\(\int \frac {1}{x^2 \sqrt {c+d x} (a-b x^2)} \, dx\) [590]

Optimal result

Integrand size = 23, antiderivative size = 180 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )} \, dx=-\frac {\sqrt {c+d x}}{a c x}+\frac {d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a c^{3/2}}-\frac {b^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{a^{3/2} \sqrt {\sqrt {b} c-\sqrt {a} d}}+\frac {b^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{a^{3/2} \sqrt {\sqrt {b} c+\sqrt {a} d}} \] Output:

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

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )} \, dx=\frac {-\frac {\sqrt {a} \sqrt {c+d x}}{c x}+\frac {b \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {-b c-\sqrt {a} \sqrt {b} d}}-\frac {b \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {-b c+\sqrt {a} \sqrt {b} d}}+\frac {\sqrt {a} d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{3/2}}}{a^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {561, 27, 1484, 2009}

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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1484

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 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 1484

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symb 
ol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x] /; Fre 
eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 
 0] && IntegerQ[q]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(\frac {-\frac {\sqrt {d x +c}}{c x}+\frac {d \,b^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {d \,b^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {d \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{c^{\frac {3}{2}}}}{a}\)	\(150\)
risch	\(-\frac {\sqrt {d x +c}}{a c x}+\frac {d \,b^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {d \,b^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {d \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a \,c^{\frac {3}{2}}}\)	\(158\)
derivativedivides	\(-2 d^{3} \left (\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \,d^{2}}-\frac {-\frac {\sqrt {d x +c}}{2 c d x}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 c^{\frac {3}{2}}}}{a \,d^{2}}\right )\)	\(168\)
default	\(2 d^{3} \left (-\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \,d^{2}}+\frac {-\frac {\sqrt {d x +c}}{2 c d x}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 c^{\frac {3}{2}}}}{a \,d^{2}}\right )\)	\(168\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1111 vs. \(2 (132) = 264\).

Time = 0.21 (sec) , antiderivative size = 2230, normalized size of antiderivative = 12.39 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*(a*c^2*x*sqrt((b^2*c + (a^3*b*c^2 - a^4*d^2)*sqrt(b^3*d^2/(a^5*b^2*c^ 
4 - 2*a^6*b*c^2*d^2 + a^7*d^4)))/(a^3*b*c^2 - a^4*d^2))*log(sqrt(d*x + c)* 
b^2*d + (a^2*b*d^2 - (a^4*b*c^3 - a^5*c*d^2)*sqrt(b^3*d^2/(a^5*b^2*c^4 - 2 
*a^6*b*c^2*d^2 + a^7*d^4)))*sqrt((b^2*c + (a^3*b*c^2 - a^4*d^2)*sqrt(b^3*d 
^2/(a^5*b^2*c^4 - 2*a^6*b*c^2*d^2 + a^7*d^4)))/(a^3*b*c^2 - a^4*d^2))) - a 
*c^2*x*sqrt((b^2*c + (a^3*b*c^2 - a^4*d^2)*sqrt(b^3*d^2/(a^5*b^2*c^4 - 2*a 
^6*b*c^2*d^2 + a^7*d^4)))/(a^3*b*c^2 - a^4*d^2))*log(sqrt(d*x + c)*b^2*d - 
 (a^2*b*d^2 - (a^4*b*c^3 - a^5*c*d^2)*sqrt(b^3*d^2/(a^5*b^2*c^4 - 2*a^6*b* 
c^2*d^2 + a^7*d^4)))*sqrt((b^2*c + (a^3*b*c^2 - a^4*d^2)*sqrt(b^3*d^2/(a^5 
*b^2*c^4 - 2*a^6*b*c^2*d^2 + a^7*d^4)))/(a^3*b*c^2 - a^4*d^2))) + a*c^2*x* 
sqrt((b^2*c - (a^3*b*c^2 - a^4*d^2)*sqrt(b^3*d^2/(a^5*b^2*c^4 - 2*a^6*b*c^ 
2*d^2 + a^7*d^4)))/(a^3*b*c^2 - a^4*d^2))*log(sqrt(d*x + c)*b^2*d + (a^2*b 
*d^2 + (a^4*b*c^3 - a^5*c*d^2)*sqrt(b^3*d^2/(a^5*b^2*c^4 - 2*a^6*b*c^2*d^2 
 + a^7*d^4)))*sqrt((b^2*c - (a^3*b*c^2 - a^4*d^2)*sqrt(b^3*d^2/(a^5*b^2*c^ 
4 - 2*a^6*b*c^2*d^2 + a^7*d^4)))/(a^3*b*c^2 - a^4*d^2))) - a*c^2*x*sqrt((b 
^2*c - (a^3*b*c^2 - a^4*d^2)*sqrt(b^3*d^2/(a^5*b^2*c^4 - 2*a^6*b*c^2*d^2 + 
 a^7*d^4)))/(a^3*b*c^2 - a^4*d^2))*log(sqrt(d*x + c)*b^2*d - (a^2*b*d^2 + 
(a^4*b*c^3 - a^5*c*d^2)*sqrt(b^3*d^2/(a^5*b^2*c^4 - 2*a^6*b*c^2*d^2 + a^7* 
d^4)))*sqrt((b^2*c - (a^3*b*c^2 - a^4*d^2)*sqrt(b^3*d^2/(a^5*b^2*c^4 - 2*a 
^6*b*c^2*d^2 + a^7*d^4)))/(a^3*b*c^2 - a^4*d^2))) + sqrt(c)*d*x*log((d*...
 

