\(\int \frac {(c+d x)^{3/2}}{x^3 (a-b x^2)} \, dx\) [576]

Optimal result

Integrand size = 23, antiderivative size = 210 \[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )} \, dx=-\frac {c \sqrt {c+d x}}{2 a x^2}-\frac {5 d \sqrt {c+d x}}{4 a x}-\frac {\left (8 b c^2+3 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 a^2 \sqrt {c}}+\frac {\sqrt [4]{b} \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{a^2}+\frac {\sqrt [4]{b} \left (\sqrt {b} c+\sqrt {a} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{a^2} \] Output:

-1/2*c*(d*x+c)^(1/2)/a/x^2-5/4*d*(d*x+c)^(1/2)/a/x-1/4*(3*a*d^2+8*b*c^2)*a 
rctanh((d*x+c)^(1/2)/c^(1/2))/a^2/c^(1/2)+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^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^2
 

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.22 \[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )} \, dx=-\frac {a \sqrt {c+d x} (2 c+5 d x)+4 \left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {-b c-\sqrt {a} \sqrt {b} d} x^2 \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )+4 \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {-b c+\sqrt {a} \sqrt {b} d} x^2 \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{4 a^2 x^2}-\frac {\left (8 b c^2+3 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 a^2 \sqrt {c}} \] Input:

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

Output:

-1/4*(a*Sqrt[c + d*x]*(2*c + 5*d*x) + 4*(Sqrt[b]*c + Sqrt[a]*d)*Sqrt[-(b*c 
) - Sqrt[a]*Sqrt[b]*d]*x^2*ArcTan[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c 
 + d*x])/(Sqrt[b]*c + Sqrt[a]*d)] + 4*(Sqrt[b]*c - Sqrt[a]*d)*Sqrt[-(b*c) 
+ Sqrt[a]*Sqrt[b]*d]*x^2*ArcTan[(Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + 
 d*x])/(Sqrt[b]*c - Sqrt[a]*d)])/(a^2*x^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 = 0.85 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.23, 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.

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 2009

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

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)*(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^2*d^2) - (b^(1/4)*(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^2*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 = 0.46 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.26

method	result	size
risch	\(-\frac {\sqrt {d x +c}\, \left (5 d x +2 c \right )}{4 a \,x^{2}}-\frac {d^{2} \left (\frac {8 b^{2} \left (-\frac {\left (2 a b c \,d^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-2 a b c \,d^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \,d^{2}}-\frac {\left (-3 a \,d^{2}-8 b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a \,d^{2} \sqrt {c}}\right )}{4 a}\)	\(264\)
derivativedivides	\(-2 d^{4} \left (\frac {\frac {\frac {5 a \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{8}-\frac {3 a c \,d^{2} \sqrt {d x +c}}{8}}{d^{2} x^{2}}+\frac {\left (3 a \,d^{2}+8 b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{8 \sqrt {c}}}{a^{2} d^{4}}+\frac {b^{2} \left (-\frac {\left (2 a b c \,d^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-2 a b c \,d^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a^{2} d^{4}}\right )\)	\(273\)
default	\(2 d^{4} \left (-\frac {\frac {\frac {5 a \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{8}-\frac {3 a c \,d^{2} \sqrt {d x +c}}{8}}{d^{2} x^{2}}+\frac {\left (3 a \,d^{2}+8 b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{8 \sqrt {c}}}{a^{2} d^{4}}-\frac {b^{2} \left (-\frac {\left (2 a b c \,d^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-2 a b c \,d^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a^{2} d^{4}}\right )\)	\(275\)
pseudoelliptic	\(\frac {-\sqrt {c}\, x^{2} b \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {a b \,d^{2}}-2 a b c \,d^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\left (\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {a b \,d^{2}}+2 a b c \,d^{2}\right ) \sqrt {c}\, x^{2} b \,\operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-\frac {\sqrt {a b \,d^{2}}\, \left (x^{2} \left (\frac {3 a \,d^{2}}{2}+4 b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+\left (c +\frac {5 d x}{2}\right ) \sqrt {c}\, a \sqrt {d x +c}\right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}{2}\right )}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {c}\, \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, a^{2} x^{2}}\)	\(290\)

Input:

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

Output:

-1/4*(d*x+c)^(1/2)*(5*d*x+2*c)/a/x^2-1/4/a*d^2*(8*b^2/a/d^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)*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)))-(-3*a*d^2-8*b*c^2)/a/d^2/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. 1046 vs. \(2 (158) = 316\).

