\(\int \frac {\sqrt {c+d x}}{x^3 (a-b x^2)} \, dx\) [569]

Optimal result

Integrand size = 23, antiderivative size = 212 \[ \int \frac {\sqrt {c+d x}}{x^3 \left (a-b x^2\right )} \, dx=-\frac {\sqrt {c+d x}}{2 a x^2}-\frac {d \sqrt {c+d x}}{4 a c x}-\frac {\left (8 b c^2-a d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 a^2 c^{3/2}}+\frac {b^{3/4} \sqrt {\sqrt {b} c-\sqrt {a} d} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{a^2}+\frac {b^{3/4} \sqrt {\sqrt {b} c+\sqrt {a} d} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{a^2} \] Output:

-1/2*(d*x+c)^(1/2)/a/x^2-1/4*d*(d*x+c)^(1/2)/a/c/x-1/4*(-a*d^2+8*b*c^2)*ar 
ctanh((d*x+c)^(1/2)/c^(1/2))/a^2/c^(3/2)+b^(3/4)*(b^(1/2)*c-a^(1/2)*d)^(1/ 
2)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/a^2+b^(3/4)* 
(b^(1/2)*c+a^(1/2)*d)^(1/2)*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 = 0.84 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {c+d x}}{x^3 \left (a-b x^2\right )} \, dx=-\frac {\sqrt {c+d x} (2 c+d x)}{4 a c x^2}-\frac {\sqrt {b} \sqrt {-\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )} \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{a^2}-\frac {\sqrt {b} \sqrt {-\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )} \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{a^2}+\frac {\left (-8 b c^2+a d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 a^2 c^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.18, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1610

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-2*d^2*(Sqrt[c + d*x]/(4*a*d^2*x^2) + Sqrt[c + d*x]/(8*a*c*d*x) - ArcTanh[ 
Sqrt[c + d*x]/Sqrt[c]]/(8*a*c^(3/2)) + (b*Sqrt[c]*ArcTanh[Sqrt[c + d*x]/Sq 
rt[c]])/(a^2*d^2) - (b^(3/4)*Sqrt[Sqrt[b]*c - Sqrt[a]*d]*ArcTanh[(b^(1/4)* 
Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*a^2*d^2) - (b^(3/4)*Sqrt[S 
qrt[b]*c + Sqrt[a]*d]*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqr 
t[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 = 226, normalized size of antiderivative = 1.07

method	result	size
risch	\(-\frac {\sqrt {d x +c}\, \left (d x +2 c \right )}{4 a \,x^{2} c}-\frac {d^{2} \left (\frac {8 b^{2} c \left (\frac {\left (-a \,d^{2}+\sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (a \,d^{2}+\sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \,d^{2}}-\frac {\left (a \,d^{2}-8 b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a \,d^{2} \sqrt {c}}\right )}{4 a c}\)	\(226\)
derivativedivides	\(-2 d^{4} \left (\frac {b^{2} \left (\frac {\left (-a \,d^{2}+\sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (a \,d^{2}+\sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a^{2} d^{4}}+\frac {\frac {\frac {a \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{8 c}+\frac {a \,d^{2} \sqrt {d x +c}}{8}}{d^{2} x^{2}}-\frac {\left (a \,d^{2}-8 b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{8 c^{\frac {3}{2}}}}{a^{2} d^{4}}\right )\)	\(231\)
default	\(2 d^{4} \left (-\frac {b^{2} \left (\frac {\left (-a \,d^{2}+\sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (a \,d^{2}+\sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a^{2} d^{4}}-\frac {\frac {\frac {a \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{8 c}+\frac {a \,d^{2} \sqrt {d x +c}}{8}}{d^{2} x^{2}}-\frac {\left (a \,d^{2}-8 b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{8 c^{\frac {3}{2}}}}{a^{2} d^{4}}\right )\)	\(233\)
pseudoelliptic	\(\frac {-x^{2} \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, b^{2} c^{\frac {5}{2}} \left (-a \,d^{2}+\sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (x^{2} b^{2} \left (a \,d^{2}+\sqrt {a b \,d^{2}}\, c \right ) c^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-\frac {\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-\frac {\left (a \,d^{2}-8 b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) c \,x^{2}}{2}+\left (\frac {d x}{2}+c \right ) a \,c^{\frac {3}{2}} \sqrt {d x +c}\right ) \sqrt {a b \,d^{2}}}{2}\right )}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, a^{2} c^{\frac {5}{2}} x^{2}}\)	\(270\)

Input:

int((d*x+c)^(1/2)/x^3/(-b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

-1/4*(d*x+c)^(1/2)*(d*x+2*c)/a/x^2/c-1/4/a/c*d^2*(8*b^2*c/a/d^2*(1/2*(-a*d 
^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)*arc 
tan(b*(d*x+c)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2))-1/2*(a*d^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)))-(a*d^2-8*b*c^2)/a/d^2/c^(1/2)*ar 
ctanh((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. 407 vs. \(2 (160) = 320\).

Time = 0.35 (sec) , antiderivative size = 824, normalized size of antiderivative = 3.89 \[ \int \frac {\sqrt {c+d x}}{x^3 \left (a-b x^2\right )} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^(1/2)/x^3/(-b*x^2+a),x, algorithm="fricas")
 

Output:

[1/8*(4*a^2*c^2*x^2*sqrt((a^4*sqrt(b^3*d^2/a^7) + b^2*c)/a^4)*log(a^2*sqrt 
((a^4*sqrt(b^3*d^2/a^7) + b^2*c)/a^4) + sqrt(d*x + c)*b) - 4*a^2*c^2*x^2*s 
qrt((a^4*sqrt(b^3*d^2/a^7) + b^2*c)/a^4)*log(-a^2*sqrt((a^4*sqrt(b^3*d^2/a 
^7) + b^2*c)/a^4) + sqrt(d*x + c)*b) + 4*a^2*c^2*x^2*sqrt(-(a^4*sqrt(b^3*d 
^2/a^7) - b^2*c)/a^4)*log(a^2*sqrt(-(a^4*sqrt(b^3*d^2/a^7) - b^2*c)/a^4) + 
 sqrt(d*x + c)*b) - 4*a^2*c^2*x^2*sqrt(-(a^4*sqrt(b^3*d^2/a^7) - b^2*c)/a^ 
4)*log(-a^2*sqrt(-(a^4*sqrt(b^3*d^2/a^7) - b^2*c)/a^4) + sqrt(d*x + c)*b) 
- (8*b*c^2 - a*d^2)*sqrt(c)*x^2*log((d*x + 2*sqrt(d*x + c)*sqrt(c) + 2*c)/ 
x) - 2*(a*c*d*x + 2*a*c^2)*sqrt(d*x + c))/(a^2*c^2*x^2), 1/4*(2*a^2*c^2*x^ 
2*sqrt((a^4*sqrt(b^3*d^2/a^7) + b^2*c)/a^4)*log(a^2*sqrt((a^4*sqrt(b^3*d^2 
/a^7) + b^2*c)/a^4) + sqrt(d*x + c)*b) - 2*a^2*c^2*x^2*sqrt((a^4*sqrt(b^3* 
d^2/a^7) + b^2*c)/a^4)*log(-a^2*sqrt((a^4*sqrt(b^3*d^2/a^7) + b^2*c)/a^4) 
+ sqrt(d*x + c)*b) + 2*a^2*c^2*x^2*sqrt(-(a^4*sqrt(b^3*d^2/a^7) - b^2*c)/a 
^4)*log(a^2*sqrt(-(a^4*sqrt(b^3*d^2/a^7) - b^2*c)/a^4) + sqrt(d*x + c)*b) 
- 2*a^2*c^2*x^2*sqrt(-(a^4*sqrt(b^3*d^2/a^7) - b^2*c)/a^4)*log(-a^2*sqrt(- 
(a^4*sqrt(b^3*d^2/a^7) - b^2*c)/a^4) + sqrt(d*x + c)*b) + (8*b*c^2 - a*d^2 
)*sqrt(-c)*x^2*arctan(sqrt(-c)/sqrt(d*x + c)) - (a*c*d*x + 2*a*c^2)*sqrt(d 
*x + c))/(a^2*c^2*x^2)]
 

Sympy [F]

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

integrate((d*x+c)**(1/2)/x**3/(-b*x**2+a),x)
 

Output:

-Integral(sqrt(c + d*x)/(-a*x**3 + b*x**5), x)
 

Maxima [F]

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

integrate((d*x+c)^(1/2)/x^3/(-b*x^2+a),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt {c+d x}}{x^3 \left (a-b x^2\right )} \, dx=\frac {\sqrt {-b^{2} c - \sqrt {a b} b d} {\left (b c^{2} - a d^{2}\right )} {\left | b \right |} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a^{2} b c + \sqrt {a^{4} b^{2} c^{2} - {\left (a^{2} b c^{2} - a^{3} d^{2}\right )} a^{2} b}}{a^{2} b}}}\right )}{a^{2} b^{2} c^{2} - a^{3} b d^{2}} + \frac {\sqrt {-b^{2} c + \sqrt {a b} b d} {\left (b c^{2} - a d^{2}\right )} {\left | b \right |} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a^{2} b c - \sqrt {a^{4} b^{2} c^{2} - {\left (a^{2} b c^{2} - a^{3} d^{2}\right )} a^{2} b}}{a^{2} b}}}\right )}{a^{2} b^{2} c^{2} - a^{3} b d^{2}} + \frac {{\left (8 \, b c^{2} - a d^{2}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{4 \, a^{2} \sqrt {-c} c} - \frac {{\left (d x + c\right )}^{\frac {3}{2}} d^{2} + \sqrt {d x + c} c d^{2}}{4 \, a c d^{2} x^{2}} \] Input:

integrate((d*x+c)^(1/2)/x^3/(-b*x^2+a),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 8.95 (sec) , antiderivative size = 3367, normalized size of antiderivative = 15.88 \[ \int \frac {\sqrt {c+d x}}{x^3 \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((b^6*d^14*((b^2*c)/(4*a^4) + (d*(a^9*b^3)^(1/2))/(4*a^8))^(1/2)*(c + 
d*x)^(1/2)*2i)/((16*b^8*c^3*d^12)/a^3 - (16*b^9*c^5*d^10)/a^4 - (b^5*d^15* 
(a^9*b^3)^(1/2))/a^6 + (17*b^6*c^2*d^13*(a^9*b^3)^(1/2))/a^7 - (64*b^7*c^4 
*d^11*(a^9*b^3)^(1/2))/a^8 + (48*b^8*c^6*d^9*(a^9*b^3)^(1/2))/a^9) - (b^7* 
d^12*((b^2*c)/(4*a^4) + (d*(a^9*b^3)^(1/2))/(4*a^8))^(1/2)*(c + d*x)^(1/2) 
*32i)/((16*b^8*c*d^12)/a^2 - (16*b^9*c^3*d^10)/a^3 + (17*b^6*d^13*(a^9*b^3 
)^(1/2))/a^6 - (b^5*d^15*(a^9*b^3)^(1/2))/(a^5*c^2) - (64*b^7*c^2*d^11*(a^ 
9*b^3)^(1/2))/a^7 + (48*b^8*c^4*d^9*(a^9*b^3)^(1/2))/a^8) + (b^5*d^13*((b^ 
2*c)/(4*a^4) + (d*(a^9*b^3)^(1/2))/(4*a^8))^(1/2)*(a^9*b^3)^(1/2)*(c + d*x 
)^(1/2)*2i)/(16*a*b^9*c^4*d^10 - 16*a^2*b^8*c^2*d^12 - (17*b^6*c*d^13*(a^9 
*b^3)^(1/2))/a^2 + (b^5*d^15*(a^9*b^3)^(1/2))/(a*c) + (64*b^7*c^3*d^11*(a^ 
9*b^3)^(1/2))/a^3 - (48*b^8*c^5*d^9*(a^9*b^3)^(1/2))/a^4) + (b^8*c^2*d^10* 
((b^2*c)/(4*a^4) + (d*(a^9*b^3)^(1/2))/(4*a^8))^(1/2)*(c + d*x)^(1/2)*128i 
)/((16*b^8*c*d^12)/a - (16*b^9*c^3*d^10)/a^2 + (17*b^6*d^13*(a^9*b^3)^(1/2 
))/a^5 - (b^5*d^15*(a^9*b^3)^(1/2))/(a^4*c^2) - (64*b^7*c^2*d^11*(a^9*b^3) 
^(1/2))/a^6 + (48*b^8*c^4*d^9*(a^9*b^3)^(1/2))/a^7) - (b^7*c^3*d^9*((b^2*c 
)/(4*a^4) + (d*(a^9*b^3)^(1/2))/(4*a^8))^(1/2)*(a^9*b^3)^(1/2)*(c + d*x)^( 
1/2)*96i)/(17*b^6*d^13*(a^9*b^3)^(1/2) + 16*a^4*b^8*c*d^12 - 16*a^3*b^9*c^ 
3*d^10 - (a*b^5*d^15*(a^9*b^3)^(1/2))/c^2 - (64*b^7*c^2*d^11*(a^9*b^3)^(1/ 
2))/a + (48*b^8*c^4*d^9*(a^9*b^3)^(1/2))/a^2))*((d*(a^9*b^3)^(1/2) + a^...
 

Reduce [B] (verification not implemented)

Time = 0.78 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {c+d x}}{x^3 \left (a-b x^2\right )} \, dx=\frac {8 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) c^{2} x^{2}-4 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) c^{2} x^{2}+4 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) c^{2} x^{2}-4 \sqrt {d x +c}\, a \,c^{2}-2 \sqrt {d x +c}\, a c d x -\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a \,d^{2} x^{2}+8 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) b \,c^{2} x^{2}+\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a \,d^{2} x^{2}-8 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) b \,c^{2} x^{2}}{8 a^{2} c^{2} x^{2}} \] Input:

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

Output:

(8*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*s 
qrt(sqrt(b)*sqrt(a)*d - b*c)))*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(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 - 2*s 
qrt(c + d*x)*a*c*d*x - 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 + 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**2*x**2)