Sympy [F]

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

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

Output:

-Integral(1/(-a*x**2*sqrt(c + d*x) + b*x**4*sqrt(c + d*x)), x)
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (132) = 264\).

Time = 0.14 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.83 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )} \, dx=-\frac {d \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a \sqrt {-c} c} - \frac {{\left (\sqrt {-b^{2} c - \sqrt {a b} b d} d {\left | a \right |} {\left | b \right |} {\left | d \right |} - \sqrt {-b^{2} c - \sqrt {a b} b d} \sqrt {a b} c d {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b c + \sqrt {a^{2} b^{2} c^{2} - {\left (a b c^{2} - a^{2} d^{2}\right )} a b}}{a b}}}\right )}{{\left (a b^{2} c^{2} - a^{2} b d^{2}\right )} {\left | a \right |} {\left | d \right |}} - \frac {{\left (\sqrt {-b^{2} c + \sqrt {a b} b d} d {\left | a \right |} {\left | b \right |} {\left | d \right |} + \sqrt {-b^{2} c + \sqrt {a b} b d} \sqrt {a b} c d {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b c - \sqrt {a^{2} b^{2} c^{2} - {\left (a b c^{2} - a^{2} d^{2}\right )} a b}}{a b}}}\right )}{{\left (a b^{2} c^{2} - a^{2} b d^{2}\right )} {\left | a \right |} {\left | d \right |}} - \frac {\sqrt {d x + c}}{a c x} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 8.82 (sec) , antiderivative size = 3815, normalized size of antiderivative = 21.19 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

- atan(((((((16*(16*a^7*b^4*c*d^11 - 8*a^6*b^5*c^3*d^9))/(a^3*c^2) - (16*( 
48*a^6*b^5*c^4*d^8 - 32*a^7*b^4*c^2*d^10)*(c + d*x)^(1/2)*(-(d*(a^7*b^3)^( 
1/2) + a^3*b^2*c)/(4*(a^7*d^2 - a^6*b*c^2)))^(1/2))/(a^2*c^2))*(-(d*(a^7*b 
^3)^(1/2) + a^3*b^2*c)/(4*(a^7*d^2 - a^6*b*c^2)))^(1/2) + (16*(4*a^4*b^5*c 
*d^10 + 20*a^3*b^6*c^3*d^8)*(c + d*x)^(1/2))/(a^2*c^2))*(-(d*(a^7*b^3)^(1/ 
2) + a^3*b^2*c)/(4*(a^7*d^2 - a^6*b*c^2)))^(1/2) - (16*(2*a^4*b^5*d^11 - 8 
*a^3*b^6*c^2*d^9))/(a^3*c^2))*(-(d*(a^7*b^3)^(1/2) + a^3*b^2*c)/(4*(a^7*d^ 
2 - a^6*b*c^2)))^(1/2) - (16*(a*b^6*d^10 + 2*b^7*c^2*d^8)*(c + d*x)^(1/2)) 
/(a^2*c^2))*(-(d*(a^7*b^3)^(1/2) + a^3*b^2*c)/(4*(a^7*d^2 - a^6*b*c^2)))^( 
1/2)*1i - (((((16*(16*a^7*b^4*c*d^11 - 8*a^6*b^5*c^3*d^9))/(a^3*c^2) + (16 
*(48*a^6*b^5*c^4*d^8 - 32*a^7*b^4*c^2*d^10)*(c + d*x)^(1/2)*(-(d*(a^7*b^3) 
^(1/2) + a^3*b^2*c)/(4*(a^7*d^2 - a^6*b*c^2)))^(1/2))/(a^2*c^2))*(-(d*(a^7 
*b^3)^(1/2) + a^3*b^2*c)/(4*(a^7*d^2 - a^6*b*c^2)))^(1/2) - (16*(4*a^4*b^5 
*c*d^10 + 20*a^3*b^6*c^3*d^8)*(c + d*x)^(1/2))/(a^2*c^2))*(-(d*(a^7*b^3)^( 
1/2) + a^3*b^2*c)/(4*(a^7*d^2 - a^6*b*c^2)))^(1/2) - (16*(2*a^4*b^5*d^11 - 
 8*a^3*b^6*c^2*d^9))/(a^3*c^2))*(-(d*(a^7*b^3)^(1/2) + a^3*b^2*c)/(4*(a^7* 
d^2 - a^6*b*c^2)))^(1/2) + (16*(a*b^6*d^10 + 2*b^7*c^2*d^8)*(c + d*x)^(1/2 
))/(a^2*c^2))*(-(d*(a^7*b^3)^(1/2) + a^3*b^2*c)/(4*(a^7*d^2 - a^6*b*c^2))) 
^(1/2)*1i)/((((((16*(16*a^7*b^4*c*d^11 - 8*a^6*b^5*c^3*d^9))/(a^3*c^2) - ( 
16*(48*a^6*b^5*c^4*d^8 - 32*a^7*b^4*c^2*d^10)*(c + d*x)^(1/2)*(-(d*(a^7...
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 409, normalized size of antiderivative = 2.27 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )} \, dx=\frac {2 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) b \,c^{3} x +2 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) a \,c^{2} d x +\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b \,c^{3} x -\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b \,c^{3} x -\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a \,c^{2} d x +\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a \,c^{2} d x -2 \sqrt {d x +c}\, a^{2} c \,d^{2}+2 \sqrt {d x +c}\, a b \,c^{3}-\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a^{2} d^{3} x +\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a b \,c^{2} d x +\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a^{2} d^{3} x -\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a b \,c^{2} d x}{2 a^{2} c^{2} x \left (a \,d^{2}-b \,c^{2}\right )} \] Input:

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