\(\int \frac {1}{x^2 \sqrt {c+d x} (a+b x^2)} \, dx\) [625]

Optimal result

Integrand size = 22, antiderivative size = 456 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a+b x^2\right )} \, dx=-\frac {\sqrt {c+d x}}{a c x}+\frac {b^{3/4} d \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{\sqrt {2} a \sqrt {b c^2+a d^2} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}-\frac {b^{3/4} d \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{\sqrt {2} a \sqrt {b c^2+a d^2} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}+\frac {d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a c^{3/2}}-\frac {b^{3/4} d \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt {b c^2+a d^2}+\sqrt {b} (c+d x)}\right )}{\sqrt {2} a \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.72 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.49 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\frac {-\frac {\sqrt {a} \sqrt {c+d x}}{c x}-\frac {i b \arctan \left (\frac {\sqrt {-b c-i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+i \sqrt {a} d}\right )}{\sqrt {-b c-i \sqrt {a} \sqrt {b} d}}+\frac {i b \arctan \left (\frac {\sqrt {-b c+i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-i \sqrt {a} d}\right )}{\sqrt {-b c+i \sqrt {a} \sqrt {b} d}}+\frac {\sqrt {a} d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{3/2}}}{a^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.43 (sec) , antiderivative size = 607, normalized size of antiderivative = 1.33, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, 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

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

\(\Big \downarrow \) 2009

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

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[(Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]] - Sqrt[2] 
*b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d^2]]])/(2*Sqrt[2] 
*a*Sqrt[b*c^2 + a*d^2]*Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d^2]]) + (b^(3/4)*A 
rcTanh[(Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]] + Sqrt[2]*b^(1/4)*Sqrt[c + d 
*x])/Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d^2]]])/(2*Sqrt[2]*a*Sqrt[b*c^2 + a*d 
^2]*Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d^2]]) + (b^(3/4)*Log[Sqrt[b*c^2 + a*d 
^2] - Sqrt[2]*b^(1/4)*Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]]*Sqrt[c + d*x] 
+ Sqrt[b]*(c + d*x)])/(4*Sqrt[2]*a*Sqrt[b*c^2 + a*d^2]*Sqrt[Sqrt[b]*c + Sq 
rt[b*c^2 + a*d^2]]) - (b^(3/4)*Log[Sqrt[b*c^2 + a*d^2] + Sqrt[2]*b^(1/4)*S 
qrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]]*Sqrt[c + d*x] + Sqrt[b]*(c + d*x)])/( 
4*Sqrt[2]*a*Sqrt[b*c^2 + a*d^2]*Sqrt[Sqrt[b]*c + Sqrt[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 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 = 0.72 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.39

method	result	size
pseudoelliptic	\(\frac {\frac {x \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}\, \sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}\, \left (c^{\frac {3}{2}} \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-c^{\frac {5}{2}} b \right ) \ln \left (\sqrt {b}\, \left (d x +c \right )-\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{4}-\frac {x \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}\, \sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}\, \left (c^{\frac {3}{2}} \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-c^{\frac {5}{2}} b \right ) \ln \left (\sqrt {b}\, \left (d x +c \right )+\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{4}+d a \left (-\sqrt {a \,d^{2}+b \,c^{2}}\, \left (-\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) d x +\sqrt {d x +c}\, \sqrt {c}\right ) \sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}+d x b \,c^{\frac {3}{2}} \left (\arctan \left (\frac {-2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )-\arctan \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )\right )\right )}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}\, c^{\frac {3}{2}} \sqrt {a \,d^{2}+b \,c^{2}}\, d x \,a^{2}}\)	\(634\)
risch	\(-\frac {\sqrt {d x +c}}{a c x}-\frac {d \left (2 b c \left (\frac {\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \ln \left (\sqrt {b}\, \left (d x +c \right )+\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{2 \sqrt {b}}+\frac {2 \left (2 a \,d^{2} \sqrt {b}-\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{2 \sqrt {b}}\right ) \arctan \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}}{4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}\, a \,d^{2}}+\frac {\frac {\left (\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c -\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \ln \left (\sqrt {b}\, \left (d x +c \right )-\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{2 \sqrt {b}}+\frac {2 \left (2 a \,d^{2} \sqrt {b}+\frac {\left (\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c -\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{2 \sqrt {b}}\right ) \arctan \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}-\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}}{4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}\, a \,d^{2}}\right )-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{a c}\)	\(843\)
derivativedivides	\(2 d^{3} \left (\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}}-\frac {b \left (\frac {-\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \ln \left (-\sqrt {b}\, \left (d x +c \right )+\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}-\sqrt {a \,d^{2}+b \,c^{2}}\right )}{2 \sqrt {b}}+\frac {2 \left (-2 a \,d^{2} \sqrt {b}+\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{2 \sqrt {b}}\right ) \arctan \left (\frac {-2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}}{4 \sqrt {b}\, \sqrt {a \,d^{2}+b \,c^{2}}\, a \,d^{2}}+\frac {\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \ln \left (\sqrt {b}\, \left (d x +c \right )+\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{2 \sqrt {b}}+\frac {2 \left (2 a \,d^{2} \sqrt {b}-\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{2 \sqrt {b}}\right ) \arctan \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}}{4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}\, a \,d^{2}}\right )}{a \,d^{2}}\right )\)	\(851\)
default	\(2 d^{3} \left (\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}}-\frac {b \left (\frac {-\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \ln \left (-\sqrt {b}\, \left (d x +c \right )+\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}-\sqrt {a \,d^{2}+b \,c^{2}}\right )}{2 \sqrt {b}}+\frac {2 \left (-2 a \,d^{2} \sqrt {b}+\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{2 \sqrt {b}}\right ) \arctan \left (\frac {-2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}}{4 \sqrt {b}\, \sqrt {a \,d^{2}+b \,c^{2}}\, a \,d^{2}}+\frac {\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \ln \left (\sqrt {b}\, \left (d x +c \right )+\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{2 \sqrt {b}}+\frac {2 \left (2 a \,d^{2} \sqrt {b}-\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{2 \sqrt {b}}\right ) \arctan \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}}{4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}\, a \,d^{2}}\right )}{a \,d^{2}}\right )\)	\(851\)

