\(\int \frac {1}{x (c+d x)^{3/2} (a+b x^2)} \, dx\) [631]

Optimal result

Integrand size = 22, antiderivative size = 566 \[ \int \frac {1}{x (c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\frac {2 d^2}{c \left (b c^2+a d^2\right ) \sqrt {c+d x}}-\frac {b^{3/4} \left (b c^2-a d^2-\sqrt {b} c \sqrt {b c^2+a d^2}\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{\sqrt {2} a \left (b c^2+a d^2\right )^{3/2} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}+\frac {b^{3/4} \left (b c^2-a d^2-\sqrt {b} c \sqrt {b c^2+a d^2}\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{\sqrt {2} a \left (b c^2+a d^2\right )^{3/2} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a c^{3/2}}+\frac {b^{3/4} \left (b c^2-a d^2+\sqrt {b} c \sqrt {b c^2+a d^2}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt {b c^2+a d^2}+\sqrt {b} (c+d x)}\right )}{\sqrt {2} a \left (b c^2+a d^2\right )^{3/2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}} \] Output:

2*d^2/c/(a*d^2+b*c^2)/(d*x+c)^(1/2)-1/2*b^(3/4)*(b*c^2-a*d^2-b^(1/2)*c*(a* 
d^2+b*c^2)^(1/2))*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)^(3/2)/(-b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)+1/2*b^(3/4)*(b*c^2 
-a*d^2-b^(1/2)*c*(a*d^2+b*c^2)^(1/2))*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)^(3/2)/(-b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2 
)-2*arctanh((d*x+c)^(1/2)/c^(1/2))/a/c^(3/2)+1/2*b^(3/4)*(b*c^2-a*d^2+b^(1 
/2)*c*(a*d^2+b*c^2)^(1/2))*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)^(3/2)/(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 = 0.73 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.46 \[ \int \frac {1}{x (c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\frac {2 d^2}{\left (b c^3+a c d^2\right ) \sqrt {c+d x}}+\frac {b \arctan \left (\frac {\sqrt {-b c-i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+i \sqrt {a} d}\right )}{\left (a \sqrt {b} c+i a^{3/2} d\right ) \sqrt {-b c-i \sqrt {a} \sqrt {b} d}}+\frac {b \arctan \left (\frac {\sqrt {-b c+i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-i \sqrt {a} d}\right )}{\left (a \sqrt {b} c-i a^{3/2} d\right ) \sqrt {-b c+i \sqrt {a} \sqrt {b} d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a c^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.99 (sec) , antiderivative size = 746, normalized size of antiderivative = 1.32, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {561, 25, 27, 1610, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1610

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-2*(-(d^2/(c*(b*c^2 + a*d^2)*Sqrt[c + d*x])) + ArcTanh[Sqrt[c + d*x]/Sqrt[ 
c]]/(a*c^(3/2)) - (b^(3/4)*(b*c^2 - a*d^2 - Sqrt[b]*c*Sqrt[b*c^2 + a*d^2]) 
*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*(b*c^2 + a*d^2 
)^(3/2)*Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d^2]]) + (b^(3/4)*(b*c^2 - a*d^2 - 
 Sqrt[b]*c*Sqrt[b*c^2 + a*d^2])*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*(b*c^2 + a*d^2)^(3/2)*Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d^2 
]]) + (b^(3/4)*(b*c^2 - a*d^2 + Sqrt[b]*c*Sqrt[b*c^2 + a*d^2])*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*(b*c^2 + a*d^2)^(3/2)*Sqrt[Sqr 
t[b]*c + Sqrt[b*c^2 + a*d^2]]) - (b^(3/4)*(b*c^2 - a*d^2 + Sqrt[b]*c*Sqrt[ 
b*c^2 + a*d^2])*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*(b* 
c^2 + a*d^2)^(3/2)*Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*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 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 1610

