\(\int \frac {x}{\sqrt {c+d x} (a-b x^2)} \, dx\) [587]

Optimal result

Integrand size = 21, antiderivative size = 123 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{b^{3/4} \sqrt {\sqrt {b} c-\sqrt {a} d}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{b^{3/4} \sqrt {\sqrt {b} c+\sqrt {a} d}} \] Output:

arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/b^(3/4)/(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))/b^(3/4)/(b^(1/2)*c+a^(1/2)*d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.24 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )} \, dx=\frac {\frac {\arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {-b c-\sqrt {a} \sqrt {b} d}}+\frac {\arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {-b c+\sqrt {a} \sqrt {b} d}}}{\sqrt {b}} \] Input:

Integrate[x/(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[b]
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {561, 25, 27, 1480, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

\(\displaystyle -\frac {2 \left (-\frac {d^2 \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 b^{3/4} \sqrt {\sqrt {b} c-\sqrt {a} d}}-\frac {d^2 \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 b^{3/4} \sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{d^2}\)

Input:

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

Output:

(-2*(-1/2*(d^2*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d] 
])/(b^(3/4)*Sqrt[Sqrt[b]*c - Sqrt[a]*d]) - (d^2*ArcTanh[(b^(1/4)*Sqrt[c + 
d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*b^(3/4)*Sqrt[Sqrt[b]*c + Sqrt[a]*d] 
)))/d^2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 561

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k/d   Subst[Int[x^(k*(n + 1) - 1)*(-c 
/d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), 
 x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac 
tionQ[n] && IntegerQ[p] && IntegerQ[m]
 

rule 1480

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.74

method	result	size
pseudoelliptic	\(\frac {\operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\)	\(91\)
derivativedivides	\(-2 b \left (-\frac {\operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )\)	\(101\)
default	\(2 b \left (\frac {\operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )\)	\(101\)

Input:

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

Output:

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)*arctan(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. 925 vs. \(2 (87) = 174\).

Time = 0.11 (sec) , antiderivative size = 925, normalized size of antiderivative = 7.52 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*sqrt(((b^2*c^2 - a*b*d^2)*sqrt(a*d^2/(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2* 
b^3*d^4)) + c)/(b^2*c^2 - a*b*d^2))*log((b*c - (b^3*c^2 - a*b^2*d^2)*sqrt( 
a*d^2/(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4)))*sqrt(((b^2*c^2 - a*b*d^2 
)*sqrt(a*d^2/(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4)) + c)/(b^2*c^2 - a* 
b*d^2)) + sqrt(d*x + c)) - 1/2*sqrt(((b^2*c^2 - a*b*d^2)*sqrt(a*d^2/(b^5*c 
^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4)) + c)/(b^2*c^2 - a*b*d^2))*log(-(b*c - 
 (b^3*c^2 - a*b^2*d^2)*sqrt(a*d^2/(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4 
)))*sqrt(((b^2*c^2 - a*b*d^2)*sqrt(a*d^2/(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2* 
b^3*d^4)) + c)/(b^2*c^2 - a*b*d^2)) + sqrt(d*x + c)) + 1/2*sqrt(-((b^2*c^2 
 - a*b*d^2)*sqrt(a*d^2/(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4)) - c)/(b^ 
2*c^2 - a*b*d^2))*log((b*c + (b^3*c^2 - a*b^2*d^2)*sqrt(a*d^2/(b^5*c^4 - 2 
*a*b^4*c^2*d^2 + a^2*b^3*d^4)))*sqrt(-((b^2*c^2 - a*b*d^2)*sqrt(a*d^2/(b^5 
*c^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4)) - c)/(b^2*c^2 - a*b*d^2)) + sqrt(d* 
x + c)) - 1/2*sqrt(-((b^2*c^2 - a*b*d^2)*sqrt(a*d^2/(b^5*c^4 - 2*a*b^4*c^2 
*d^2 + a^2*b^3*d^4)) - c)/(b^2*c^2 - a*b*d^2))*log(-(b*c + (b^3*c^2 - a*b^ 
2*d^2)*sqrt(a*d^2/(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4)))*sqrt(-((b^2* 
c^2 - a*b*d^2)*sqrt(a*d^2/(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4)) - c)/ 
(b^2*c^2 - a*b*d^2)) + sqrt(d*x + c))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-integrate(x/((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. 205 vs. \(2 (87) = 174\).

