\(\int \frac {1}{x^3 (c+d x)^2 \sqrt {a x+b x^2}} \, dx\) [156]

Optimal result

Integrand size = 24, antiderivative size = 281 \[ \int \frac {1}{x^3 (c+d x)^2 \sqrt {a x+b x^2}} \, dx=-\frac {(2 b c-7 a d) \sqrt {a x+b x^2}}{5 a c^2 (b c-a d) x^3}+\frac {\left (8 b^2 c^2+12 a b c d-35 a^2 d^2\right ) \sqrt {a x+b x^2}}{15 a^2 c^3 (b c-a d) x^2}-\frac {\left (16 b^3 c^3+24 a b^2 c^2 d+50 a^2 b c d^2-105 a^3 d^3\right ) \sqrt {a x+b x^2}}{15 a^3 c^4 (b c-a d) x}-\frac {d \sqrt {a x+b x^2}}{c (b c-a d) x^3 (c+d x)}-\frac {d^3 (8 b c-7 a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{c^{9/2} (b c-a d)^{3/2}} \] Output:

-1/5*(-7*a*d+2*b*c)*(b*x^2+a*x)^(1/2)/a/c^2/(-a*d+b*c)/x^3+1/15*(-35*a^2*d 
^2+12*a*b*c*d+8*b^2*c^2)*(b*x^2+a*x)^(1/2)/a^2/c^3/(-a*d+b*c)/x^2-1/15*(-1 
05*a^3*d^3+50*a^2*b*c*d^2+24*a*b^2*c^2*d+16*b^3*c^3)*(b*x^2+a*x)^(1/2)/a^3 
/c^4/(-a*d+b*c)/x-d*(b*x^2+a*x)^(1/2)/c/(-a*d+b*c)/x^3/(d*x+c)-d^3*(-7*a*d 
+8*b*c)*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a*x)^(1/2))/c^(9/2)/(-a* 
d+b*c)^(3/2)
 

Mathematica [A] (verified)

Time = 1.79 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^3 (c+d x)^2 \sqrt {a x+b x^2}} \, dx=\frac {\frac {\sqrt {c} (a+b x) \left (16 b^3 c^3 x^2 (c+d x)-8 a b^2 c^2 x \left (c^2-2 c d x-3 d^2 x^2\right )+2 a^2 b c \left (3 c^3-3 c^2 d x+19 c d^2 x^2+25 d^3 x^3\right )-a^3 d \left (6 c^3-14 c^2 d x+70 c d^2 x^2+105 d^3 x^3\right )\right )}{a^3 (-b c+a d) (c+d x)}-\frac {15 d^3 (8 b c-7 a d) x^{5/2} \sqrt {a+b x} \arctan \left (\frac {-d \sqrt {x} \sqrt {a+b x}+\sqrt {b} (c+d x)}{\sqrt {c} \sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}}{15 c^{9/2} x^2 \sqrt {x (a+b x)}} \] Input:

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

Output:

((Sqrt[c]*(a + b*x)*(16*b^3*c^3*x^2*(c + d*x) - 8*a*b^2*c^2*x*(c^2 - 2*c*d 
*x - 3*d^2*x^2) + 2*a^2*b*c*(3*c^3 - 3*c^2*d*x + 19*c*d^2*x^2 + 25*d^3*x^3 
) - a^3*d*(6*c^3 - 14*c^2*d*x + 70*c*d^2*x^2 + 105*d^3*x^3)))/(a^3*(-(b*c) 
 + a*d)*(c + d*x)) - (15*d^3*(8*b*c - 7*a*d)*x^(5/2)*Sqrt[a + b*x]*ArcTan[ 
(-(d*Sqrt[x]*Sqrt[a + b*x]) + Sqrt[b]*(c + d*x))/(Sqrt[c]*Sqrt[-(b*c) + a* 
d])])/(-(b*c) + a*d)^(3/2))/(15*c^(9/2)*x^2*Sqrt[x*(a + b*x)])
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {1261, 114, 27, 169, 27, 169, 27, 169, 27, 104, 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.

