\(\int \frac {(c+d x)^{3/2}}{x (a-b x^2)} \, dx\) [574]

Optimal result

Integrand size = 23, antiderivative size = 155 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )} \, dx=-\frac {2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a}+\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{a b^{3/4}}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{a b^{3/4}} \] Output:

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

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.39 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )} \, dx=\frac {-\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {-b c-\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{b}+\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^2 \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {b} \sqrt {-b c+\sqrt {a} \sqrt {b} d}}-2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.05, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1610

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 0.40 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.42

method	result	size
derivativedivides	\(-2 d^{2} \left (\frac {b \left (-\frac {\left (2 a b c \,d^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-2 a b c \,d^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \,d^{2}}+\frac {c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a \,d^{2}}\right )\)	\(220\)
default	\(2 d^{2} \left (\frac {b \left (\frac {\left (2 a b c \,d^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-2 a b c \,d^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \,d^{2}}-\frac {c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a \,d^{2}}\right )\)	\(225\)
pseudoelliptic	\(-\frac {2 \left (-\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\frac {\left (-a \,d^{2}-b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{2}+a b c \,d^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\left (\frac {\left (-a \,d^{2}-b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{2}-a b c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+c^{\frac {3}{2}} \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) \sqrt {a b \,d^{2}}\right )\right )}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, a}\)	\(242\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 978 vs. \(2 (113) = 226\).

Time = 0.37 (sec) , antiderivative size = 1964, normalized size of antiderivative = 12.67 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/2*(a*sqrt((b*c^3 + 3*a*c*d^2 + a^2*b*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d 
^4 + a^2*d^6)/(a^3*b^3)))/(a^2*b))*log(-(3*b^2*c^4 - 2*a*b*c^2*d^2 - a^2*d 
^4)*sqrt(d*x + c) + (3*a*b^2*c^3 + a^2*b*c*d^2 - a^3*b^2*sqrt((9*b^2*c^4*d 
^2 + 6*a*b*c^2*d^4 + a^2*d^6)/(a^3*b^3)))*sqrt((b*c^3 + 3*a*c*d^2 + a^2*b* 
sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6)/(a^3*b^3)))/(a^2*b))) - a*s 
qrt((b*c^3 + 3*a*c*d^2 + a^2*b*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d 
^6)/(a^3*b^3)))/(a^2*b))*log(-(3*b^2*c^4 - 2*a*b*c^2*d^2 - a^2*d^4)*sqrt(d 
*x + c) - (3*a*b^2*c^3 + a^2*b*c*d^2 - a^3*b^2*sqrt((9*b^2*c^4*d^2 + 6*a*b 
*c^2*d^4 + a^2*d^6)/(a^3*b^3)))*sqrt((b*c^3 + 3*a*c*d^2 + a^2*b*sqrt((9*b^ 
2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6)/(a^3*b^3)))/(a^2*b))) + a*sqrt((b*c^3 
 + 3*a*c*d^2 - a^2*b*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6)/(a^3*b 
^3)))/(a^2*b))*log(-(3*b^2*c^4 - 2*a*b*c^2*d^2 - a^2*d^4)*sqrt(d*x + c) + 
(3*a*b^2*c^3 + a^2*b*c*d^2 + a^3*b^2*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + 
 a^2*d^6)/(a^3*b^3)))*sqrt((b*c^3 + 3*a*c*d^2 - a^2*b*sqrt((9*b^2*c^4*d^2 
+ 6*a*b*c^2*d^4 + a^2*d^6)/(a^3*b^3)))/(a^2*b))) - a*sqrt((b*c^3 + 3*a*c*d 
^2 - a^2*b*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6)/(a^3*b^3)))/(a^2 
*b))*log(-(3*b^2*c^4 - 2*a*b*c^2*d^2 - a^2*d^4)*sqrt(d*x + c) - (3*a*b^2*c 
^3 + a^2*b*c*d^2 + a^3*b^2*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6)/ 
(a^3*b^3)))*sqrt((b*c^3 + 3*a*c*d^2 - a^2*b*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^ 
2*d^4 + a^2*d^6)/(a^3*b^3)))/(a^2*b))) - 2*c^(3/2)*log((d*x - 2*sqrt(d*...
 

Sympy [F]

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

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

Output:

-Integral(c*sqrt(c + d*x)/(-a*x + b*x**3), x) - Integral(d*x*sqrt(c + d*x) 
/(-a*x + b*x**3), x)
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (113) = 226\).

Time = 0.18 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.46 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )} \, dx=\frac {2 \, c^{2} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a \sqrt {-c}} - \frac {{\left (2 \, \sqrt {a b} a^{2} b c^{2} d^{2} {\left | b \right |} - {\left (\sqrt {a b} b c^{2} + \sqrt {a b} a d^{2}\right )} a^{2} d^{2} {\left | b \right |} + {\left (a b^{2} c^{3} - a^{2} b c d^{2}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b c + \sqrt {a^{2} b^{2} c^{2} - {\left (a b c^{2} - a^{2} d^{2}\right )} a b}}{a b}}}\right )}{{\left (a^{2} b^{2} c - \sqrt {a b} a^{2} b d\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} + \frac {{\left (2 \, a^{2} b c^{2} d^{2} {\left | b \right |} - {\left (b c^{2} + a d^{2}\right )} a^{2} d^{2} {\left | b \right |} - {\left (\sqrt {a b} b c^{3} - \sqrt {a b} a c d^{2}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b c - \sqrt {a^{2} b^{2} c^{2} - {\left (a b c^{2} - a^{2} d^{2}\right )} a b}}{a b}}}\right )}{{\left (a^{2} b d + \sqrt {a b} a b c\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} \] Input:

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

Output:

2*c^2*arctan(sqrt(d*x + c)/sqrt(-c))/(a*sqrt(-c)) - (2*sqrt(a*b)*a^2*b*c^2 
*d^2*abs(b) - (sqrt(a*b)*b*c^2 + sqrt(a*b)*a*d^2)*a^2*d^2*abs(b) + (a*b^2* 
c^3 - a^2*b*c*d^2)*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)))/((a^2*b^2*c - sqrt 
(a*b)*a^2*b*d)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(a)*abs(d)) + (2*a^2*b*c^2* 
d^2*abs(b) - (b*c^2 + a*d^2)*a^2*d^2*abs(b) - (sqrt(a*b)*b*c^3 - sqrt(a*b) 
*a*c*d^2)*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)))/((a^2*b*d + sqrt(a*b)*a*b*c 
)*sqrt(-b^2*c + sqrt(a*b)*b*d)*abs(a)*abs(d))
 

Mupad [B] (verification not implemented)

Time = 8.12 (sec) , antiderivative size = 3387, normalized size of antiderivative = 21.85 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

2*atanh((32*a*c*d^15*(a^5*b^3)^(1/2)*(c + d*x)^(1/2)*(c^3/(4*a^2) + (3*c*d 
^2)/(4*a*b) + (d^3*(a^5*b^3)^(1/2))/(4*a^3*b^3) + (3*c^2*d*(a^5*b^3)^(1/2) 
)/(4*a^4*b^2))^(1/2))/(128*c^3*d^15*(a^5*b^3)^(1/2) - 192*a*b^3*c^6*d^12 + 
 64*a^3*b*c^2*d^16 + 128*a^2*b^2*c^4*d^14 + (96*b*c^5*d^13*(a^5*b^3)^(1/2) 
)/a - (384*b^2*c^7*d^11*(a^5*b^3)^(1/2))/a^2 + (144*b^3*c^9*d^9*(a^5*b^3)^ 
(1/2))/a^3 + (16*a*c*d^17*(a^5*b^3)^(1/2))/b) - (576*a*b^4*c^6*d^10*(c + d 
*x)^(1/2)*(c^3/(4*a^2) + (3*c*d^2)/(4*a*b) + (d^3*(a^5*b^3)^(1/2))/(4*a^3* 
b^3) + (3*c^2*d*(a^5*b^3)^(1/2))/(4*a^4*b^2))^(1/2))/(128*c^3*d^15*(a^5*b^ 
3)^(1/2) - 192*a*b^3*c^6*d^12 + 64*a^3*b*c^2*d^16 + 128*a^2*b^2*c^4*d^14 + 
 (96*b*c^5*d^13*(a^5*b^3)^(1/2))/a - (384*b^2*c^7*d^11*(a^5*b^3)^(1/2))/a^ 
2 + (144*b^3*c^9*d^9*(a^5*b^3)^(1/2))/a^3 + (16*a*c*d^17*(a^5*b^3)^(1/2))/ 
b) + (96*b^2*c^5*d^11*(a^5*b^3)^(1/2)*(c + d*x)^(1/2)*(c^3/(4*a^2) + (3*c* 
d^2)/(4*a*b) + (d^3*(a^5*b^3)^(1/2))/(4*a^3*b^3) + (3*c^2*d*(a^5*b^3)^(1/2 
))/(4*a^4*b^2))^(1/2))/(64*a^4*b*c^2*d^16 - 192*a^2*b^3*c^6*d^12 + 128*a^3 
*b^2*c^4*d^14 + 128*a*c^3*d^15*(a^5*b^3)^(1/2) + 96*b*c^5*d^13*(a^5*b^3)^( 
1/2) + (16*a^2*c*d^17*(a^5*b^3)^(1/2))/b - (384*b^2*c^7*d^11*(a^5*b^3)^(1/ 
2))/a + (144*b^3*c^9*d^9*(a^5*b^3)^(1/2))/a^2) + (288*b^3*c^7*d^9*(a^5*b^3 
)^(1/2)*(c + d*x)^(1/2)*(c^3/(4*a^2) + (3*c*d^2)/(4*a*b) + (d^3*(a^5*b^3)^ 
(1/2))/(4*a^3*b^3) + (3*c^2*d*(a^5*b^3)^(1/2))/(4*a^4*b^2))^(1/2))/(64*a^5 
*b*c^2*d^16 - 192*a^3*b^3*c^6*d^12 + 128*a^4*b^2*c^4*d^14 + 128*a^2*c^3...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.84 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )} \, dx=\frac {-2 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) d +2 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) c -\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) d +\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) d -\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) c +\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) c +2 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) b c -2 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) b c}{2 a b} \] Input:

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