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*((d + e*x^2)^q/(a 
+ b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4 
*a*c, 0] && IntegerQ[q] && IntegerQ[m]
 

rule 2009

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

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 833, normalized size of antiderivative = 1.47

method	result	size
pseudoelliptic	\(-\frac {-\frac {\left (\left (c^{\frac {3}{2}} a \,d^{2}-c^{\frac {5}{2}} \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-c^{\frac {7}{2}} b \right ) \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-a b \,c^{\frac {5}{2}} d^{2}+b^{\frac {3}{2}} c^{\frac {7}{2}} \sqrt {a \,d^{2}+b \,c^{2}}+b^{2} c^{\frac {9}{2}}\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}\, \sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}\, \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 {\left (\left (c^{\frac {3}{2}} a \,d^{2}-c^{\frac {5}{2}} \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-c^{\frac {7}{2}} b \right ) \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-a b \,c^{\frac {5}{2}} d^{2}+b^{\frac {3}{2}} c^{\frac {7}{2}} \sqrt {a \,d^{2}+b \,c^{2}}+b^{2} c^{\frac {9}{2}}\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}\, \sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}\, \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^{2} a \left (-2 \sqrt {a \,d^{2}+b \,c^{2}}\, \left (-\sqrt {d x +c}\, \left (a \,d^{2}+b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+a \,d^{2} \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}+\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 ) \left (b \,c^{\frac {3}{2}} a \,d^{2}+\sqrt {a \,d^{2}+b \,c^{2}}\, c^{\frac {5}{2}} b^{\frac {3}{2}}-c^{\frac {7}{2}} b^{2}\right ) \sqrt {d x +c}\right )}{c^{\frac {3}{2}} \left (a \,d^{2}+b \,c^{2}\right )^{\frac {3}{2}} \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}\, \sqrt {d x +c}\, a^{2} d^{2}}\)	\(833\)
derivativedivides	\(\text {Expression too large to display}\)	\(2156\)
default	\(\text {Expression too large to display}\)	\(2156\)

Input:

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

Output:

-1/c^(3/2)/(a*d^2+b*c^2)^(3/2)*(-1/4*((c^(3/2)*a*d^2-c^(5/2)*(a*d^2+b*c^2) 
^(1/2)*b^(1/2)-c^(7/2)*b)*((a*d^2+b*c^2)*b)^(1/2)-a*b*c^(5/2)*d^2+b^(3/2)* 
c^(7/2)*(a*d^2+b*c^2)^(1/2)+b^2*c^(9/2))*(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+c)^(1/2)*(2*((a*d^2+b*c^2)*b)^(1 
/2)+2*b*c)^(1/2)*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*((c^(3/2)*a*d^2-c^(5/2)*(a*d^2+b* 
c^2)^(1/2)*b^(1/2)-c^(7/2)*b)*((a*d^2+b*c^2)*b)^(1/2)-a*b*c^(5/2)*d^2+b^(3 
/2)*c^(7/2)*(a*d^2+b*c^2)^(1/2)+b^2*c^(9/2))*(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+c)^(1/2)*(2*((a*d^2+b*c^2)*b 
)^(1/2)+2*b*c)^(1/2)*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^2*a*(-2*(a*d^2+b*c^2)^(1/2)*(-( 
d*x+c)^(1/2)*(a*d^2+b*c^2)*arctanh((d*x+c)^(1/2)/c^(1/2))+a*d^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)+(arct 
an((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)))*(b*c^(3/2) 
*a*d^2+(a*d^2+b*c^2)^(1/2)*c^(5/2)*b^(3/2)-c^(7/2)*b^2)*(d*x+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+c 
)^(1/2)/a^2/d^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3107 vs. \(2 (462) = 924\).