Time = 0.16 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.67 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )} \, dx=-\frac {\frac {{\left (b c d {\left | b \right |} - \sqrt {a b} d {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b c + \sqrt {b^{2} c^{2} - {\left (b c^{2} - a d^{2}\right )} b}}{b}}}\right )}{{\left (b^{2} c - \sqrt {a b} b d\right )} \sqrt {-b^{2} c - \sqrt {a b} b d}} + \frac {{\left (b c d {\left | b \right |} + \sqrt {a b} d {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b c - \sqrt {b^{2} c^{2} - {\left (b c^{2} - a d^{2}\right )} b}}{b}}}\right )}{{\left (b^{2} c + \sqrt {a b} b d\right )} \sqrt {-b^{2} c + \sqrt {a b} b d}}}{d} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 8.38 (sec) , antiderivative size = 1367, normalized size of antiderivative = 11.11 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

2*atanh((32*a*b^6*c^2*d^2*((b^2*c)/(4*(b^4*c^2 - a*b^3*d^2)) + (d*(a*b^3)^ 
(1/2))/(4*(b^4*c^2 - a*b^3*d^2)))^(1/2)*(c + d*x)^(1/2))/(16*a^2*b^4*d^4 - 
 16*a*b^5*c^2*d^2 - (16*a^2*b^8*c^2*d^4)/(b^4*c^2 - a*b^3*d^2) + (16*a*b^9 
*c^4*d^2)/(b^4*c^2 - a*b^3*d^2) + (16*a*b^7*c^3*d^3*(a*b^3)^(1/2))/(b^4*c^ 
2 - a*b^3*d^2) - (16*a^2*b^6*c*d^5*(a*b^3)^(1/2))/(b^4*c^2 - a*b^3*d^2)) - 
 (32*a*b^2*d^2*((b^2*c)/(4*(b^4*c^2 - a*b^3*d^2)) + (d*(a*b^3)^(1/2))/(4*( 
b^4*c^2 - a*b^3*d^2)))^(1/2)*(c + d*x)^(1/2))/((16*a*b^5*c^2*d^2)/(b^4*c^2 
 - a*b^3*d^2) - 16*a*b*d^2 + (16*a*b^3*c*d^3*(a*b^3)^(1/2))/(b^4*c^2 - a*b 
^3*d^2)) + (32*a*b^4*c*d^3*((b^2*c)/(4*(b^4*c^2 - a*b^3*d^2)) + (d*(a*b^3) 
^(1/2))/(4*(b^4*c^2 - a*b^3*d^2)))^(1/2)*(a*b^3)^(1/2)*(c + d*x)^(1/2))/(1 
6*a^2*b^4*d^4 - 16*a*b^5*c^2*d^2 - (16*a^2*b^8*c^2*d^4)/(b^4*c^2 - a*b^3*d 
^2) + (16*a*b^9*c^4*d^2)/(b^4*c^2 - a*b^3*d^2) + (16*a*b^7*c^3*d^3*(a*b^3) 
^(1/2))/(b^4*c^2 - a*b^3*d^2) - (16*a^2*b^6*c*d^5*(a*b^3)^(1/2))/(b^4*c^2 
- a*b^3*d^2)))*((b^2*c + d*(a*b^3)^(1/2))/(4*(b^4*c^2 - a*b^3*d^2)))^(1/2) 
 + 2*atanh((32*a*b^2*d^2*((b^2*c)/(4*(b^4*c^2 - a*b^3*d^2)) - (d*(a*b^3)^( 
1/2))/(4*(b^4*c^2 - a*b^3*d^2)))^(1/2)*(c + d*x)^(1/2))/(16*a*b*d^2 - (16* 
a*b^5*c^2*d^2)/(b^4*c^2 - a*b^3*d^2) + (16*a*b^3*c*d^3*(a*b^3)^(1/2))/(b^4 
*c^2 - a*b^3*d^2)) + (32*a*b^6*c^2*d^2*((b^2*c)/(4*(b^4*c^2 - a*b^3*d^2)) 
- (d*(a*b^3)^(1/2))/(4*(b^4*c^2 - a*b^3*d^2)))^(1/2)*(c + d*x)^(1/2))/(16* 
a^2*b^4*d^4 - 16*a*b^5*c^2*d^2 - (16*a^2*b^8*c^2*d^4)/(b^4*c^2 - a*b^3*...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.13 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )} \, dx=\frac {-2 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) d -2 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) c -\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) d +\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) d +\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) c -\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) c}{2 b \left (a \,d^{2}-b \,c^{2}\right )} \] Input:

int(x/(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 
)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*d - 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)))*c - sq 
rt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + 
 sqrt(b)*sqrt(c + d*x))*d + sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log(sqrt 
(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*d + sqrt(b)*sqrt(sqrt(b 
)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + 
 d*x))*c - sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log(sqrt(sqrt(b)*sqrt(a)* 
d + b*c) + sqrt(b)*sqrt(c + d*x))*c)/(2*b*(a*d**2 - b*c**2))