\(\int \frac {1}{x^3 (c+d x)^2 \sqrt {a x^2+b x^3}} \, dx\) [254]

Optimal result

Integrand size = 26, antiderivative size = 362 \[ \int \frac {1}{x^3 (c+d x)^2 \sqrt {a x^2+b x^3}} \, dx=-\frac {\sqrt {a x^2+b x^3}}{3 a c x^4 (c+d x)}+\frac {(5 b c+8 a d) \sqrt {a x^2+b x^3}}{12 a^2 c^2 x^3 (c+d x)}-\frac {\left (15 b^2 c^2+26 a b c d+48 a^2 d^2\right ) \sqrt {a x^2+b x^3}}{24 a^3 c^3 x^2 (c+d x)}-\frac {d \left (5 b^3 c^3+7 a b^2 c^2 d+12 a^2 b c d^2-32 a^3 d^3\right ) \sqrt {a x^2+b x^3}}{8 a^3 c^4 (b c-a d) x (c+d x)}+\frac {d^{7/2} (9 b c-8 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a x^2+b x^3}}{\sqrt {b c-a d} x}\right )}{c^5 (b c-a d)^{3/2}}+\frac {\left (5 b^3 c^3+12 a b^2 c^2 d+24 a^2 b c d^2+64 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {a x^2+b x^3}}{\sqrt {a} x}\right )}{8 a^{7/2} c^5} \] Output:

-1/3*(b*x^3+a*x^2)^(1/2)/a/c/x^4/(d*x+c)+1/12*(8*a*d+5*b*c)*(b*x^3+a*x^2)^ 
(1/2)/a^2/c^2/x^3/(d*x+c)-1/24*(48*a^2*d^2+26*a*b*c*d+15*b^2*c^2)*(b*x^3+a 
*x^2)^(1/2)/a^3/c^3/x^2/(d*x+c)-1/8*d*(-32*a^3*d^3+12*a^2*b*c*d^2+7*a*b^2* 
c^2*d+5*b^3*c^3)*(b*x^3+a*x^2)^(1/2)/a^3/c^4/(-a*d+b*c)/x/(d*x+c)+d^(7/2)* 
(-8*a*d+9*b*c)*arctan(d^(1/2)*(b*x^3+a*x^2)^(1/2)/(-a*d+b*c)^(1/2)/x)/c^5/ 
(-a*d+b*c)^(3/2)+1/8*(64*a^3*d^3+24*a^2*b*c*d^2+12*a*b^2*c^2*d+5*b^3*c^3)* 
arctanh((b*x^3+a*x^2)^(1/2)/a^(1/2)/x)/a^(7/2)/c^5
 

Mathematica [A] (verified)

Time = 1.70 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^3 (c+d x)^2 \sqrt {a x^2+b x^3}} \, dx=\frac {\frac {c (a+b x) \left (15 b^3 c^3 x^2 (c+d x)+a b^2 c^2 x \left (-10 c^2+11 c d x+21 d^2 x^2\right )-8 a^3 d \left (c^3-2 c^2 d x+6 c d^2 x^2+12 d^3 x^3\right )+2 a^2 b c \left (4 c^3-3 c^2 d x+11 c d^2 x^2+18 d^3 x^3\right )\right )}{a^3 (-b c+a d) (c+d x)}+\frac {24 d^{7/2} (9 b c-8 a d) x^3 \sqrt {a+b x} \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}+\frac {3 \left (5 b^3 c^3+12 a b^2 c^2 d+24 a^2 b c d^2+64 a^3 d^3\right ) x^3 \sqrt {a+b x} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{7/2}}}{24 c^5 x^2 \sqrt {x^2 (a+b x)}} \] Input:

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

Output:

((c*(a + b*x)*(15*b^3*c^3*x^2*(c + d*x) + a*b^2*c^2*x*(-10*c^2 + 11*c*d*x 
+ 21*d^2*x^2) - 8*a^3*d*(c^3 - 2*c^2*d*x + 6*c*d^2*x^2 + 12*d^3*x^3) + 2*a 
^2*b*c*(4*c^3 - 3*c^2*d*x + 11*c*d^2*x^2 + 18*d^3*x^3)))/(a^3*(-(b*c) + a* 
d)*(c + d*x)) + (24*d^(7/2)*(9*b*c - 8*a*d)*x^3*Sqrt[a + b*x]*ArcTan[(Sqrt 
[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2) + (3*(5*b^3*c^3 + 1 
2*a*b^2*c^2*d + 24*a^2*b*c*d^2 + 64*a^3*d^3)*x^3*Sqrt[a + b*x]*ArcTanh[Sqr 
t[a + b*x]/Sqrt[a]])/a^(7/2))/(24*c^5*x^2*Sqrt[x^2*(a + b*x)])
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1948, 114, 27, 168, 27, 168, 27, 168, 25, 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.

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

\(\Big \downarrow \) 1948

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

\(\Big \downarrow \) 114

\(\displaystyle \frac {x \sqrt {a+b x} \left (-\frac {\int \frac {5 b c+8 a d+7 b d x}{2 x^3 \sqrt {a+b x} (c+d x)^2}dx}{3 a c}-\frac {\sqrt {a+b x}}{3 a c x^3 (c+d x)}\right )}{\sqrt {a x^2+b x^3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x \sqrt {a+b x} \left (-\frac {\int \frac {5 b c+8 a d+7 b d x}{x^3 \sqrt {a+b x} (c+d x)^2}dx}{6 a c}-\frac {\sqrt {a+b x}}{3 a c x^3 (c+d x)}\right )}{\sqrt {a x^2+b x^3}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {x \sqrt {a+b x} \left (-\frac {-\frac {\int \frac {15 b^2 c^2+26 a b d c+48 a^2 d^2+5 b d (5 b c+8 a d) x}{2 x^2 \sqrt {a+b x} (c+d x)^2}dx}{2 a c}-\frac {\sqrt {a+b x} (8 a d+5 b c)}{2 a c x^2 (c+d x)}}{6 a c}-\frac {\sqrt {a+b x}}{3 a c x^3 (c+d x)}\right )}{\sqrt {a x^2+b x^3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x \sqrt {a+b x} \left (-\frac {-\frac {\int \frac {15 b^2 c^2+26 a b d c+48 a^2 d^2+5 b d (5 b c+8 a d) x}{x^2 \sqrt {a+b x} (c+d x)^2}dx}{4 a c}-\frac {\sqrt {a+b x} (8 a d+5 b c)}{2 a c x^2 (c+d x)}}{6 a c}-\frac {\sqrt {a+b x}}{3 a c x^3 (c+d x)}\right )}{\sqrt {a x^2+b x^3}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {x \sqrt {a+b x} \left (-\frac {-\frac {-\frac {\int \frac {3 \left (5 b^3 c^3+12 a b^2 d c^2+24 a^2 b d^2 c+64 a^3 d^3+b d \left (15 b^2 c^2+26 a b d c+48 a^2 d^2\right ) x\right )}{2 x \sqrt {a+b x} (c+d x)^2}dx}{a c}-\frac {\sqrt {a+b x} \left (\frac {15 b^2 c}{a}+\frac {48 a d^2}{c}+26 b d\right )}{x (c+d x)}}{4 a c}-\frac {\sqrt {a+b x} (8 a d+5 b c)}{2 a c x^2 (c+d x)}}{6 a c}-\frac {\sqrt {a+b x}}{3 a c x^3 (c+d x)}\right )}{\sqrt {a x^2+b x^3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x \sqrt {a+b x} \left (-\frac {-\frac {-\frac {3 \int \frac {5 b^3 c^3+12 a b^2 d c^2+24 a^2 b d^2 c+64 a^3 d^3+b d \left (15 b^2 c^2+26 a b d c+48 a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^2}dx}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {15 b^2 c}{a}+\frac {48 a d^2}{c}+26 b d\right )}{x (c+d x)}}{4 a c}-\frac {\sqrt {a+b x} (8 a d+5 b c)}{2 a c x^2 (c+d x)}}{6 a c}-\frac {\sqrt {a+b x}}{3 a c x^3 (c+d x)}\right )}{\sqrt {a x^2+b x^3}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {x \sqrt {a+b x} \left (-\frac {-\frac {-\frac {3 \left (\frac {2 d \sqrt {a+b x} \left (-32 a^3 d^3+12 a^2 b c d^2+7 a b^2 c^2 d+5 b^3 c^3\right )}{c (c+d x) (b c-a d)}-\frac {\int -\frac {(b c-a d) \left (5 b^3 c^3+12 a b^2 d c^2+24 a^2 b d^2 c+64 a^3 d^3\right )+b d \left (5 b^3 c^3+7 a b^2 d c^2+12 a^2 b d^2 c-32 a^3 d^3\right ) x}{x \sqrt {a+b x} (c+d x)}dx}{c (b c-a d)}\right )}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {15 b^2 c}{a}+\frac {48 a d^2}{c}+26 b d\right )}{x (c+d x)}}{4 a c}-\frac {\sqrt {a+b x} (8 a d+5 b c)}{2 a c x^2 (c+d x)}}{6 a c}-\frac {\sqrt {a+b x}}{3 a c x^3 (c+d x)}\right )}{\sqrt {a x^2+b x^3}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x \sqrt {a+b x} \left (-\frac {-\frac {-\frac {3 \left (\frac {\int \frac {(b c-a d) \left (5 b^3 c^3+12 a b^2 d c^2+24 a^2 b d^2 c+64 a^3 d^3\right )+b d \left (5 b^3 c^3+7 a b^2 d c^2+12 a^2 b d^2 c-32 a^3 d^3\right ) x}{x \sqrt {a+b x} (c+d x)}dx}{c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (-32 a^3 d^3+12 a^2 b c d^2+7 a b^2 c^2 d+5 b^3 c^3\right )}{c (c+d x) (b c-a d)}\right )}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {15 b^2 c}{a}+\frac {48 a d^2}{c}+26 b d\right )}{x (c+d x)}}{4 a c}-\frac {\sqrt {a+b x} (8 a d+5 b c)}{2 a c x^2 (c+d x)}}{6 a c}-\frac {\sqrt {a+b x}}{3 a c x^3 (c+d x)}\right )}{\sqrt {a x^2+b x^3}}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {x \sqrt {a+b x} \left (-\frac {-\frac {-\frac {3 \left (\frac {\frac {(b c-a d) \left (64 a^3 d^3+24 a^2 b c d^2+12 a b^2 c^2 d+5 b^3 c^3\right ) \int \frac {1}{x \sqrt {a+b x}}dx}{c}-\frac {8 a^3 d^4 (9 b c-8 a d) \int \frac {1}{\sqrt {a+b x} (c+d x)}dx}{c}}{c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (-32 a^3 d^3+12 a^2 b c d^2+7 a b^2 c^2 d+5 b^3 c^3\right )}{c (c+d x) (b c-a d)}\right )}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {15 b^2 c}{a}+\frac {48 a d^2}{c}+26 b d\right )}{x (c+d x)}}{4 a c}-\frac {\sqrt {a+b x} (8 a d+5 b c)}{2 a c x^2 (c+d x)}}{6 a c}-\frac {\sqrt {a+b x}}{3 a c x^3 (c+d x)}\right )}{\sqrt {a x^2+b x^3}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {x \sqrt {a+b x} \left (-\frac {-\frac {-\frac {3 \left (\frac {\frac {2 (b c-a d) \left (64 a^3 d^3+24 a^2 b c d^2+12 a b^2 c^2 d+5 b^3 c^3\right ) \int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{b c}-\frac {16 a^3 d^4 (9 b c-8 a d) \int \frac {1}{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}d\sqrt {a+b x}}{b c}}{c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (-32 a^3 d^3+12 a^2 b c d^2+7 a b^2 c^2 d+5 b^3 c^3\right )}{c (c+d x) (b c-a d)}\right )}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {15 b^2 c}{a}+\frac {48 a d^2}{c}+26 b d\right )}{x (c+d x)}}{4 a c}-\frac {\sqrt {a+b x} (8 a d+5 b c)}{2 a c x^2 (c+d x)}}{6 a c}-\frac {\sqrt {a+b x}}{3 a c x^3 (c+d x)}\right )}{\sqrt {a x^2+b x^3}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {x \sqrt {a+b x} \left (-\frac {-\frac {-\frac {3 \left (\frac {\frac {2 (b c-a d) \left (64 a^3 d^3+24 a^2 b c d^2+12 a b^2 c^2 d+5 b^3 c^3\right ) \int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{b c}-\frac {16 a^3 d^{7/2} (9 b c-8 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}}{c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (-32 a^3 d^3+12 a^2 b c d^2+7 a b^2 c^2 d+5 b^3 c^3\right )}{c (c+d x) (b c-a d)}\right )}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {15 b^2 c}{a}+\frac {48 a d^2}{c}+26 b d\right )}{x (c+d x)}}{4 a c}-\frac {\sqrt {a+b x} (8 a d+5 b c)}{2 a c x^2 (c+d x)}}{6 a c}-\frac {\sqrt {a+b x}}{3 a c x^3 (c+d x)}\right )}{\sqrt {a x^2+b x^3}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {x \sqrt {a+b x} \left (-\frac {-\frac {-\frac {3 \left (\frac {-\frac {16 a^3 d^{7/2} (9 b c-8 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) (b c-a d) \left (64 a^3 d^3+24 a^2 b c d^2+12 a b^2 c^2 d+5 b^3 c^3\right )}{\sqrt {a} c}}{c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (-32 a^3 d^3+12 a^2 b c d^2+7 a b^2 c^2 d+5 b^3 c^3\right )}{c (c+d x) (b c-a d)}\right )}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {15 b^2 c}{a}+\frac {48 a d^2}{c}+26 b d\right )}{x (c+d x)}}{4 a c}-\frac {\sqrt {a+b x} (8 a d+5 b c)}{2 a c x^2 (c+d x)}}{6 a c}-\frac {\sqrt {a+b x}}{3 a c x^3 (c+d x)}\right )}{\sqrt {a x^2+b x^3}}\)

Input:

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

Output:

(x*Sqrt[a + b*x]*(-1/3*Sqrt[a + b*x]/(a*c*x^3*(c + d*x)) - (-1/2*((5*b*c + 
 8*a*d)*Sqrt[a + b*x])/(a*c*x^2*(c + d*x)) - (-((((15*b^2*c)/a + 26*b*d + 
(48*a*d^2)/c)*Sqrt[a + b*x])/(x*(c + d*x))) - (3*((2*d*(5*b^3*c^3 + 7*a*b^ 
2*c^2*d + 12*a^2*b*c*d^2 - 32*a^3*d^3)*Sqrt[a + b*x])/(c*(b*c - a*d)*(c + 
d*x)) + ((-16*a^3*d^(7/2)*(9*b*c - 8*a*d)*ArcTan[(Sqrt[d]*Sqrt[a + b*x])/S 
qrt[b*c - a*d]])/(c*Sqrt[b*c - a*d]) - (2*(b*c - a*d)*(5*b^3*c^3 + 12*a*b^ 
2*c^2*d + 24*a^2*b*c*d^2 + 64*a^3*d^3)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(Sq 
rt[a]*c))/(c*(b*c - a*d))))/(2*a*c))/(4*a*c))/(6*a*c)))/Sqrt[a*x^2 + b*x^3 
]
 

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 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 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S 
imp[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*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

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 1948

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + 
(d_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Simp[e^IntPart[m]*(e*x)^FracPart[m]*( 
(a*x^j + b*x^(j + n))^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x 
^n)^FracPart[p]))   Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; 
FreeQ[{a, b, c, d, e, j, m, n, p, q}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] 
 && NeQ[b*c - a*d, 0] &&  !(EqQ[n, 1] && EqQ[j, 1])
 

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.43

method	result	size
pseudoelliptic	\(\frac {-\frac {c \sqrt {b x +a}\, \left (-8 a d x -3 c b x +2 a c \right )}{4 a^{2} x^{2}}-\frac {\left (24 a^{2} d^{2}+8 a b c d +3 b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{4 a^{\frac {5}{2}}}-\frac {d^{3} \left (-\frac {c \sqrt {b x +a}}{d x +c}-\frac {\left (6 a d -7 b c \right ) \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {d \left (a d -b c \right )}}\right )}{\sqrt {d \left (a d -b c \right )}}\right )}{a d -b c}}{c^{4}}\)	\(155\)
risch	\(-\frac {\left (b x +a \right ) \left (72 a^{2} d^{2} x^{2}+36 a b c d \,x^{2}+15 b^{2} c^{2} x^{2}-24 a^{2} c d x -10 a b \,c^{2} x +8 a^{2} c^{2}\right )}{24 a^{3} c^{4} x^{2} \sqrt {x^{2} \left (b x +a \right )}}-\frac {b \left (-\frac {\left (64 a^{3} d^{3}+24 a^{2} b c \,d^{2}+12 a \,b^{2} c^{2} d +5 b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{b c \sqrt {a}}-\frac {16 a^{3} d^{4} \left (-\frac {b c \sqrt {b x +a}}{2 \left (a d -b c \right ) \left (d \left (b x +a \right )-a d +b c \right )}-\frac {\left (8 a d -9 b c \right ) \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {d \left (a d -b c \right )}}\right )}{2 \left (a d -b c \right ) \sqrt {d \left (a d -b c \right )}}\right )}{b c}\right ) \sqrt {b x +a}\, x}{8 c^{4} a^{3} \sqrt {x^{2} \left (b x +a \right )}}\)	\(282\)
default	\(\text {Expression too large to display}\)	\(1391\)

Input:

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

Output:

1/c^4*(-1/4*c*(b*x+a)^(1/2)*(-8*a*d*x-3*b*c*x+2*a*c)/a^2/x^2-1/4*(24*a^2*d 
^2+8*a*b*c*d+3*b^2*c^2)/a^(5/2)*arctanh((b*x+a)^(1/2)/a^(1/2))-d^3/(a*d-b* 
c)*(-c*(b*x+a)^(1/2)/(d*x+c)-(6*a*d-7*b*c)/(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 = 0.66 (sec) , antiderivative size = 1979, normalized size of antiderivative = 5.47 \[ \int \frac {1}{x^3 (c+d x)^2 \sqrt {a x^2+b x^3}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/48*(24*((9*a^4*b*c*d^4 - 8*a^5*d^5)*x^5 + (9*a^4*b*c^2*d^3 - 8*a^5*c*d^ 
4)*x^4)*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)) + 3*((5*b^4*c^4 
*d + 7*a*b^3*c^3*d^2 + 12*a^2*b^2*c^2*d^3 + 40*a^3*b*c*d^4 - 64*a^4*d^5)*x 
^5 + (5*b^4*c^5 + 7*a*b^3*c^4*d + 12*a^2*b^2*c^3*d^2 + 40*a^3*b*c^2*d^3 - 
64*a^4*c*d^4)*x^4)*sqrt(a)*log((b*x^2 + 2*a*x + 2*sqrt(b*x^3 + a*x^2)*sqrt 
(a))/x^2) - 2*(8*a^3*b*c^5 - 8*a^4*c^4*d + 3*(5*a*b^3*c^4*d + 7*a^2*b^2*c^ 
3*d^2 + 12*a^3*b*c^2*d^3 - 32*a^4*c*d^4)*x^3 + (15*a*b^3*c^5 + 11*a^2*b^2* 
c^4*d + 22*a^3*b*c^3*d^2 - 48*a^4*c^2*d^3)*x^2 - 2*(5*a^2*b^2*c^5 + 3*a^3* 
b*c^4*d - 8*a^4*c^3*d^2)*x)*sqrt(b*x^3 + a*x^2))/((a^4*b*c^6*d - a^5*c^5*d 
^2)*x^5 + (a^4*b*c^7 - a^5*c^6*d)*x^4), 1/48*(48*((9*a^4*b*c*d^4 - 8*a^5*d 
^5)*x^5 + (9*a^4*b*c^2*d^3 - 8*a^5*c*d^4)*x^4)*sqrt(d/(b*c - a*d))*arctan( 
sqrt(b*x^3 + a*x^2)*sqrt(d/(b*c - a*d))/x) + 3*((5*b^4*c^4*d + 7*a*b^3*c^3 
*d^2 + 12*a^2*b^2*c^2*d^3 + 40*a^3*b*c*d^4 - 64*a^4*d^5)*x^5 + (5*b^4*c^5 
+ 7*a*b^3*c^4*d + 12*a^2*b^2*c^3*d^2 + 40*a^3*b*c^2*d^3 - 64*a^4*c*d^4)*x^ 
4)*sqrt(a)*log((b*x^2 + 2*a*x + 2*sqrt(b*x^3 + a*x^2)*sqrt(a))/x^2) - 2*(8 
*a^3*b*c^5 - 8*a^4*c^4*d + 3*(5*a*b^3*c^4*d + 7*a^2*b^2*c^3*d^2 + 12*a^3*b 
*c^2*d^3 - 32*a^4*c*d^4)*x^3 + (15*a*b^3*c^5 + 11*a^2*b^2*c^4*d + 22*a^3*b 
*c^3*d^2 - 48*a^4*c^2*d^3)*x^2 - 2*(5*a^2*b^2*c^5 + 3*a^3*b*c^4*d - 8*a^4* 
c^3*d^2)*x)*sqrt(b*x^3 + a*x^2))/((a^4*b*c^6*d - a^5*c^5*d^2)*x^5 + (a^...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(1/x^3/(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^3), x)
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1238, normalized size of antiderivative = 3.42 \[ \int \frac {1}{x^3 (c+d x)^2 \sqrt {a x^2+b x^3}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 384*sqrt(d)*sqrt( - a*d + b*c)*atan((sqrt(a + b*x)*d)/(sqrt(d)*sqrt( - 
 a*d + b*c)))*a**5*c*d**4*x**3 - 384*sqrt(d)*sqrt( - a*d + b*c)*atan((sqrt 
(a + b*x)*d)/(sqrt(d)*sqrt( - a*d + b*c)))*a**5*d**5*x**4 + 432*sqrt(d)*sq 
rt( - a*d + b*c)*atan((sqrt(a + b*x)*d)/(sqrt(d)*sqrt( - a*d + b*c)))*a**4 
*b*c**2*d**3*x**3 + 432*sqrt(d)*sqrt( - a*d + b*c)*atan((sqrt(a + b*x)*d)/ 
(sqrt(d)*sqrt( - a*d + b*c)))*a**4*b*c*d**4*x**4 - 16*sqrt(a + b*x)*a**5*c 
**4*d**2 + 32*sqrt(a + b*x)*a**5*c**3*d**3*x - 96*sqrt(a + b*x)*a**5*c**2* 
d**4*x**2 - 192*sqrt(a + b*x)*a**5*c*d**5*x**3 + 32*sqrt(a + b*x)*a**4*b*c 
**5*d - 44*sqrt(a + b*x)*a**4*b*c**4*d**2*x + 140*sqrt(a + b*x)*a**4*b*c** 
3*d**3*x**2 + 264*sqrt(a + b*x)*a**4*b*c**2*d**4*x**3 - 16*sqrt(a + b*x)*a 
**3*b**2*c**6 - 8*sqrt(a + b*x)*a**3*b**2*c**5*d*x - 22*sqrt(a + b*x)*a**3 
*b**2*c**4*d**2*x**2 - 30*sqrt(a + b*x)*a**3*b**2*c**3*d**3*x**3 + 20*sqrt 
(a + b*x)*a**2*b**3*c**6*x + 8*sqrt(a + b*x)*a**2*b**3*c**5*d*x**2 - 12*sq 
rt(a + b*x)*a**2*b**3*c**4*d**2*x**3 - 30*sqrt(a + b*x)*a*b**4*c**6*x**2 - 
 30*sqrt(a + b*x)*a*b**4*c**5*d*x**3 - 192*sqrt(a)*log(sqrt(a + b*x) - sqr 
t(a))*a**5*c*d**5*x**3 - 192*sqrt(a)*log(sqrt(a + b*x) - sqrt(a))*a**5*d** 
6*x**4 + 312*sqrt(a)*log(sqrt(a + b*x) - sqrt(a))*a**4*b*c**2*d**4*x**3 + 
312*sqrt(a)*log(sqrt(a + b*x) - sqrt(a))*a**4*b*c*d**5*x**4 - 84*sqrt(a)*l 
og(sqrt(a + b*x) - sqrt(a))*a**3*b**2*c**3*d**3*x**3 - 84*sqrt(a)*log(sqrt 
(a + b*x) - sqrt(a))*a**3*b**2*c**2*d**4*x**4 - 15*sqrt(a)*log(sqrt(a +...