\(\int \frac {(c+d x)^{3/2}}{x^3 (a+b x^2)} \, dx\) [617]

Optimal result

Integrand size = 22, antiderivative size = 547 \[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a+b x^2\right )} \, dx=-\frac {c \sqrt {c+d x}}{2 a x^2}-\frac {5 d \sqrt {c+d x}}{4 a x}-\frac {\sqrt [4]{b} \left (b c^2-a d^2-\sqrt {b} c \sqrt {b c^2+a d^2}\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{\sqrt {2} a^2 \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}+\frac {\sqrt [4]{b} \left (b c^2-a d^2-\sqrt {b} c \sqrt {b c^2+a d^2}\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{\sqrt {2} a^2 \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}+\frac {\left (8 b c^2-3 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 a^2 \sqrt {c}}-\frac {\sqrt [4]{b} \left (b c^2-a d^2+\sqrt {b} c \sqrt {b c^2+a d^2}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt {b c^2+a d^2}+\sqrt {b} (c+d x)}\right )}{\sqrt {2} a^2 \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}} \] Output:

-1/2*c*(d*x+c)^(1/2)/a/x^2-5/4*d*(d*x+c)^(1/2)/a/x-1/2*b^(1/4)*(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^2/(-b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)+1/2*b^(1/4)*(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^2/(-b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)+1/4*(-3*a*d^2+8*b*c 
^2)*arctanh((d*x+c)^(1/2)/c^(1/2))/a^2/c^(1/2)-1/2*b^(1/4)*(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^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 = 0.93 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.50 \[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a+b x^2\right )} \, dx=-\frac {\sqrt {c+d x} (2 c+5 d x)}{4 a x^2}+\frac {\left (\sqrt {b} c+i \sqrt {a} d\right ) \sqrt {-b c-i \sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c-i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+i \sqrt {a} d}\right )}{a^2}+\frac {\left (\sqrt {b} c-i \sqrt {a} d\right ) \sqrt {-b c+i \sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c+i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-i \sqrt {a} d}\right )}{a^2}+\frac {\left (8 b c^2-3 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 a^2 \sqrt {c}} \] Input:

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

Output:

-1/4*(Sqrt[c + d*x]*(2*c + 5*d*x))/(a*x^2) + ((Sqrt[b]*c + I*Sqrt[a]*d)*Sq 
rt[-(b*c) - I*Sqrt[a]*Sqrt[b]*d]*ArcTan[(Sqrt[-(b*c) - I*Sqrt[a]*Sqrt[b]*d 
]*Sqrt[c + d*x])/(Sqrt[b]*c + I*Sqrt[a]*d)])/a^2 + ((Sqrt[b]*c - I*Sqrt[a] 
*d)*Sqrt[-(b*c) + I*Sqrt[a]*Sqrt[b]*d]*ArcTan[(Sqrt[-(b*c) + I*Sqrt[a]*Sqr 
t[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - I*Sqrt[a]*d)])/a^2 + ((8*b*c^2 - 3*a*d 
^2)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(4*a^2*Sqrt[c])
 

Rubi [A] (verified)

Time = 2.87 (sec) , antiderivative size = 846, normalized size of antiderivative = 1.55, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1610

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

\(\Big \downarrow \) 2009

\(\displaystyle -2 d^2 \left (\frac {\sqrt {c+d x} c}{4 a d^2 x^2}+\frac {\sqrt [4]{b} \left (b^{3/2} c^3+a \sqrt {b} d^2 c-\left (b c^2-a d^2\right ) \sqrt {b c^2+a d^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {b c^2+a d^2}}}\right )}{2 \sqrt {2} a^2 d^2 \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c-\sqrt {b c^2+a d^2}}}-\frac {\sqrt [4]{b} \left (b^{3/2} c^3+a \sqrt {b} d^2 c-\left (b c^2-a d^2\right ) \sqrt {b c^2+a d^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {b c^2+a d^2}}}\right )}{2 \sqrt {2} a^2 d^2 \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c-\sqrt {b c^2+a d^2}}}-\frac {\sqrt [4]{b} \left (b c^2+\sqrt {b} \sqrt {b c^2+a d^2} c-a d^2\right ) \log \left (\sqrt {b} (c+d x)-\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}+\sqrt {b c^2+a d^2}\right )}{4 \sqrt {2} a^2 d^2 \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}+\frac {b^{3/4} \left (b c^3+a d^2 c+\frac {\left (b c^2-a d^2\right ) \sqrt {b c^2+a d^2}}{\sqrt {b}}\right ) \log \left (\sqrt {b} (c+d x)+\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}+\sqrt {b c^2+a d^2}\right )}{4 \sqrt {2} a^2 d^2 \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}+\frac {5 \sqrt {c+d x}}{8 a d x}-\frac {\left (b c^2-a d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2 d^2 \sqrt {c}}-\frac {5 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{8 a \sqrt {c}}\right )\)