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

\(\Big \downarrow \) 1261

\(\displaystyle \frac {\sqrt {x} \sqrt {a+b x} \int \frac {1}{x^{7/2} \sqrt {a+b x} (c+d x)^2}dx}{\sqrt {a x+b x^2}}\)

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

\(\displaystyle \frac {\sqrt {x} \sqrt {a+b x} \left (\frac {-\frac {2 \int \frac {8 b^2 c^2+12 a b d c-35 a^2 d^2+4 b d (2 b c-7 a d) x}{2 x^{5/2} \sqrt {a+b x} (c+d x)}dx}{5 a c}-\frac {2 \sqrt {a+b x} (2 b c-7 a d)}{5 a c x^{5/2}}}{2 c (b c-a d)}-\frac {d \sqrt {a+b x}}{c x^{5/2} (c+d x) (b c-a d)}\right )}{\sqrt {a x+b x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {x} \sqrt {a+b x} \left (\frac {-\frac {\int \frac {8 b^2 c^2+12 a b d c-35 a^2 d^2+4 b d (2 b c-7 a d) x}{x^{5/2} \sqrt {a+b x} (c+d x)}dx}{5 a c}-\frac {2 \sqrt {a+b x} (2 b c-7 a d)}{5 a c x^{5/2}}}{2 c (b c-a d)}-\frac {d \sqrt {a+b x}}{c x^{5/2} (c+d x) (b c-a d)}\right )}{\sqrt {a x+b x^2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\sqrt {x} \sqrt {a+b x} \left (\frac {-\frac {-\frac {2 \int \frac {16 b^3 c^3+24 a b^2 d c^2+50 a^2 b d^2 c-105 a^3 d^3+2 b d \left (8 b^2 c^2+12 a b d c-35 a^2 d^2\right ) x}{2 x^{3/2} \sqrt {a+b x} (c+d x)}dx}{3 a c}-\frac {2 \sqrt {a+b x} \left (\frac {8 b^2 c}{a}-\frac {35 a d^2}{c}+12 b d\right )}{3 x^{3/2}}}{5 a c}-\frac {2 \sqrt {a+b x} (2 b c-7 a d)}{5 a c x^{5/2}}}{2 c (b c-a d)}-\frac {d \sqrt {a+b x}}{c x^{5/2} (c+d x) (b c-a d)}\right )}{\sqrt {a x+b x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {x} \sqrt {a+b x} \left (\frac {-\frac {-\frac {\int \frac {16 b^3 c^3+24 a b^2 d c^2+50 a^2 b d^2 c-105 a^3 d^3+2 b d \left (8 b^2 c^2+12 a b d c-35 a^2 d^2\right ) x}{x^{3/2} \sqrt {a+b x} (c+d x)}dx}{3 a c}-\frac {2 \sqrt {a+b x} \left (\frac {8 b^2 c}{a}-\frac {35 a d^2}{c}+12 b d\right )}{3 x^{3/2}}}{5 a c}-\frac {2 \sqrt {a+b x} (2 b c-7 a d)}{5 a c x^{5/2}}}{2 c (b c-a d)}-\frac {d \sqrt {a+b x}}{c x^{5/2} (c+d x) (b c-a d)}\right )}{\sqrt {a x+b x^2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\sqrt {x} \sqrt {a+b x} \left (\frac {-\frac {-\frac {-\frac {2 \int \frac {15 a^3 d^3 (8 b c-7 a d)}{2 \sqrt {x} \sqrt {a+b x} (c+d x)}dx}{a c}-\frac {2 \sqrt {a+b x} \left (-105 a^3 d^3+50 a^2 b c d^2+24 a b^2 c^2 d+16 b^3 c^3\right )}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \left (\frac {8 b^2 c}{a}-\frac {35 a d^2}{c}+12 b d\right )}{3 x^{3/2}}}{5 a c}-\frac {2 \sqrt {a+b x} (2 b c-7 a d)}{5 a c x^{5/2}}}{2 c (b c-a d)}-\frac {d \sqrt {a+b x}}{c x^{5/2} (c+d x) (b c-a d)}\right )}{\sqrt {a x+b x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {x} \sqrt {a+b x} \left (\frac {-\frac {-\frac {-\frac {15 a^2 d^3 (8 b c-7 a d) \int \frac {1}{\sqrt {x} \sqrt {a+b x} (c+d x)}dx}{c}-\frac {2 \sqrt {a+b x} \left (-105 a^3 d^3+50 a^2 b c d^2+24 a b^2 c^2 d+16 b^3 c^3\right )}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \left (\frac {8 b^2 c}{a}-\frac {35 a d^2}{c}+12 b d\right )}{3 x^{3/2}}}{5 a c}-\frac {2 \sqrt {a+b x} (2 b c-7 a d)}{5 a c x^{5/2}}}{2 c (b c-a d)}-\frac {d \sqrt {a+b x}}{c x^{5/2} (c+d x) (b c-a d)}\right )}{\sqrt {a x+b x^2}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\sqrt {x} \sqrt {a+b x} \left (\frac {-\frac {-\frac {-\frac {30 a^2 d^3 (8 b c-7 a d) \int \frac {1}{c-\frac {(b c-a d) x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{c}-\frac {2 \sqrt {a+b x} \left (-105 a^3 d^3+50 a^2 b c d^2+24 a b^2 c^2 d+16 b^3 c^3\right )}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \left (\frac {8 b^2 c}{a}-\frac {35 a d^2}{c}+12 b d\right )}{3 x^{3/2}}}{5 a c}-\frac {2 \sqrt {a+b x} (2 b c-7 a d)}{5 a c x^{5/2}}}{2 c (b c-a d)}-\frac {d \sqrt {a+b x}}{c x^{5/2} (c+d x) (b c-a d)}\right )}{\sqrt {a x+b x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {x} \sqrt {a+b x} \left (\frac {-\frac {-\frac {-\frac {30 a^2 d^3 (8 b c-7 a d) \text {arctanh}\left (\frac {\sqrt {x} \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x}}\right )}{c^{3/2} \sqrt {b c-a d}}-\frac {2 \sqrt {a+b x} \left (-105 a^3 d^3+50 a^2 b c d^2+24 a b^2 c^2 d+16 b^3 c^3\right )}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \left (\frac {8 b^2 c}{a}-\frac {35 a d^2}{c}+12 b d\right )}{3 x^{3/2}}}{5 a c}-\frac {2 \sqrt {a+b x} (2 b c-7 a d)}{5 a c x^{5/2}}}{2 c (b c-a d)}-\frac {d \sqrt {a+b x}}{c x^{5/2} (c+d x) (b c-a d)}\right )}{\sqrt {a x+b x^2}}\)