Time = 0.74 (sec) , antiderivative size = 6223, normalized size of antiderivative = 10.99 \[ \int \frac {1}{x (c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 760, normalized size of antiderivative = 1.34 \[ \int \frac {1}{x (c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\frac {2 \, d^{2}}{{\left (b c^{3} + a c d^{2}\right )} \sqrt {d x + c}} + \frac {{\left ({\left (a b c^{2} d + a^{2} d^{3}\right )}^{2} b c {\left | b \right |} + {\left (\sqrt {-a b} b^{2} c^{4} - \sqrt {-a b} a^{2} d^{4}\right )} {\left | -a b c^{2} d - a^{2} d^{3} \right |} {\left | b \right |} + {\left (a^{2} b^{3} c^{5} d^{2} + 2 \, a^{3} b^{2} c^{3} d^{4} + a^{4} b c d^{6}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b^{2} c^{3} + a^{2} b c d^{2} + \sqrt {{\left (a b^{2} c^{3} + a^{2} b c d^{2}\right )}^{2} - {\left (a b^{2} c^{4} + 2 \, a^{2} b c^{2} d^{2} + a^{3} d^{4}\right )} {\left (a b^{2} c^{2} + a^{2} b d^{2}\right )}}}{a b^{2} c^{2} + a^{2} b d^{2}}}}\right )}{{\left (a^{2} b^{2} c^{4} d + 2 \, a^{3} b c^{2} d^{3} + a^{4} d^{5} - \sqrt {-a b} a b^{2} c^{5} - 2 \, \sqrt {-a b} a^{2} b c^{3} d^{2} - \sqrt {-a b} a^{3} c d^{4}\right )} \sqrt {-b^{2} c + \sqrt {-a b} b d} {\left | -a b c^{2} d - a^{2} d^{3} \right |}} + \frac {{\left ({\left (a b c^{2} d + a^{2} d^{3}\right )}^{2} b c {\left | b \right |} - {\left (\sqrt {-a b} b^{2} c^{4} - \sqrt {-a b} a^{2} d^{4}\right )} {\left | -a b c^{2} d - a^{2} d^{3} \right |} {\left | b \right |} + {\left (a^{2} b^{3} c^{5} d^{2} + 2 \, a^{3} b^{2} c^{3} d^{4} + a^{4} b c d^{6}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b^{2} c^{3} + a^{2} b c d^{2} - \sqrt {{\left (a b^{2} c^{3} + a^{2} b c d^{2}\right )}^{2} - {\left (a b^{2} c^{4} + 2 \, a^{2} b c^{2} d^{2} + a^{3} d^{4}\right )} {\left (a b^{2} c^{2} + a^{2} b d^{2}\right )}}}{a b^{2} c^{2} + a^{2} b d^{2}}}}\right )}{{\left (a^{2} b^{2} c^{4} d + 2 \, a^{3} b c^{2} d^{3} + a^{4} d^{5} + \sqrt {-a b} a b^{2} c^{5} + 2 \, \sqrt {-a b} a^{2} b c^{3} d^{2} + \sqrt {-a b} a^{3} c d^{4}\right )} \sqrt {-b^{2} c - \sqrt {-a b} b d} {\left | -a b c^{2} d - a^{2} d^{3} \right |}} + \frac {2 \, \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a \sqrt {-c} c} \] Input:

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

Output:

2*d^2/((b*c^3 + a*c*d^2)*sqrt(d*x + c)) + ((a*b*c^2*d + a^2*d^3)^2*b*c*abs 
(b) + (sqrt(-a*b)*b^2*c^4 - sqrt(-a*b)*a^2*d^4)*abs(-a*b*c^2*d - a^2*d^3)* 
abs(b) + (a^2*b^3*c^5*d^2 + 2*a^3*b^2*c^3*d^4 + a^4*b*c*d^6)*abs(b))*arcta 
n(sqrt(d*x + c)/sqrt(-(a*b^2*c^3 + a^2*b*c*d^2 + sqrt((a*b^2*c^3 + a^2*b*c 
*d^2)^2 - (a*b^2*c^4 + 2*a^2*b*c^2*d^2 + a^3*d^4)*(a*b^2*c^2 + a^2*b*d^2)) 
)/(a*b^2*c^2 + a^2*b*d^2)))/((a^2*b^2*c^4*d + 2*a^3*b*c^2*d^3 + a^4*d^5 - 
sqrt(-a*b)*a*b^2*c^5 - 2*sqrt(-a*b)*a^2*b*c^3*d^2 - sqrt(-a*b)*a^3*c*d^4)* 
sqrt(-b^2*c + sqrt(-a*b)*b*d)*abs(-a*b*c^2*d - a^2*d^3)) + ((a*b*c^2*d + a 
^2*d^3)^2*b*c*abs(b) - (sqrt(-a*b)*b^2*c^4 - sqrt(-a*b)*a^2*d^4)*abs(-a*b* 
c^2*d - a^2*d^3)*abs(b) + (a^2*b^3*c^5*d^2 + 2*a^3*b^2*c^3*d^4 + a^4*b*c*d 
^6)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(a*b^2*c^3 + a^2*b*c*d^2 - sqrt((a* 
b^2*c^3 + a^2*b*c*d^2)^2 - (a*b^2*c^4 + 2*a^2*b*c^2*d^2 + a^3*d^4)*(a*b^2* 
c^2 + a^2*b*d^2)))/(a*b^2*c^2 + a^2*b*d^2)))/((a^2*b^2*c^4*d + 2*a^3*b*c^2 
*d^3 + a^4*d^5 + sqrt(-a*b)*a*b^2*c^5 + 2*sqrt(-a*b)*a^2*b*c^3*d^2 + sqrt( 
-a*b)*a^3*c*d^4)*sqrt(-b^2*c - sqrt(-a*b)*b*d)*abs(-a*b*c^2*d - a^2*d^3)) 
+ 2*arctan(sqrt(d*x + c)/sqrt(-c))/(a*sqrt(-c)*c)
 

Mupad [B] (verification not implemented)

Time = 10.82 (sec) , antiderivative size = 12456, normalized size of antiderivative = 22.01 \[ \int \frac {1}{x (c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

(atan((b^13*c^21*d^10*(c + d*x)^(1/2)*8640i)/((c^3)^(1/2)*(8640*b^13*c^20* 
d^10 + 47424*a*b^12*c^18*d^12 + 139456*a^2*b^11*c^16*d^14 + 253120*a^3*b^1 
0*c^14*d^16 + 302400*a^4*b^9*c^12*d^18 + 244160*a^5*b^8*c^10*d^20 + 133184 
*a^6*b^7*c^8*d^22 + 47616*a^7*b^6*c^6*d^24 + 10240*a^8*b^5*c^4*d^26 + 1024 
*a^9*b^4*c^2*d^28 + (576*b^14*c^22*d^8)/a)) + (b^14*c^23*d^8*(c + d*x)^(1/ 
2)*576i)/((c^3)^(1/2)*(576*b^14*c^22*d^8 + 8640*a*b^13*c^20*d^10 + 47424*a 
^2*b^12*c^18*d^12 + 139456*a^3*b^11*c^16*d^14 + 253120*a^4*b^10*c^14*d^16 
+ 302400*a^5*b^9*c^12*d^18 + 244160*a^6*b^8*c^10*d^20 + 133184*a^7*b^7*c^8 
*d^22 + 47616*a^8*b^6*c^6*d^24 + 10240*a^9*b^5*c^4*d^26 + 1024*a^10*b^4*c^ 
2*d^28)) + (a*b^12*c^19*d^12*(c + d*x)^(1/2)*47424i)/((c^3)^(1/2)*(8640*b^ 
13*c^20*d^10 + 47424*a*b^12*c^18*d^12 + 139456*a^2*b^11*c^16*d^14 + 253120 
*a^3*b^10*c^14*d^16 + 302400*a^4*b^9*c^12*d^18 + 244160*a^5*b^8*c^10*d^20 
+ 133184*a^6*b^7*c^8*d^22 + 47616*a^7*b^6*c^6*d^24 + 10240*a^8*b^5*c^4*d^2 
6 + 1024*a^9*b^4*c^2*d^28 + (576*b^14*c^22*d^8)/a)) + (a^2*b^11*c^17*d^14* 
(c + d*x)^(1/2)*139456i)/((c^3)^(1/2)*(8640*b^13*c^20*d^10 + 47424*a*b^12* 
c^18*d^12 + 139456*a^2*b^11*c^16*d^14 + 253120*a^3*b^10*c^14*d^16 + 302400 
*a^4*b^9*c^12*d^18 + 244160*a^5*b^8*c^10*d^20 + 133184*a^6*b^7*c^8*d^22 + 
47616*a^7*b^6*c^6*d^24 + 10240*a^8*b^5*c^4*d^26 + 1024*a^9*b^4*c^2*d^28 + 
(576*b^14*c^22*d^8)/a)) + (a^3*b^10*c^15*d^16*(c + d*x)^(1/2)*253120i)/((c 
^3)^(1/2)*(8640*b^13*c^20*d^10 + 47424*a*b^12*c^18*d^12 + 139456*a^2*b^...
 

Reduce [F]

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

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

Output:

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