*b^5*c^2*d^2*((d*(a^3*b)^(1/2))/(4*(a^3*b*d^2 - a^2*b^2*c^2 
)) - (a*b*c)/(4*(a^3*b*d^2 - a^2*b^2*c^2)))^(1/2)*(c + d*x)^(1/2))/((16*a^ 
4*b^6*c^3*d^3)/(a^3*b*d^2 - a^2*b^2*c^2) + (16*a^4*b^4*d^6*(a^3*b)^(1/2...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.02 \[ \int \frac {1}{\sqrt {c+d x} \left (a-b x^2\right )} \, dx=\frac {2 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) b c +2 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) a d +\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b c -\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b c -\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a d +\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a d}{2 a b \left (a \,d^{2}-b \,c^{2}\right )} \] Input:

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

Output:

(2*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*s 
qrt(sqrt(b)*sqrt(a)*d - b*c)))*b*c + 2*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b* 
c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*d + s 
qrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) 
+ sqrt(b)*sqrt(c + d*x))*b*c - sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log(s 
qrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*b*c - sqrt(b)*sqrt(s 
qrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqr 
t(c + d*x))*a*d + sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log(sqrt(sqrt(b)*s 
qrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a*d)/(2*a*b*(a*d**2 - b*c**2))