Time = 1.18 (sec) , antiderivative size = 2102, normalized size of antiderivative = 10.01 \[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )} \, dx=\text {Too large to display} \] 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)*sq 
rt(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)) - (8*b*c^2 + 3*a*d^2)*sq...
 

Sympy [F]

\[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )} \, dx=- \int \frac {c \sqrt {c + d x}}{- a x^{3} + b x^{5}}\, dx - \int \frac {d x \sqrt {c + d x}}{- a x^{3} + b x^{5}}\, dx \] Input:

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

Output:

-Integral(c*sqrt(c + d*x)/(-a*x**3 + b*x**5), x) - Integral(d*x*sqrt(c + d 
*x)/(-a*x**3 + b*x**5), 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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (158) = 316\).

Time = 0.18 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.00 \[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )} \, dx=\frac {{\left (8 \, b c^{2} + 3 \, a d^{2}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{4 \, a^{2} \sqrt {-c}} - \frac {{\left (2 \, \sqrt {a b} b c^{2} d^{2} {\left | b \right |} - {\left (\sqrt {a b} b c^{2} + \sqrt {a b} a d^{2}\right )} d^{2} {\left | b \right |} + {\left (b^{2} c^{3} - a b c d^{2}\right )} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a^{2} b c + \sqrt {a^{4} b^{2} c^{2} - {\left (a^{2} b c^{2} - a^{3} d^{2}\right )} a^{2} b}}{a^{2} b}}}\right )}{{\left (a^{2} b c - \sqrt {a b} a^{2} d\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | d \right |}} + \frac {{\left (2 \, \sqrt {a b} b c^{2} d^{2} {\left | b \right |} - {\left (\sqrt {a b} b c^{2} + \sqrt {a b} a d^{2}\right )} d^{2} {\left | b \right |} - {\left (b^{2} c^{3} - a b c d^{2}\right )} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a^{2} b c - \sqrt {a^{4} b^{2} c^{2} - {\left (a^{2} b c^{2} - a^{3} d^{2}\right )} a^{2} b}}{a^{2} b}}}\right )}{{\left (a^{2} b c + \sqrt {a b} a^{2} d\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | d \right |}} - \frac {5 \, {\left (d x + c\right )}^{\frac {3}{2}} d^{2} - 3 \, \sqrt {d x + c} c d^{2}}{4 \, a d^{2} x^{2}} \] 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 + sqrt(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 = 8.81 (sec) , antiderivative size = 5492, normalized size of antiderivative = 26.15 \[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )} \, dx=\text {Too large to display} \] 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)*18i 
)/((59*b^7*c^4*d^14)/a^3 - (2*b^6*c^2*d^16)/a^2 - (9*b^5*d^18)/a - (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)/((59*b^7*c^4*d 
^14)/a^2 - (2*b^6*c^2*d^16)/a - 9*b^5*d^18 - (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 - (24 
0*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)/((59*b^7*c^4*d^14)/a - 2*b^6*c^2*d^16 - 9*a*b^5*d^18 - (48*b^...
 

Reduce [B] (verification not implemented)

Time = 3.04 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.88 \[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )} \, dx=\frac {-8 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) c d \,x^{2}+8 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) c^{2} x^{2}-4 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) c d \,x^{2}+4 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) c d \,x^{2}-4 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) c^{2} x^{2}+4 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) c^{2} x^{2}-4 \sqrt {d x +c}\, a \,c^{2}-10 \sqrt {d x +c}\, a c d x +3 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a \,d^{2} x^{2}+8 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) b \,c^{2} x^{2}-3 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a \,d^{2} x^{2}-8 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) b \,c^{2} x^{2}}{8 a^{2} c \,x^{2}} \] Input:

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

Output:

( - 8*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)))*c*d*x**2 + 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))) 
*c**2*x**2 - 4*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*s 
qrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*c*d*x**2 + 4*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 
))*c*d*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))*c**2*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))*c**2*x**2 - 4*sqrt(c + d*x)*a*c**2 - 10*sqrt(c + d*x)*a*c*d*x + 3*sqr 
t(c)*log(sqrt(c + d*x) - sqrt(c))*a*d**2*x**2 + 8*sqrt(c)*log(sqrt(c + d*x 
) - sqrt(c))*b*c**2*x**2 - 3*sqrt(c)*log(sqrt(c + d*x) + sqrt(c))*a*d**2*x 
**2 - 8*sqrt(c)*log(sqrt(c + d*x) + sqrt(c))*b*c**2*x**2)/(8*a**2*c*x**2)