\(\int \frac {(c+d x)^{3/2}}{x^2 (a-b x^2)} \, dx\) [575]

Optimal result

Integrand size = 23, antiderivative size = 179 \[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )} \, dx=-\frac {c \sqrt {c+d x}}{a x}-\frac {3 \sqrt {c} d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{a^{3/2} \sqrt [4]{b}}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{a^{3/2} \sqrt [4]{b}} \] Output:

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

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.32 \[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )} \, dx=\frac {-\frac {\sqrt {a} c \sqrt {c+d x}}{x}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right )^2 \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {-b c-\sqrt {a} \sqrt {b} d}}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^2 \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {-b c+\sqrt {a} \sqrt {b} d}}-3 \sqrt {a} \sqrt {c} d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {561, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1610

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

2*d*(-1/2*(c*Sqrt[c + d*x])/(a*d*x) - (3*Sqrt[c]*ArcTanh[Sqrt[c + d*x]/Sqr 
t[c]])/(2*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^(3/2)*b^(1/4)*d) + ((Sqrt[b]*c + Sq 
rt[a]*d)^(3/2)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d] 
])/(2*a^(3/2)*b^(1/4)*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 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 = 0.45 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.11

method	result	size
risch	\(-\frac {c \sqrt {d x +c}}{a x}-\frac {d \left (2 b \left (\frac {\left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+3 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )\right )}{a}\)	\(198\)
derivativedivides	\(-2 d^{3} \left (\frac {c \left (\frac {\sqrt {d x +c}}{2 d x}+\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a \,d^{2}}+\frac {b \left (\frac {\left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \,d^{2}}\right )\)	\(209\)
default	\(2 d^{3} \left (-\frac {c \left (\frac {\sqrt {d x +c}}{2 d x}+\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a \,d^{2}}+\frac {b \left (-\frac {\left (-a \,d^{2}-b \,c^{2}-2 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (a \,d^{2}+b \,c^{2}-2 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \,d^{2}}\right )\)	\(210\)
pseudoelliptic	\(-\frac {-\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, d x b \left (a \,d^{2}+b \,c^{2}-2 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\left (-b d x \left (a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (3 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) d x \sqrt {c}+c \sqrt {d x +c}\right ) \sqrt {a b \,d^{2}}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, a x}\)	\(247\)

Input:

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

Output:

-c*(d*x+c)^(1/2)/a/x-1/a*d*(2*b*(1/2*(-a*d^2-b*c^2+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))-1/2*(a*d^2+b*c^2+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)))+3*c^(1/2)*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. 968 vs. \(2 (131) = 262\).

Time = 0.32 (sec) , antiderivative size = 1945, normalized size of antiderivative = 10.87 \[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*(a*x*sqrt((b*c^3 + 3*a*c*d^2 + a^3*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^ 
4 + a^2*d^6)/(a^5*b)))/a^3)*log(-(3*b^2*c^4*d - 2*a*b*c^2*d^3 - a^2*d^5)*s 
qrt(d*x + c) + (3*a^2*b*c^2*d^2 + a^3*d^4 - a^4*b*c*sqrt((9*b^2*c^4*d^2 + 
6*a*b*c^2*d^4 + a^2*d^6)/(a^5*b)))*sqrt((b*c^3 + 3*a*c*d^2 + a^3*sqrt((9*b 
^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6)/(a^5*b)))/a^3)) - a*x*sqrt((b*c^3 + 
3*a*c*d^2 + a^3*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6)/(a^5*b)))/a 
^3)*log(-(3*b^2*c^4*d - 2*a*b*c^2*d^3 - a^2*d^5)*sqrt(d*x + c) - (3*a^2*b* 
c^2*d^2 + a^3*d^4 - a^4*b*c*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6) 
/(a^5*b)))*sqrt((b*c^3 + 3*a*c*d^2 + a^3*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d 
^4 + a^2*d^6)/(a^5*b)))/a^3)) + a*x*sqrt((b*c^3 + 3*a*c*d^2 - a^3*sqrt((9* 
b^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6)/(a^5*b)))/a^3)*log(-(3*b^2*c^4*d - 
2*a*b*c^2*d^3 - a^2*d^5)*sqrt(d*x + c) + (3*a^2*b*c^2*d^2 + a^3*d^4 + a^4* 
b*c*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6)/(a^5*b)))*sqrt((b*c^3 + 
 3*a*c*d^2 - a^3*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6)/(a^5*b)))/ 
a^3)) - a*x*sqrt((b*c^3 + 3*a*c*d^2 - a^3*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2* 
d^4 + a^2*d^6)/(a^5*b)))/a^3)*log(-(3*b^2*c^4*d - 2*a*b*c^2*d^3 - a^2*d^5) 
*sqrt(d*x + c) - (3*a^2*b*c^2*d^2 + a^3*d^4 + a^4*b*c*sqrt((9*b^2*c^4*d^2 
+ 6*a*b*c^2*d^4 + a^2*d^6)/(a^5*b)))*sqrt((b*c^3 + 3*a*c*d^2 - a^3*sqrt((9 
*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6)/(a^5*b)))/a^3)) + 3*sqrt(c)*d*x*lo 
g((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) - 2*sqrt(d*x + c)*c)/(a*x), ...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (131) = 262\).

Time = 0.19 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.16 \[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )} \, dx=\frac {3 \, c d \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a \sqrt {-c}} - \frac {\sqrt {d x + c} c}{a x} + \frac {{\left (2 \, \sqrt {a b} a^{2} c d^{3} {\left | b \right |} - {\left (a b c^{2} d - a^{2} d^{3}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |} - {\left (\sqrt {a b} a b c^{3} d + \sqrt {a b} a^{2} c d^{3}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b c + \sqrt {a^{2} b^{2} c^{2} - {\left (a b c^{2} - a^{2} d^{2}\right )} a b}}{a b}}}\right )}{{\left (a^{2} b c - \sqrt {a b} a^{2} d\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} - \frac {{\left (2 \, a^{2} b c d^{3} {\left | b \right |} + {\left (\sqrt {a b} b c^{2} d - \sqrt {a b} a d^{3}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |} - {\left (a b^{2} c^{3} d + a^{2} b c d^{3}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b c - \sqrt {a^{2} b^{2} c^{2} - {\left (a b c^{2} - a^{2} d^{2}\right )} a b}}{a b}}}\right )}{{\left (a^{2} b d + \sqrt {a b} a b c\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 7.47 (sec) , antiderivative size = 3762, normalized size of antiderivative = 21.02 \[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((b^6*c^6*d^10*(c + d*x)^(1/2)*((b*c^3)/(4*a^3) + (3*c*d^2)/(4*a^2) - 
(3*c^2*d*(a^7*b)^(1/2))/(4*a^6) - (d^3*(a^7*b)^(1/2))/(4*a^5*b))^(1/2)*192 
i)/(16*a^2*b^3*c*d^17 - 16*a*b^4*c^3*d^15 - 128*b^5*c^5*d^13 + (176*b^6*c^ 
7*d^11)/a - (48*b^7*c^9*d^9)/a^2 + (80*b^3*c^2*d^16*(a^7*b)^(1/2))/a^2 + ( 
96*b^4*c^4*d^14*(a^7*b)^(1/2))/a^3 - (368*b^5*c^6*d^12*(a^7*b)^(1/2))/a^4 
+ (192*b^6*c^8*d^10*(a^7*b)^(1/2))/a^5) + (b^5*c^4*d^12*(c + d*x)^(1/2)*(( 
b*c^3)/(4*a^3) + (3*c*d^2)/(4*a^2) - (3*c^2*d*(a^7*b)^(1/2))/(4*a^6) - (d^ 
3*(a^7*b)^(1/2))/(4*a^5*b))^(1/2)*736i)/((176*b^6*c^7*d^11)/a^2 - (128*b^5 
*c^5*d^13)/a - 16*b^4*c^3*d^15 - (48*b^7*c^9*d^9)/a^3 + 16*a*b^3*c*d^17 + 
(80*b^3*c^2*d^16*(a^7*b)^(1/2))/a^3 + (96*b^4*c^4*d^14*(a^7*b)^(1/2))/a^4 
- (368*b^5*c^6*d^12*(a^7*b)^(1/2))/a^5 + (192*b^6*c^8*d^10*(a^7*b)^(1/2))/ 
a^6) - (b^6*c^7*d^9*(a^7*b)^(1/2)*(c + d*x)^(1/2)*((b*c^3)/(4*a^3) + (3*c* 
d^2)/(4*a^2) - (3*c^2*d*(a^7*b)^(1/2))/(4*a^6) - (d^3*(a^7*b)^(1/2))/(4*a^ 
5*b))^(1/2)*96i)/(16*a^6*b^3*c*d^17 - 48*a^2*b^7*c^9*d^9 + 176*a^3*b^6*c^7 
*d^11 - 128*a^4*b^5*c^5*d^13 - 16*a^5*b^4*c^3*d^15 - 368*b^5*c^6*d^12*(a^7 
*b)^(1/2) + 80*a^2*b^3*c^2*d^16*(a^7*b)^(1/2) + (192*b^6*c^8*d^10*(a^7*b)^ 
(1/2))/a + 96*a*b^4*c^4*d^14*(a^7*b)^(1/2)) + (b^4*c^3*d^13*(a^7*b)^(1/2)* 
(c + d*x)^(1/2)*((b*c^3)/(4*a^3) + (3*c*d^2)/(4*a^2) - (3*c^2*d*(a^7*b)^(1 
/2))/(4*a^6) - (d^3*(a^7*b)^(1/2))/(4*a^5*b))^(1/2)*384i)/(176*a*b^6*c^7*d 
^11 - 48*b^7*c^9*d^9 + 16*a^4*b^3*c*d^17 - 128*a^2*b^5*c^5*d^13 - 16*a^...
 

Reduce [B] (verification not implemented)

Time = 0.71 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.76 \[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )} \, dx=\frac {-2 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) b c x +2 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) a d x -\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b c x +\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b c x -\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a d x +\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a d x -2 \sqrt {d x +c}\, a b c +3 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a b d x -3 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a b d x}{2 a^{2} b x} \] Input:

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