\(\int \frac {1}{x (c+d x)^{3/2} (a-b x^2)} \, dx\) [596]

Optimal result

Integrand size = 23, antiderivative size = 186 \[ \int \frac {1}{x (c+d x)^{3/2} \left (a-b x^2\right )} \, dx=-\frac {2 d^2}{c \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a c^{3/2}}+\frac {b^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{a \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2}}+\frac {b^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{a \left (\sqrt {b} c+\sqrt {a} d\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x (c+d x)^{3/2} \left (a-b x^2\right )} \, dx=-\frac {2 d^2}{\left (b c^3-a c d^2\right ) \sqrt {c+d x}}+\frac {b \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\left (a \sqrt {b} c+a^{3/2} d\right ) \sqrt {-b c-\sqrt {a} \sqrt {b} d}}+\frac {b \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\left (a \sqrt {b} c-a^{3/2} d\right ) \sqrt {-b c+\sqrt {a} \sqrt {b} d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a c^{3/2}} \] 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) - Sqrt 
[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt[a]*d)])/((a*Sqrt[b]*c + a^ 
(3/2)*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)])/((a*Sqrt[b]*c - a^ 
(3/2)*d)*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]) - (2*ArcTanh[Sqrt[c + d*x]/Sqrt 
[c]])/(a*c^(3/2))
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, 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

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

\(\Big \downarrow \) 2009

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

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

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 = 0.43 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.19

method	result	size
derivativedivides	\(-2 d^{2} \left (\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{c^{\frac {3}{2}} a \,d^{2}}+\frac {b^{2} \left (-\frac {\left (a \,d^{2}-\sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-a \,d^{2}-\sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \,d^{2} \left (a \,d^{2}-b \,c^{2}\right )}-\frac {1}{c \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {d x +c}}\right )\)	\(221\)
default	\(2 d^{2} \left (-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{c^{\frac {3}{2}} a \,d^{2}}-\frac {b^{2} \left (-\frac {\left (a \,d^{2}-\sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-a \,d^{2}-\sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \,d^{2} \left (a \,d^{2}-b \,c^{2}\right )}+\frac {1}{c \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {d x +c}}\right )\)	\(222\)
pseudoelliptic	\(-\frac {-b^{2} \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {d x +c}\, \left (a \,d^{2}+\sqrt {a b \,d^{2}}\, c \right ) c^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\left (\left (-a \,d^{2}+\sqrt {a b \,d^{2}}\, c \right ) b^{2} \sqrt {d x +c}\, c^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-2 \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-\sqrt {d x +c}\, c \left (a \,d^{2}-b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+c^{\frac {3}{2}} a \,d^{2}\right ) \sqrt {a b \,d^{2}}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}{\sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {d x +c}\, \left (a \,d^{2}-b \,c^{2}\right ) a \,c^{\frac {5}{2}}}\)	\(291\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3119 vs. \(2 (142) = 284\).

Time = 0.78 (sec) , antiderivative size = 6247, normalized size of antiderivative = 33.59 \[ \int \frac {1}{x (c+d x)^{3/2} \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {1}{x (c+d x)^{3/2} \left (a-b x^2\right )} \, dx=- \int \frac {1}{- a c x \sqrt {c + d x} - a d x^{2} \sqrt {c + d x} + b c x^{3} \sqrt {c + d x} + b d x^{4} \sqrt {c + d x}}\, dx \] Input:

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

Output:

-Integral(1/(-a*c*x*sqrt(c + d*x) - a*d*x**2*sqrt(c + d*x) + b*c*x**3*sqrt 
(c + d*x) + b*d*x**4*sqrt(c + d*x)), 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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 770 vs. \(2 (142) = 284\).