Input:

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

Output:

(Sqrt[x]*Sqrt[a + b*x]*(-((d*Sqrt[a + b*x])/(c*(b*c - a*d)*x^(5/2)*(c + d* 
x))) + ((-2*(2*b*c - 7*a*d)*Sqrt[a + b*x])/(5*a*c*x^(5/2)) - ((-2*((8*b^2* 
c)/a + 12*b*d - (35*a*d^2)/c)*Sqrt[a + b*x])/(3*x^(3/2)) - ((-2*(16*b^3*c^ 
3 + 24*a*b^2*c^2*d + 50*a^2*b*c*d^2 - 105*a^3*d^3)*Sqrt[a + b*x])/(a*c*Sqr 
t[x]) - (30*a^2*d^3*(8*b*c - 7*a*d)*ArcTanh[(Sqrt[b*c - a*d]*Sqrt[x])/(Sqr 
t[c]*Sqrt[a + b*x])])/(c^(3/2)*Sqrt[b*c - a*d]))/(3*a*c))/(5*a*c))/(2*c*(b 
*c - a*d))))/Sqrt[a*x + b*x^2]
 

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 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*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 169

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] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

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 1261

Int[((e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((b_.)*(x_) + (c_.)*(x_)^2) 
^(p_), x_Symbol] :> Simp[(e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) 
  Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m, 
 n}, x] &&  !IGtQ[n, 0]
 

