\(\int \frac {(c+d x)^{3/2}}{x (a-b x^2)^2} \, dx\) [651]

Optimal result

Integrand size = 23, antiderivative size = 220 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^2} \, dx=\frac {(c+d x)^{3/2}}{2 a \left (a-b x^2\right )}-\frac {2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2}+\frac {\sqrt {\sqrt {b} c-\sqrt {a} d} \left (4 \sqrt {b} c-\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{4 a^2 b^{3/4}}+\frac {\sqrt {\sqrt {b} c+\sqrt {a} d} \left (4 \sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{4 a^2 b^{3/4}} \] Output:

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

Mathematica [A] (verified)

Time = 1.43 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.08 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^2} \, dx=-\frac {\frac {2 a (c+d x)^{3/2}}{-a+b x^2}+\frac {\left (4 \sqrt {b} c+\sqrt {a} d\right ) \sqrt {-b c-\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{b}+\frac {\left (4 \sqrt {b} c-\sqrt {a} d\right ) \sqrt {-b c+\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{b}+8 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 a^2} \] Input:

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

Output:

-1/4*((2*a*(c + d*x)^(3/2))/(-a + b*x^2) + ((4*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 + ((4*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 + 8*c^(3/2)*ArcTanh[Sqrt[c + d*x]/Sqr 
t[c]])/a^2
 

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.83, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {561, 25, 27, 1650, 1484, 1492, 27, 1480, 221, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1650

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

\(\Big \downarrow \) 1484

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

\(\Big \downarrow \) 1492

\(\displaystyle -2 \left (\frac {c^2 \int \left (-\frac {b d x}{a \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )}-\frac {1}{a d x}\right )d\sqrt {c+d x}}{a}-\frac {\frac {d^4 \int \frac {2 a b \left (b c^2-a d^2\right ) (5 c+d x)}{d^4 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}+\frac {(c+d x)^{3/2}}{4 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{a}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -2 \left (\frac {c^2 \int \left (-\frac {b d x}{a \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )}-\frac {1}{a d x}\right )d\sqrt {c+d x}}{a}-\frac {\frac {1}{4} \int \frac {5 c+d x}{-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}d\sqrt {c+d x}+\frac {(c+d x)^{3/2}}{4 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{a}\right )\)

\(\Big \downarrow \) 1480

\(\displaystyle -2 \left (\frac {c^2 \int \left (-\frac {b d x}{a \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )}-\frac {1}{a d x}\right )d\sqrt {c+d x}}{a}-\frac {\frac {1}{4} \left (\frac {1}{2} \left (1-\frac {5 \sqrt {b} c}{\sqrt {a} d}\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}+\frac {1}{2} \left (\frac {5 \sqrt {b} c}{\sqrt {a} d}+1\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}\right )+\frac {(c+d x)^{3/2}}{4 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{a}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle -2 \left (\frac {c^2 \int \left (-\frac {b d x}{a \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )}-\frac {1}{a d x}\right )d\sqrt {c+d x}}{a}-\frac {\frac {1}{4} \left (\frac {d^2 \left (1-\frac {5 \sqrt {b} c}{\sqrt {a} d}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 b^{3/4} \sqrt {\sqrt {b} c-\sqrt {a} d}}+\frac {d^2 \left (\frac {5 \sqrt {b} c}{\sqrt {a} d}+1\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 b^{3/4} \sqrt {\sqrt {a} d+\sqrt {b} c}}\right )+\frac {(c+d x)^{3/2}}{4 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{a}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (\frac {c^2 \left (-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 a \sqrt {\sqrt {b} c-\sqrt {a} d}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 a \sqrt {\sqrt {a} d+\sqrt {b} c}}+\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a \sqrt {c}}\right )}{a}-\frac {\frac {1}{4} \left (\frac {d^2 \left (1-\frac {5 \sqrt {b} c}{\sqrt {a} d}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 b^{3/4} \sqrt {\sqrt {b} c-\sqrt {a} d}}+\frac {d^2 \left (\frac {5 \sqrt {b} c}{\sqrt {a} d}+1\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 b^{3/4} \sqrt {\sqrt {a} d+\sqrt {b} c}}\right )+\frac {(c+d x)^{3/2}}{4 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{a}\right )\)

Input:

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

Output:

-2*((c^2*(ArcTanh[Sqrt[c + d*x]/Sqrt[c]]/(a*Sqrt[c]) - (b^(1/4)*ArcTanh[(b 
^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*a*Sqrt[Sqrt[b]*c - 
Sqrt[a]*d]) - (b^(1/4)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sq 
rt[a]*d]])/(2*a*Sqrt[Sqrt[b]*c + Sqrt[a]*d])))/a - ((c + d*x)^(3/2)/(4*(a 
- (b*c^2)/d^2 + (2*b*c*(c + d*x))/d^2 - (b*(c + d*x)^2)/d^2)) + (((1 - (5* 
Sqrt[b]*c)/(Sqrt[a]*d))*d^2*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c 
 - Sqrt[a]*d]])/(2*b^(3/4)*Sqrt[Sqrt[b]*c - Sqrt[a]*d]) + ((1 + (5*Sqrt[b] 
*c)/(Sqrt[a]*d))*d^2*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt 
[a]*d]])/(2*b^(3/4)*Sqrt[Sqrt[b]*c + Sqrt[a]*d]))/4)/a)
 

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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

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 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

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 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + 
 c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 
 - 4*a*c))   Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 LtQ[p, -1] && IntegerQ[2*p]
 

rule 1650

Int[(((f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_))/((d_) + 
 (e_.)*(x_)^2), x_Symbol] :> Simp[-f^4/(c*d^2 - b*d*e + a*e^2)   Int[(f*x)^ 
(m - 4)*(a*d + (b*d - a*e)*x^2)*(a + b*x^2 + c*x^4)^p, x], x] + Simp[d^2*(f 
^4/(c*d^2 - b*d*e + a*e^2))   Int[(f*x)^(m - 4)*((a + b*x^2 + c*x^4)^(p + 1 
)/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 
0] && LtQ[p, -1] && GtQ[m, 2]
 

rule 2009

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

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.25

method	result	size
derivativedivides	\(2 d^{4} \left (-\frac {c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{2} d^{4}}+\frac {\frac {a \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{-4 b \left (d x +c \right )^{2}+8 b c \left (d x +c \right )+4 a \,d^{2}-4 b \,c^{2}}+\frac {b \left (\frac {\left (5 a b c \,d^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-4 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-5 a b c \,d^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-4 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}}{a^{2} d^{4}}\right )\)	\(274\)
default	\(2 d^{4} \left (-\frac {c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{2} d^{4}}+\frac {\frac {a \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{-4 b \left (d x +c \right )^{2}+8 b c \left (d x +c \right )+4 a \,d^{2}-4 b \,c^{2}}+\frac {b \left (\frac {\left (5 a b c \,d^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-4 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-5 a b c \,d^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-4 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}}{a^{2} d^{4}}\right )\)	\(274\)
pseudoelliptic	\(\frac {\frac {5 \left (\frac {\left (-a \,d^{2}-4 b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{5}+a b c \,d^{2}\right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-b \,x^{2}+a \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}+\frac {5 \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\left (\frac {\left (a \,d^{2}+4 b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{5}+a b c \,d^{2}\right ) \left (-b \,x^{2}+a \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\frac {2 \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \left (-4 c^{\frac {3}{2}} \left (-b \,x^{2}+a \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+a \left (d x +c \right )^{\frac {3}{2}}\right )}{5}\right )}{4}}{a^{2} \left (-b \,x^{2}+a \right ) \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\)	\(286\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1088 vs. \(2 (163) = 326\).

Time = 0.41 (sec) , antiderivative size = 2185, normalized size of antiderivative = 9.93 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/8*((a^2*b*x^2 - a^3)*sqrt((a^4*b*sqrt((576*b^2*c^4*d^2 + 48*a*b*c^2*d^4 
 + a^2*d^6)/(a^7*b^3)) + 16*b*c^3 + 9*a*c*d^2)/(a^4*b))*log(-(384*b^2*c^4 
- 8*a*b*c^2*d^2 - a^2*d^4)*sqrt(d*x + c) + (a^6*b^2*sqrt((576*b^2*c^4*d^2 
+ 48*a*b*c^2*d^4 + a^2*d^6)/(a^7*b^3)) - 96*a^2*b^2*c^3 - 4*a^3*b*c*d^2)*s 
qrt((a^4*b*sqrt((576*b^2*c^4*d^2 + 48*a*b*c^2*d^4 + a^2*d^6)/(a^7*b^3)) + 
16*b*c^3 + 9*a*c*d^2)/(a^4*b))) - (a^2*b*x^2 - a^3)*sqrt((a^4*b*sqrt((576* 
b^2*c^4*d^2 + 48*a*b*c^2*d^4 + a^2*d^6)/(a^7*b^3)) + 16*b*c^3 + 9*a*c*d^2) 
/(a^4*b))*log(-(384*b^2*c^4 - 8*a*b*c^2*d^2 - a^2*d^4)*sqrt(d*x + c) - (a^ 
6*b^2*sqrt((576*b^2*c^4*d^2 + 48*a*b*c^2*d^4 + a^2*d^6)/(a^7*b^3)) - 96*a^ 
2*b^2*c^3 - 4*a^3*b*c*d^2)*sqrt((a^4*b*sqrt((576*b^2*c^4*d^2 + 48*a*b*c^2* 
d^4 + a^2*d^6)/(a^7*b^3)) + 16*b*c^3 + 9*a*c*d^2)/(a^4*b))) - (a^2*b*x^2 - 
 a^3)*sqrt(-(a^4*b*sqrt((576*b^2*c^4*d^2 + 48*a*b*c^2*d^4 + a^2*d^6)/(a^7* 
b^3)) - 16*b*c^3 - 9*a*c*d^2)/(a^4*b))*log(-(384*b^2*c^4 - 8*a*b*c^2*d^2 - 
 a^2*d^4)*sqrt(d*x + c) + (a^6*b^2*sqrt((576*b^2*c^4*d^2 + 48*a*b*c^2*d^4 
+ a^2*d^6)/(a^7*b^3)) + 96*a^2*b^2*c^3 + 4*a^3*b*c*d^2)*sqrt(-(a^4*b*sqrt( 
(576*b^2*c^4*d^2 + 48*a*b*c^2*d^4 + a^2*d^6)/(a^7*b^3)) - 16*b*c^3 - 9*a*c 
*d^2)/(a^4*b))) + (a^2*b*x^2 - a^3)*sqrt(-(a^4*b*sqrt((576*b^2*c^4*d^2 + 4 
8*a*b*c^2*d^4 + a^2*d^6)/(a^7*b^3)) - 16*b*c^3 - 9*a*c*d^2)/(a^4*b))*log(- 
(384*b^2*c^4 - 8*a*b*c^2*d^2 - a^2*d^4)*sqrt(d*x + c) - (a^6*b^2*sqrt((576 
*b^2*c^4*d^2 + 48*a*b*c^2*d^4 + a^2*d^6)/(a^7*b^3)) + 96*a^2*b^2*c^3 + ...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (163) = 326\).

Time = 0.24 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.95 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^2} \, dx=-\frac {{\left (d x + c\right )}^{\frac {3}{2}} d^{2}}{2 \, {\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} - a d^{2}\right )} a} + \frac {2 \, c^{2} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c}} - \frac {{\left (5 \, \sqrt {a b} b c^{2} d^{2} {\left | b \right |} - {\left (4 \, \sqrt {a b} b c^{2} + \sqrt {a b} a d^{2}\right )} d^{2} {\left | b \right |} + 4 \, {\left (b^{2} c^{3} - a b c d^{2}\right )} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a^{2} b c + \sqrt {a^{4} b^{2} c^{2} - {\left (a^{2} b c^{2} - a^{3} d^{2}\right )} a^{2} b}}{a^{2} b}}}\right )}{4 \, {\left (a^{2} b^{2} c - \sqrt {a b} a^{2} b d\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | d \right |}} + \frac {{\left (5 \, \sqrt {a b} b c^{2} d^{2} {\left | b \right |} - {\left (4 \, \sqrt {a b} b c^{2} + \sqrt {a b} a d^{2}\right )} d^{2} {\left | b \right |} - 4 \, {\left (b^{2} c^{3} - a b c d^{2}\right )} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a^{2} b c - \sqrt {a^{4} b^{2} c^{2} - {\left (a^{2} b c^{2} - a^{3} d^{2}\right )} a^{2} b}}{a^{2} b}}}\right )}{4 \, {\left (a^{2} b^{2} c + \sqrt {a b} a^{2} b d\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | d \right |}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 8.87 (sec) , antiderivative size = 4123, normalized size of antiderivative = 18.74 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(atan((b*d^16*(c^3)^(1/2)*(c + d*x)^(1/2)*1i)/(4*((b*c^2*d^16)/4 - (17*b^2 
*c^4*d^14)/(2*a) + (1473*b^3*c^6*d^12)/(4*a^2) - (936*b^4*c^8*d^10)/a^3 + 
(576*b^5*c^10*d^8)/a^4)) - (b^2*c^2*d^14*(c^3)^(1/2)*(c + d*x)^(1/2)*17i)/ 
(2*((1473*b^3*c^6*d^12)/(4*a) - (17*b^2*c^4*d^14)/2 - (936*b^4*c^8*d^10)/a 
^2 + (576*b^5*c^10*d^8)/a^3 + (a*b*c^2*d^16)/4)) + (b^3*c^4*d^12*(c^3)^(1/ 
2)*(c + d*x)^(1/2)*1473i)/(4*((1473*b^3*c^6*d^12)/4 - (17*a*b^2*c^4*d^14)/ 
2 + (a^2*b*c^2*d^16)/4 - (936*b^4*c^8*d^10)/a + (576*b^5*c^10*d^8)/a^2)) - 
 (b^4*c^6*d^10*(c^3)^(1/2)*(c + d*x)^(1/2)*936i)/((1473*a*b^3*c^6*d^12)/4 
- 936*b^4*c^8*d^10 + (a^3*b*c^2*d^16)/4 - (17*a^2*b^2*c^4*d^14)/2 + (576*b 
^5*c^10*d^8)/a) + (b^5*c^8*d^8*(c^3)^(1/2)*(c + d*x)^(1/2)*576i)/(576*b^5* 
c^10*d^8 - 936*a*b^4*c^8*d^10 + (a^4*b*c^2*d^16)/4 + (1473*a^2*b^3*c^6*d^1 
2)/4 - (17*a^3*b^2*c^4*d^14)/2))*(c^3)^(1/2)*2i)/a^2 - atan((b^4*c^6*d^10* 
(c + d*x)^(1/2)*(c^3/(4*a^4) + (9*c*d^2)/(64*a^3*b) - (d^3*(a^9*b^3)^(1/2) 
)/(64*a^7*b^3) - (3*c^2*d*(a^9*b^3)^(1/2))/(8*a^8*b^2))^(1/2)*216i)/((18*b 
^4*c^8*d^10)/a^2 - (99*b^3*c^6*d^12)/(8*a) - (347*b^2*c^4*d^14)/64 - (13*a 
*b*c^2*d^16)/64 + (83*c^3*d^15*(a^9*b^3)^(1/2))/(64*a^4) + (c*d^17*(a^9*b^ 
3)^(1/2))/(64*a^3*b) + (291*b*c^5*d^13*(a^9*b^3)^(1/2))/(16*a^5) - (183*b^ 
2*c^7*d^11*(a^9*b^3)^(1/2))/(2*a^6) + (72*b^3*c^9*d^9*(a^9*b^3)^(1/2))/a^7 
) + (c*d^15*(a^9*b^3)^(1/2)*(c + d*x)^(1/2)*(c^3/(4*a^4) + (9*c*d^2)/(64*a 
^3*b) - (d^3*(a^9*b^3)^(1/2))/(64*a^7*b^3) - (3*c^2*d*(a^9*b^3)^(1/2))/...
 

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 638, normalized size of antiderivative = 2.90 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^(3/2)/x/(-b*x^2+a)^2,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)))*a*d + 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*d* 
x**2 + 8*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqr 
t(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*c - 8*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)))*b 
*c*x**2 - 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))*a*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))*b*d*x* 
*2 + 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))*a*d - sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*lo 
g(sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*b*d*x**2 - 4*sqrt 
(b)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + s 
qrt(b)*sqrt(c + d*x))*a*c + 4*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))*b*c*x**2 + 4*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))*a*c - 4*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log(sqrt(s 
qrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*b*c*x**2 + 4*sqrt(c + d*x 
)*a*b*c + 4*sqrt(c + d*x)*a*b*d*x + 8*sqrt(c)*log(sqrt(c + d*x) - sqrt(c)) 
*a*b*c - 8*sqrt(c)*log(sqrt(c + d*x) - sqrt(c))*b**2*c*x**2 - 8*sqrt(c)...