Input:

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

Output:

-2*d^2*((c*Sqrt[c + d*x])/(4*a*d^2*x^2) + (5*Sqrt[c + d*x])/(8*a*d*x) - (5 
*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(8*a*Sqrt[c]) - ((b*c^2 - a*d^2)*ArcTanh[ 
Sqrt[c + d*x]/Sqrt[c]])/(a^2*Sqrt[c]*d^2) + (b^(1/4)*(b^(3/2)*c^3 + a*Sqrt 
[b]*c*d^2 - (b*c^2 - a*d^2)*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 - Sq 
rt[b*c^2 + a*d^2]]])/(2*Sqrt[2]*a^2*d^2*Sqrt[b*c^2 + a*d^2]*Sqrt[Sqrt[b]*c 
 - Sqrt[b*c^2 + a*d^2]]) - (b^(1/4)*(b^(3/2)*c^3 + a*Sqrt[b]*c*d^2 - (b*c^ 
2 - a*d^2)*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^2*d^2*Sqrt[b*c^2 + a*d^2]*Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a 
*d^2]]) - (b^(1/4)*(b*c^2 - a*d^2 + Sqrt[b]*c*Sqrt[b*c^2 + a*d^2])*Log[Sqr 
t[b*c^2 + a*d^2] - Sqrt[2]*b^(1/4)*Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]]*S 
qrt[c + d*x] + Sqrt[b]*(c + d*x)])/(4*Sqrt[2]*a^2*d^2*Sqrt[Sqrt[b]*c + Sqr 
t[b*c^2 + a*d^2]]) + (b^(3/4)*(b*c^3 + a*c*d^2 + ((b*c^2 - a*d^2)*Sqrt[b*c 
^2 + a*d^2])/Sqrt[b])*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^2*d^2*Sqrt[b*c^2 + a*d^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 = 0.84 (sec) , antiderivative size = 812, normalized size of antiderivative = 1.48

method	result	size
pseudoelliptic	\(-\frac {\frac {\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}\, x^{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}\, \left (\left (a \,d^{2} \sqrt {c}-c^{\frac {3}{2}} \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-c^{\frac {5}{2}} b \right ) \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-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 ) \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 {\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}\, x^{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}\, \left (\left (a \,d^{2} \sqrt {c}-c^{\frac {3}{2}} \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-c^{\frac {5}{2}} b \right ) \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-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 ) \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 (\left (x^{2} \left (-2 b^{\frac {3}{2}} c^{2}+\frac {3 a \,d^{2} \sqrt {b}}{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+\frac {5 \left (d x \sqrt {c}+\frac {2 c^{\frac {3}{2}}}{5}\right ) \sqrt {d x +c}\, \sqrt {b}\, a}{4}\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 ) x^{2} \left (a \sqrt {c}\, b \,d^{2}+c^{\frac {3}{2}} \sqrt {a \,d^{2}+b \,c^{2}}\, b^{\frac {3}{2}}-b^{2} c^{\frac {5}{2}}\right )\right )}{\sqrt {b}\, \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 {c}\, d^{2} a^{3} x^{2}}\)	\(812\)
risch	\(\text {Expression too large to display}\)	\(1637\)
derivativedivides	\(\text {Expression too large to display}\)	\(1648\)
default	\(\text {Expression too large to display}\)	\(1648\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1063 vs. \(2 (440) = 880\).

Time = 1.00 (sec) , antiderivative size = 2135, normalized size of antiderivative = 3.90 \[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 427, normalized size of antiderivative = 0.78 \[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a+b x^2\right )} \, dx=-\frac {{\left (8 \, b c^{2} - 3 \, a d^{2}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{4 \, a^{2} \sqrt {-c}} + \frac {{\left (2 \, \sqrt {-a b} b c^{2} d^{2} {\left | b \right |} - {\left (\sqrt {-a b} b c^{2} - \sqrt {-a b} a d^{2}\right )} d^{2} {\left | b \right |} + {\left (b^{2} c^{3} + a b c d^{2}\right )} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a^{2} b c + \sqrt {a^{4} b^{2} c^{2} - {\left (a^{2} b c^{2} + a^{3} d^{2}\right )} a^{2} b}}{a^{2} b}}}\right )}{{\left (a^{2} b c - \sqrt {-a b} a^{2} d\right )} \sqrt {-b^{2} c - \sqrt {-a b} b d} {\left | d \right |}} - \frac {{\left (2 \, \sqrt {-a b} b c^{2} d^{2} {\left | b \right |} - {\left (\sqrt {-a b} b c^{2} - \sqrt {-a b} a d^{2}\right )} d^{2} {\left | b \right |} - {\left (b^{2} c^{3} + a b c d^{2}\right )} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a^{2} b c - \sqrt {a^{4} b^{2} c^{2} - {\left (a^{2} b c^{2} + a^{3} d^{2}\right )} a^{2} b}}{a^{2} b}}}\right )}{{\left (a^{2} b c + \sqrt {-a b} a^{2} d\right )} \sqrt {-b^{2} c + \sqrt {-a b} b d} {\left | d \right |}} - \frac {5 \, {\left (d x + c\right )}^{\frac {3}{2}} d^{2} - 3 \, \sqrt {d x + c} c d^{2}}{4 \, a d^{2} x^{2}} \] Input:

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

Output:

-1/4*(8*b*c^2 - 3*a*d^2)*arctan(sqrt(d*x + c)/sqrt(-c))/(a^2*sqrt(-c)) + ( 
2*sqrt(-a*b)*b*c^2*d^2*abs(b) - (sqrt(-a*b)*b*c^2 - sqrt(-a*b)*a*d^2)*d^2* 
abs(b) + (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*c - sqrt(-a*b)*a^2*d)*sqrt(-b^2*c - sqrt(-a*b)*b*d)*abs(d)) - (2*sqrt(- 
a*b)*b*c^2*d^2*abs(b) - (sqrt(-a*b)*b*c^2 - sqrt(-a*b)*a*d^2)*d^2*abs(b) - 
 (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*c + s 
qrt(-a*b)*a^2*d)*sqrt(-b^2*c + sqrt(-a*b)*b*d)*abs(d)) - 1/4*(5*(d*x + c)^ 
(3/2)*d^2 - 3*sqrt(d*x + c)*c*d^2)/(a*d^2*x^2)
 

Mupad [B] (verification not implemented)

Time = 8.91 (sec) , antiderivative size = 5616, normalized size of antiderivative = 10.27 \[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((b^5*d^16*(c + d*x)^(1/2)*((b^2*c^3)/(4*a^4) + (d^3*(-a^9*b)^(1/2))/( 
4*a^7) - (3*b*c*d^2)/(4*a^3) - (3*b*c^2*d*(-a^9*b)^(1/2))/(4*a^8))^(1/2)*1 
8i)/((2*b^6*c^2*d^16)/a^2 - (9*b^5*d^18)/a + (59*b^7*c^4*d^14)/a^3 + (48*b 
^8*c^6*d^12)/a^4 - (62*b^6*c^3*d^15*(-a^9*b)^(1/2))/a^7 + (27*b^7*c^5*d^13 
*(-a^9*b)^(1/2))/a^8 + (240*b^8*c^7*d^11*(-a^9*b)^(1/2))/a^9 + (144*b^9*c^ 
9*d^9*(-a^9*b)^(1/2))/a^10 + (7*b^5*c*d^17*(-a^9*b)^(1/2))/a^6) - (b^6*c^2 
*d^14*(c + d*x)^(1/2)*((b^2*c^3)/(4*a^4) + (d^3*(-a^9*b)^(1/2))/(4*a^7) - 
(3*b*c*d^2)/(4*a^3) - (3*b*c^2*d*(-a^9*b)^(1/2))/(4*a^8))^(1/2)*86i)/((2*b 
^6*c^2*d^16)/a - 9*b^5*d^18 + (59*b^7*c^4*d^14)/a^2 + (48*b^8*c^6*d^12)/a^ 
3 - (62*b^6*c^3*d^15*(-a^9*b)^(1/2))/a^6 + (27*b^7*c^5*d^13*(-a^9*b)^(1/2) 
)/a^7 + (240*b^8*c^7*d^11*(-a^9*b)^(1/2))/a^8 + (144*b^9*c^9*d^9*(-a^9*b)^ 
(1/2))/a^9 + (7*b^5*c*d^17*(-a^9*b)^(1/2))/a^5) - (b^8*c^6*d^10*(c + d*x)^ 
(1/2)*((b^2*c^3)/(4*a^4) + (d^3*(-a^9*b)^(1/2))/(4*a^7) - (3*b*c*d^2)/(4*a 
^3) - (3*b*c^2*d*(-a^9*b)^(1/2))/(4*a^8))^(1/2)*576i)/(59*b^7*c^4*d^14 - 9 
*a^2*b^5*d^18 + 2*a*b^6*c^2*d^16 + (48*b^8*c^6*d^12)/a - (62*b^6*c^3*d^15* 
(-a^9*b)^(1/2))/a^4 + (27*b^7*c^5*d^13*(-a^9*b)^(1/2))/a^5 + (240*b^8*c^7* 
d^11*(-a^9*b)^(1/2))/a^6 + (144*b^9*c^9*d^9*(-a^9*b)^(1/2))/a^7 + (7*b^5*c 
*d^17*(-a^9*b)^(1/2))/a^3) + (b^7*c^4*d^12*(c + d*x)^(1/2)*((b^2*c^3)/(4*a 
^4) + (d^3*(-a^9*b)^(1/2))/(4*a^7) - (3*b*c*d^2)/(4*a^3) - (3*b*c^2*d*(-a^ 
9*b)^(1/2))/(4*a^8))^(1/2)*288i)/(2*b^6*c^2*d^16 - 9*a*b^5*d^18 + (59*b...
 

Reduce [B] (verification not implemented)

Time = 44.96 (sec) , antiderivative size = 878, normalized size of antiderivative = 1.61 \[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*sqrt(a*d**2 + b*c**2)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2) 
*atan((sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) - 2*sqrt(b)*sqrt( 
c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*c*x**2 + 8* 
sqrt(b)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)*atan((sqrt(sqrt( 
b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) - 2*sqrt(b)*sqrt(c + d*x))/(sqrt(s 
qrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*c**2*x**2 - 4*sqrt(a*d**2 + 
b*c**2)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)*atan((sqrt(sqrt( 
b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) + 2*sqrt(b)*sqrt(c + d*x))/(sqrt(s 
qrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*c*x**2 - 8*sqrt(b)*sqrt(sqrt 
(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)*atan((sqrt(sqrt(b)*sqrt(a*d**2 + 
b*c**2) + b*c)*sqrt(2) + 2*sqrt(b)*sqrt(c + d*x))/(sqrt(sqrt(b)*sqrt(a*d** 
2 + b*c**2) - b*c)*sqrt(2)))*c**2*x**2 - 2*sqrt(a*d**2 + b*c**2)*sqrt(sqrt 
(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2)*log( - sqrt(c + d*x)*sqrt(sqrt(b) 
*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) + sqrt(a*d**2 + b*c**2) + sqrt(b)*c 
+ sqrt(b)*d*x)*c*x**2 + 2*sqrt(a*d**2 + b*c**2)*sqrt(sqrt(b)*sqrt(a*d**2 + 
 b*c**2) + b*c)*sqrt(2)*log(sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c** 
2) + b*c)*sqrt(2) + sqrt(a*d**2 + b*c**2) + sqrt(b)*c + sqrt(b)*d*x)*c*x** 
2 + 4*sqrt(b)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2)*log( - sqr 
t(c + d*x)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) + sqrt(a*d**2 
 + b*c**2) + sqrt(b)*c + sqrt(b)*d*x)*c**2*x**2 - 4*sqrt(b)*sqrt(sqrt(b...