Time = 0.21 (sec) , antiderivative size = 770, normalized size of antiderivative = 4.14 \[ \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} \sqrt {-b^{2} c - \sqrt {a b} b d} \sqrt {a b} c {\left | b \right |} + {\left (a b^{2} c^{4} - a^{3} d^{4}\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | -a b c^{2} d + a^{2} d^{3} \right |} {\left | b \right |} + {\left (\sqrt {a b} a^{2} b^{2} c^{5} d^{2} - 2 \, \sqrt {a b} a^{3} b c^{3} d^{4} + \sqrt {a b} a^{4} c d^{6}\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\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^{4} c^{6} - 3 \, a^{3} b^{3} c^{4} d^{2} + 3 \, a^{4} b^{2} c^{2} d^{4} - a^{5} b d^{6}\right )} {\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*sqrt(- 
b^2*c - sqrt(a*b)*b*d)*sqrt(a*b)*c*abs(b) + (a*b^2*c^4 - a^3*d^4)*sqrt(-b^ 
2*c - sqrt(a*b)*b*d)*abs(-a*b*c^2*d + a^2*d^3)*abs(b) + (sqrt(a*b)*a^2*b^2 
*c^5*d^2 - 2*sqrt(a*b)*a^3*b*c^3*d^4 + sqrt(a*b)*a^4*c*d^6)*sqrt(-b^2*c - 
sqrt(a*b)*b*d)*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^4*c^6 - 3* 
a^3*b^3*c^4*d^2 + 3*a^4*b^2*c^2*d^4 - a^5*b*d^6)*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 = 10.68 (sec) , antiderivative size = 12415, normalized size of antiderivative = 66.75 \[ \int \frac {1}{x (c+d x)^{3/2} \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(-(((c + d*x)^(1/2)*(96*b^14*c^23*d^8 - 608*a*b^13*c^21*d^10 + 1568*a^ 
2*b^12*c^19*d^12 - 2016*a^3*b^11*c^17*d^14 + 1120*a^4*b^10*c^15*d^16 + 224 
*a^5*b^9*c^13*d^18 - 672*a^6*b^8*c^11*d^20 + 352*a^7*b^7*c^9*d^22 - 64*a^8 
*b^6*c^7*d^24) + (-(a^2*b^3*c^3 + a*d^3*(a^5*b^3)^(1/2) + 3*a^3*b^2*c*d^2 
+ 3*b*c^2*d*(a^5*b^3)^(1/2))/(4*(a^7*d^6 - a^4*b^3*c^6 - 3*a^6*b*c^2*d^4 + 
 3*a^5*b^2*c^4*d^2)))^(1/2)*(96*a*b^14*c^25*d^8 - ((c + d*x)^(1/2)*(576*a^ 
2*b^14*c^26*d^8 - 3712*a^3*b^13*c^24*d^10 + 10112*a^4*b^12*c^22*d^12 - 152 
32*a^5*b^11*c^20*d^14 + 14336*a^6*b^10*c^18*d^16 - 9856*a^7*b^9*c^16*d^18 
+ 6272*a^8*b^8*c^14*d^20 - 3712*a^9*b^7*c^12*d^22 + 1472*a^10*b^6*c^10*d^2 
4 - 256*a^11*b^5*c^8*d^26) - (-(a^2*b^3*c^3 + a*d^3*(a^5*b^3)^(1/2) + 3*a^ 
3*b^2*c*d^2 + 3*b*c^2*d*(a^5*b^3)^(1/2))/(4*(a^7*d^6 - a^4*b^3*c^6 - 3*a^6 
*b*c^2*d^4 + 3*a^5*b^2*c^4*d^2)))^(1/2)*((c + d*x)^(1/2)*(-(a^2*b^3*c^3 + 
a*d^3*(a^5*b^3)^(1/2) + 3*a^3*b^2*c*d^2 + 3*b*c^2*d*(a^5*b^3)^(1/2))/(4*(a 
^7*d^6 - a^4*b^3*c^6 - 3*a^6*b*c^2*d^4 + 3*a^5*b^2*c^4*d^2)))^(1/2)*(768*a 
^4*b^14*c^29*d^8 - 7424*a^5*b^13*c^27*d^10 + 32256*a^6*b^12*c^25*d^12 - 82 
944*a^7*b^11*c^23*d^14 + 139776*a^8*b^10*c^21*d^16 - 161280*a^9*b^9*c^19*d 
^18 + 129024*a^10*b^8*c^17*d^20 - 70656*a^11*b^7*c^15*d^22 + 25344*a^12*b^ 
6*c^13*d^24 - 5376*a^13*b^5*c^11*d^26 + 512*a^14*b^4*c^9*d^28) - 384*a^3*b 
^14*c^28*d^8 + 4224*a^4*b^13*c^26*d^10 - 20480*a^5*b^12*c^24*d^12 + 57856* 
a^6*b^11*c^22*d^14 - 105728*a^7*b^10*c^20*d^16 + 130816*a^8*b^9*c^18*d^...
 

Reduce [B] (verification not implemented)

Time = 15.58 (sec) , antiderivative size = 686, normalized size of antiderivative = 3.69 \[ \int \frac {1}{x (c+d x)^{3/2} \left (a-b x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x) 
*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*b*c**3*d + 2*sqrt(b)*sqrt(c + 
 d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(s 
qrt(b)*sqrt(a)*d - b*c)))*a*c**2*d**2 + 2*sqrt(b)*sqrt(c + d*x)*sqrt(sqrt( 
b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d 
 - b*c)))*b*c**4 + 2*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*l 
og( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*b*c**3*d - 2* 
sqrt(a)*sqrt(c + d*x)*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*d - sqrt(b)*sqrt(c + d*x)*sqrt 
(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*s 
qrt(c + d*x))*a*c**2*d**2 - sqrt(b)*sqrt(c + d*x)*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**4 
 + sqrt(b)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log(sqrt(sqrt(b)*sq 
rt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a*c**2*d**2 + sqrt(b)*sqrt(c + d*x 
)*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**4 + 2*sqrt(c)*sqrt(c + d*x)*log(sqrt(c + d*x) - sqrt 
(c))*a**2*d**4 - 4*sqrt(c)*sqrt(c + d*x)*log(sqrt(c + d*x) - sqrt(c))*a*b* 
c**2*d**2 + 2*sqrt(c)*sqrt(c + d*x)*log(sqrt(c + d*x) - sqrt(c))*b**2*c**4 
 - 2*sqrt(c)*sqrt(c + d*x)*log(sqrt(c + d*x) + sqrt(c))*a**2*d**4 + 4*sqrt 
(c)*sqrt(c + d*x)*log(sqrt(c + d*x) + sqrt(c))*a*b*c**2*d**2 - 2*sqrt(c...