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.76

method	result	size
pseudoelliptic	\(\frac {7 \left (d x +c \right ) d^{3} x^{3} a^{3} \left (a d -\frac {8 b c}{7}\right ) \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )-\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {c \left (a d -b c \right )}\, \left (\left (-\frac {8}{3} b^{3} x^{2}+\frac {4}{3} x a \,b^{2}-a^{2} b \right ) c^{4}+d \left (b x +a \right ) \left (-\frac {8 b^{2} x^{2}}{3}+a^{2}\right ) c^{3}-\frac {7 d^{2} x \left (b x +a \right ) \left (\frac {12 b x}{7}+a \right ) a \,c^{2}}{3}+\frac {35 d^{3} \left (-\frac {5 b x}{7}+a \right ) x^{2} a^{2} c}{3}+\frac {35 a^{3} d^{4} x^{3}}{2}\right )}{5}}{c^{4} x^{3} \left (a d -b c \right ) \left (d x +c \right ) \sqrt {c \left (a d -b c \right )}\, a^{3}}\)	\(214\)
risch	\(-\frac {2 \left (b x +a \right ) \left (45 a^{2} d^{2} x^{2}+20 a b c d \,x^{2}+8 b^{2} c^{2} x^{2}-10 a^{2} c d x -4 a b \,c^{2} x +3 a^{2} c^{2}\right )}{15 a^{3} c^{4} \sqrt {x \left (b x +a \right )}\, x^{2}}-\frac {d^{3} \left (\frac {c \left (\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{c \left (a d -b c \right ) \left (x +\frac {c}{d}\right )}-\frac {\left (a d -2 b c \right ) d \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{2 c \left (a d -b c \right ) \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{d^{2}}-\frac {3 \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{c^{4}}\)	\(452\)
default	\(\frac {-\frac {2 \sqrt {b \,x^{2}+a x}}{5 a \,x^{3}}-\frac {4 b \left (-\frac {2 \sqrt {b \,x^{2}+a x}}{3 a \,x^{2}}+\frac {4 b \sqrt {b \,x^{2}+a x}}{3 a^{2} x}\right )}{5 a}}{c^{2}}-\frac {6 d^{2} \sqrt {b \,x^{2}+a x}}{c^{4} a x}-\frac {2 d \left (-\frac {2 \sqrt {b \,x^{2}+a x}}{3 a \,x^{2}}+\frac {4 b \sqrt {b \,x^{2}+a x}}{3 a^{2} x}\right )}{c^{3}}+\frac {3 d^{2} \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{c^{4} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}-\frac {d \left (\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{c \left (a d -b c \right ) \left (x +\frac {c}{d}\right )}-\frac {\left (a d -2 b c \right ) d \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{2 c \left (a d -b c \right ) \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{c^{3}}\)	\(507\)

Input:

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

Output:

7*((d*x+c)*d^3*x^3*a^3*(a*d-8/7*b*c)*arctan((x*(b*x+a))^(1/2)/x*c/(c*(a*d- 
b*c))^(1/2))-2/35*(x*(b*x+a))^(1/2)*(c*(a*d-b*c))^(1/2)*((-8/3*b^3*x^2+4/3 
*x*a*b^2-a^2*b)*c^4+d*(b*x+a)*(-8/3*b^2*x^2+a^2)*c^3-7/3*d^2*x*(b*x+a)*(12 
/7*b*x+a)*a*c^2+35/3*d^3*(-5/7*b*x+a)*x^2*a^2*c+35/2*a^3*d^4*x^3))/(c*(a*d 
-b*c))^(1/2)/c^4/x^3/(a*d-b*c)/(d*x+c)/a^3
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 787, normalized size of antiderivative = 2.80 \[ \int \frac {1}{x^3 (c+d x)^2 \sqrt {a x+b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/30*(15*((8*a^3*b*c*d^4 - 7*a^4*d^5)*x^4 + (8*a^3*b*c^2*d^3 - 7*a^4*c*d^ 
4)*x^3)*sqrt(b*c^2 - a*c*d)*log((a*c + (2*b*c - a*d)*x - 2*sqrt(b*c^2 - a* 
c*d)*sqrt(b*x^2 + a*x))/(d*x + c)) - 2*(6*a^2*b^2*c^6 - 12*a^3*b*c^5*d + 6 
*a^4*c^4*d^2 + (16*b^4*c^5*d + 8*a*b^3*c^4*d^2 + 26*a^2*b^2*c^3*d^3 - 155* 
a^3*b*c^2*d^4 + 105*a^4*c*d^5)*x^3 + 2*(8*b^4*c^6 + 11*a^2*b^2*c^4*d^2 - 5 
4*a^3*b*c^3*d^3 + 35*a^4*c^2*d^4)*x^2 - 2*(4*a*b^3*c^6 - a^2*b^2*c^5*d - 1 
0*a^3*b*c^4*d^2 + 7*a^4*c^3*d^3)*x)*sqrt(b*x^2 + a*x))/((a^3*b^2*c^7*d - 2 
*a^4*b*c^6*d^2 + a^5*c^5*d^3)*x^4 + (a^3*b^2*c^8 - 2*a^4*b*c^7*d + a^5*c^6 
*d^2)*x^3), 1/15*(15*((8*a^3*b*c*d^4 - 7*a^4*d^5)*x^4 + (8*a^3*b*c^2*d^3 - 
 7*a^4*c*d^4)*x^3)*sqrt(-b*c^2 + a*c*d)*arctan(sqrt(-b*c^2 + a*c*d)*sqrt(b 
*x^2 + a*x)/(b*c*x + a*c)) - (6*a^2*b^2*c^6 - 12*a^3*b*c^5*d + 6*a^4*c^4*d 
^2 + (16*b^4*c^5*d + 8*a*b^3*c^4*d^2 + 26*a^2*b^2*c^3*d^3 - 155*a^3*b*c^2* 
d^4 + 105*a^4*c*d^5)*x^3 + 2*(8*b^4*c^6 + 11*a^2*b^2*c^4*d^2 - 54*a^3*b*c^ 
3*d^3 + 35*a^4*c^2*d^4)*x^2 - 2*(4*a*b^3*c^6 - a^2*b^2*c^5*d - 10*a^3*b*c^ 
4*d^2 + 7*a^4*c^3*d^3)*x)*sqrt(b*x^2 + a*x))/((a^3*b^2*c^7*d - 2*a^4*b*c^6 
*d^2 + a^5*c^5*d^3)*x^4 + (a^3*b^2*c^8 - 2*a^4*b*c^7*d + a^5*c^6*d^2)*x^3) 
]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: NotImplementedError >> unable to parse Giac output: 1603 
70139 icas_eval sage2Psr 0, Mod 3.16228, Heu 6.4, Min0GCD dim 3 degree 1 p 
srgcdop 0 heuop 6.4 modgcdop 8,125// Using PSR gcd 160.382 NTL factor begi 
n160.382 NTL factor
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 1.81 (sec) , antiderivative size = 1494, normalized size of antiderivative = 5.32 \[ \int \frac {1}{x^3 (c+d x)^2 \sqrt {a x+b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(735*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) 
 - sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a**5*c*d**5*x**3 + 735*sqrt 
(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x 
)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a**5*d**6*x**4 - 1260*sqrt(c)*sqrt(a 
*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)* 
sqrt(b))/(sqrt(c)*sqrt(b)))*a**4*b*c**2*d**4*x**3 - 1260*sqrt(c)*sqrt(a*d 
- b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqr 
t(b))/(sqrt(c)*sqrt(b)))*a**4*b*c*d**5*x**4 + 480*sqrt(c)*sqrt(a*d - b*c)* 
atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b))/( 
sqrt(c)*sqrt(b)))*a**3*b**2*c**3*d**3*x**3 + 480*sqrt(c)*sqrt(a*d - b*c)*a 
tan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b))/(s 
qrt(c)*sqrt(b)))*a**3*b**2*c**2*d**4*x**4 + 735*sqrt(c)*sqrt(a*d - b*c)*at 
an((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sq 
rt(c)*sqrt(b)))*a**5*c*d**5*x**3 + 735*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt( 
a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqr 
t(b)))*a**5*d**6*x**4 - 1260*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) 
 + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a** 
4*b*c**2*d**4*x**3 - 1260*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + 
sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a**4*b 
*c*d**5*x**4 + 480*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt...