\(\int \frac {1}{(c+d x)^2 \sqrt {a x^2+b x^3}} \, dx\) [251]

Optimal result

Integrand size = 23, antiderivative size = 145 \[ \int \frac {1}{(c+d x)^2 \sqrt {a x^2+b x^3}} \, dx=-\frac {d \sqrt {a x^2+b x^3}}{c (b c-a d) x (c+d x)}-\frac {\sqrt {d} (3 b c-2 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a x^2+b x^3}}{\sqrt {b c-a d} x}\right )}{c^2 (b c-a d)^{3/2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a x^2+b x^3}}{\sqrt {a} x}\right )}{\sqrt {a} c^2} \] Output:

-d*(b*x^3+a*x^2)^(1/2)/c/(-a*d+b*c)/x/(d*x+c)-d^(1/2)*(-2*a*d+3*b*c)*arcta 
n(d^(1/2)*(b*x^3+a*x^2)^(1/2)/(-a*d+b*c)^(1/2)/x)/c^2/(-a*d+b*c)^(3/2)-2*a 
rctanh((b*x^3+a*x^2)^(1/2)/a^(1/2)/x)/a^(1/2)/c^2
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(c+d x)^2 \sqrt {a x^2+b x^3}} \, dx=\frac {x \left (-\frac {c d (a+b x)}{(b c-a d) (c+d x)}-\frac {\sqrt {d} (3 b c-2 a d) \sqrt {a+b x} \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}-\frac {2 \sqrt {a+b x} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}}\right )}{c^2 \sqrt {x^2 (a+b x)}} \] Input:

Integrate[1/((c + d*x)^2*Sqrt[a*x^2 + b*x^3]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2467, 114, 27, 174, 73, 218, 221}

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.

\(\displaystyle \int \frac {1}{\sqrt {a x^2+b x^3} (c+d x)^2} \, dx\)

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

Input:

Int[1/((c + d*x)^2*Sqrt[a*x^2 + b*x^3]),x]
 

Output:

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

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 218

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

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 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.47

method	result	size
pseudoelliptic	\(\frac {b \left (-\frac {\sqrt {b x +a}}{b \left (d x +c \right )}+\frac {\operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {d \left (a d -b c \right )}}\right )}{\sqrt {d \left (a d -b c \right )}}\right )}{a d -b c}\)	\(68\)
default	\(\frac {x \sqrt {b x +a}\, \left (2 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {d \left (a d -b c \right )}}\right ) d^{3} x -3 \sqrt {a}\, \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {d \left (a d -b c \right )}}\right ) b c \,d^{2} x +2 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {d \left (a d -b c \right )}}\right ) c \,d^{2}-3 \sqrt {a}\, \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {d \left (a d -b c \right )}}\right ) b \,c^{2} d -2 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) \sqrt {d \left (a d -b c \right )}\, a \,d^{2} x +2 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) \sqrt {d \left (a d -b c \right )}\, b c d x +\sqrt {a}\, \sqrt {d \left (a d -b c \right )}\, \sqrt {b x +a}\, c d -2 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) \sqrt {d \left (a d -b c \right )}\, a c d +2 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) \sqrt {d \left (a d -b c \right )}\, b \,c^{2}\right )}{\sqrt {b \,x^{3}+a \,x^{2}}\, c^{2} \sqrt {a}\, \left (a d -b c \right ) \left (d x +c \right ) \sqrt {d \left (a d -b c \right )}}\)	\(331\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 941, normalized size of antiderivative = 6.49 \[ \int \frac {1}{(c+d x)^2 \sqrt {a x^2+b x^3}} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {1}{(c+d x)^2 \sqrt {a x^2+b x^3}} \, dx=\int \frac {1}{\sqrt {x^{2} \left (a + b x\right )} \left (c + d x\right )^{2}}\, dx \] Input:

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

Output:

Integral(1/(sqrt(x**2*(a + b*x))*(c + d*x)**2), x)
 

Maxima [F]

\[ \int \frac {1}{(c+d x)^2 \sqrt {a x^2+b x^3}} \, dx=\int { \frac {1}{\sqrt {b x^{3} + a x^{2}} {\left (d x + c\right )}^{2}} \,d x } \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (127) = 254\).

Time = 0.42 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.30 \[ \int \frac {1}{(c+d x)^2 \sqrt {a x^2+b x^3}} \, dx=\frac {{\left (3 \, \sqrt {-a} b c d \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) - 2 \, \sqrt {-a} a d^{2} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) - 2 \, \sqrt {b c d - a d^{2}} b c \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) + 2 \, \sqrt {b c d - a d^{2}} a d \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) + \sqrt {b c d - a d^{2}} \sqrt {-a} \sqrt {a} d\right )} \mathrm {sgn}\left (x\right )}{\sqrt {b c d - a d^{2}} \sqrt {-a} b c^{3} - \sqrt {b c d - a d^{2}} \sqrt {-a} a c^{2} d} - \frac {\sqrt {b x + a} b d}{{\left (b c^{2} \mathrm {sgn}\left (x\right ) - a c d \mathrm {sgn}\left (x\right )\right )} {\left (b c + {\left (b x + a\right )} d - a d\right )}} - \frac {{\left (3 \, b c d - 2 \, a d^{2}\right )} \arctan \left (\frac {\sqrt {b x + a} d}{\sqrt {b c d - a d^{2}}}\right )}{{\left (b c^{3} \mathrm {sgn}\left (x\right ) - a c^{2} d \mathrm {sgn}\left (x\right )\right )} \sqrt {b c d - a d^{2}}} + \frac {2 \, \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} c^{2} \mathrm {sgn}\left (x\right )} \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(c+d x)^2 \sqrt {a x^2+b x^3}} \, dx=\int \frac {1}{\sqrt {b\,x^3+a\,x^2}\,{\left (c+d\,x\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 516, normalized size of antiderivative = 3.56 \[ \int \frac {1}{(c+d x)^2 \sqrt {a x^2+b x^3}} \, dx=\frac {2 \sqrt {d}\, \sqrt {-a d +b c}\, \mathit {atan} \left (\frac {\sqrt {b x +a}\, d}{\sqrt {d}\, \sqrt {-a d +b c}}\right ) a^{2} c d +2 \sqrt {d}\, \sqrt {-a d +b c}\, \mathit {atan} \left (\frac {\sqrt {b x +a}\, d}{\sqrt {d}\, \sqrt {-a d +b c}}\right ) a^{2} d^{2} x -3 \sqrt {d}\, \sqrt {-a d +b c}\, \mathit {atan} \left (\frac {\sqrt {b x +a}\, d}{\sqrt {d}\, \sqrt {-a d +b c}}\right ) a b \,c^{2}-3 \sqrt {d}\, \sqrt {-a d +b c}\, \mathit {atan} \left (\frac {\sqrt {b x +a}\, d}{\sqrt {d}\, \sqrt {-a d +b c}}\right ) a b c d x +\sqrt {b x +a}\, a^{2} c \,d^{2}-\sqrt {b x +a}\, a b \,c^{2} d +\sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a^{2} c \,d^{2}+\sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a^{2} d^{3} x -2 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a b \,c^{2} d -2 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a b c \,d^{2} x +\sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) b^{2} c^{3}+\sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) b^{2} c^{2} d x -\sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a^{2} c \,d^{2}-\sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a^{2} d^{3} x +2 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a b \,c^{2} d +2 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a b c \,d^{2} x -\sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) b^{2} c^{3}-\sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) b^{2} c^{2} d x}{a \,c^{2} \left (a^{2} d^{3} x -2 a b c \,d^{2} x +b^{2} c^{2} d x +a^{2} c \,d^{2}-2 a b \,c^{2} d +b^{2} c^{3}\right )} \] Input:

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

Output:

(2*sqrt(d)*sqrt( - a*d + b*c)*atan((sqrt(a + b*x)*d)/(sqrt(d)*sqrt( - a*d 
+ b*c)))*a**2*c*d + 2*sqrt(d)*sqrt( - a*d + b*c)*atan((sqrt(a + b*x)*d)/(s 
qrt(d)*sqrt( - a*d + b*c)))*a**2*d**2*x - 3*sqrt(d)*sqrt( - a*d + b*c)*ata 
n((sqrt(a + b*x)*d)/(sqrt(d)*sqrt( - a*d + b*c)))*a*b*c**2 - 3*sqrt(d)*sqr 
t( - a*d + b*c)*atan((sqrt(a + b*x)*d)/(sqrt(d)*sqrt( - a*d + b*c)))*a*b*c 
*d*x + sqrt(a + b*x)*a**2*c*d**2 - sqrt(a + b*x)*a*b*c**2*d + sqrt(a)*log( 
sqrt(a + b*x) - sqrt(a))*a**2*c*d**2 + sqrt(a)*log(sqrt(a + b*x) - sqrt(a) 
)*a**2*d**3*x - 2*sqrt(a)*log(sqrt(a + b*x) - sqrt(a))*a*b*c**2*d - 2*sqrt 
(a)*log(sqrt(a + b*x) - sqrt(a))*a*b*c*d**2*x + sqrt(a)*log(sqrt(a + b*x) 
- sqrt(a))*b**2*c**3 + sqrt(a)*log(sqrt(a + b*x) - sqrt(a))*b**2*c**2*d*x 
- sqrt(a)*log(sqrt(a + b*x) + sqrt(a))*a**2*c*d**2 - sqrt(a)*log(sqrt(a + 
b*x) + sqrt(a))*a**2*d**3*x + 2*sqrt(a)*log(sqrt(a + b*x) + sqrt(a))*a*b*c 
**2*d + 2*sqrt(a)*log(sqrt(a + b*x) + sqrt(a))*a*b*c*d**2*x - sqrt(a)*log( 
sqrt(a + b*x) + sqrt(a))*b**2*c**3 - sqrt(a)*log(sqrt(a + b*x) + sqrt(a))* 
b**2*c**2*d*x)/(a*c**2*(a**2*c*d**2 + a**2*d**3*x - 2*a*b*c**2*d - 2*a*b*c 
*d**2*x + b**2*c**3 + b**2*c**2*d*x))