Input:

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

Output:

1/(4*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b*c)^(1/2)*(1 
/4*x*(2*((a*d^2+b*c^2)*b)^(1/2)+2*b*c)^(1/2)*(4*(a*d^2+b*c^2)^(1/2)*b^(1/2 
)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b*c)^(1/2)*(c^(3/2)*((a*d^2+b*c^2)*b)^(1/2)- 
c^(5/2)*b)*ln(b^(1/2)*(d*x+c)-(d*x+c)^(1/2)*(2*((a*d^2+b*c^2)*b)^(1/2)+2*b 
*c)^(1/2)+(a*d^2+b*c^2)^(1/2))-1/4*x*(2*((a*d^2+b*c^2)*b)^(1/2)+2*b*c)^(1/ 
2)*(4*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b*c)^(1/2)*( 
c^(3/2)*((a*d^2+b*c^2)*b)^(1/2)-c^(5/2)*b)*ln(b^(1/2)*(d*x+c)+(d*x+c)^(1/2 
)*(2*((a*d^2+b*c^2)*b)^(1/2)+2*b*c)^(1/2)+(a*d^2+b*c^2)^(1/2))+d*a*(-(a*d^ 
2+b*c^2)^(1/2)*(-arctanh((d*x+c)^(1/2)/c^(1/2))*d*x+(d*x+c)^(1/2)*c^(1/2)) 
*(4*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b*c)^(1/2)+d*x 
*b*c^(3/2)*(arctan((-2*b^(1/2)*(d*x+c)^(1/2)+(2*((a*d^2+b*c^2)*b)^(1/2)+2* 
b*c)^(1/2))/(4*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b*c 
)^(1/2))-arctan((2*b^(1/2)*(d*x+c)^(1/2)+(2*((a*d^2+b*c^2)*b)^(1/2)+2*b*c) 
^(1/2))/(4*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b*c)^(1 
/2)))))/c^(3/2)/(a*d^2+b*c^2)^(1/2)/d/x/a^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1111 vs. \(2 (363) = 726\).

Time = 0.22 (sec) , antiderivative size = 2231, normalized size of antiderivative = 4.89 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] 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))) ...
 

Sympy [F]

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

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

Output:

Integral(1/(x**2*(a + b*x**2)*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 [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a+b x^2\right )} \, dx=-\frac {d \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a \sqrt {-c} c} - \frac {{\left (\sqrt {-b^{2} c + \sqrt {-a b} b d} d {\left | a \right |} {\left | b \right |} {\left | d \right |} - \sqrt {-b^{2} c + \sqrt {-a b} b d} \sqrt {-a b} c d {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b c - \sqrt {a^{2} b^{2} c^{2} - {\left (a b c^{2} + a^{2} d^{2}\right )} a b}}{a b}}}\right )}{{\left (a b^{2} c^{2} + a^{2} b d^{2}\right )} {\left | a \right |} {\left | d \right |}} - \frac {{\left (a b c d {\left | b \right |} - \sqrt {-a b} d {\left | a \right |} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b c + \sqrt {a^{2} b^{2} c^{2} - {\left (a b c^{2} + a^{2} d^{2}\right )} a b}}{a b}}}\right )}{\sqrt {-b^{2} c - \sqrt {-a b} b d} {\left (a^{2} d + \sqrt {-a b} a c\right )} {\left | a \right |} {\left | d \right |}} - \frac {\sqrt {d x + c}}{a c x} \] 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)) - (a*b* 
c*d*abs(b) - sqrt(-a*b)*d*abs(a)*abs(b)*abs(d))*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)))/(sqrt(-b^2* 
c - sqrt(-a*b)*b*d)*(a^2*d + sqrt(-a*b)*a*c)*abs(a)*abs(d)) - sqrt(d*x + c 
)/(a*c*x)
 

Mupad [B] (verification not implemented)

Time = 8.70 (sec) , antiderivative size = 3857, normalized size of antiderivative = 8.46 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] 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)*(...
 

Reduce [F]

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

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

Output:

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