\(\int \frac {(a x+b x^2)^{3/2}}{x^4 (c+d x)^3} \, dx\) [114]

Optimal result

Integrand size = 24, antiderivative size = 219 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^4 (c+d x)^3} \, dx=-\frac {5 (10 b c-21 a d) \sqrt {a x+b x^2}}{12 c^4 x}-\frac {2 a \sqrt {a x+b x^2}}{3 c x^2 (c+d x)^2}+\frac {(3 b c-7 a d) \sqrt {a x+b x^2}}{6 c^2 x (c+d x)^2}+\frac {(12 b c-35 a d) \sqrt {a x+b x^2}}{12 c^3 x (c+d x)}+\frac {\left (8 b^2 c^2-40 a b c d+35 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{4 c^{9/2} \sqrt {b c-a d}} \] Output:

-5/12*(-21*a*d+10*b*c)*(b*x^2+a*x)^(1/2)/c^4/x-2/3*a*(b*x^2+a*x)^(1/2)/c/x 
^2/(d*x+c)^2+1/6*(-7*a*d+3*b*c)*(b*x^2+a*x)^(1/2)/c^2/x/(d*x+c)^2+1/12*(-3 
5*a*d+12*b*c)*(b*x^2+a*x)^(1/2)/c^3/x/(d*x+c)+1/4*(35*a^2*d^2-40*a*b*c*d+8 
*b^2*c^2)*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)^(1/2)
 

Mathematica [A] (verified)

Time = 10.51 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^4 (c+d x)^3} \, dx=\frac {(x (a+b x))^{3/2} \left (6 d (a+b x)+\frac {(c+d x) \left (3 c^{5/2} d (6 b c-7 a d) (a+b x)^{5/2}+\left (8 b^2 c^2-40 a b c d+35 a^2 d^2\right ) (c+d x) \left (\sqrt {c} \sqrt {a+b x} (4 b c x+a (c-3 d x))-3 (b c-a d)^{3/2} x^{3/2} \text {arctanh}\left (\frac {\sqrt {b c-a d} \sqrt {x}}{\sqrt {c} \sqrt {a+b x}}\right )\right )\right )}{c^{7/2} (b c-a d) (a+b x)^{3/2}}\right )}{12 c (-b c+a d) x^3 (c+d x)^2} \] Input:

Integrate[(a*x + b*x^2)^(3/2)/(x^4*(c + d*x)^3),x]
 

Output:

((x*(a + b*x))^(3/2)*(6*d*(a + b*x) + ((c + d*x)*(3*c^(5/2)*d*(6*b*c - 7*a 
*d)*(a + b*x)^(5/2) + (8*b^2*c^2 - 40*a*b*c*d + 35*a^2*d^2)*(c + d*x)*(Sqr 
t[c]*Sqrt[a + b*x]*(4*b*c*x + a*(c - 3*d*x)) - 3*(b*c - a*d)^(3/2)*x^(3/2) 
*ArcTanh[(Sqrt[b*c - a*d]*Sqrt[x])/(Sqrt[c]*Sqrt[a + b*x])])))/(c^(7/2)*(b 
*c - a*d)*(a + b*x)^(3/2))))/(12*c*(-(b*c) + a*d)*x^3*(c + d*x)^2)
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.29, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {1261, 109, 27, 168, 27, 168, 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.

\(\displaystyle \int \frac {\left (a x+b x^2\right )^{3/2}}{x^4 (c+d x)^3} \, dx\)

\(\Big \downarrow \) 1261

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} \left (-\frac {2 \int -\frac {a (4 b c-7 a d)+3 b (b c-2 a d) x}{2 x^{3/2} \sqrt {a+b x} (c+d x)^3}dx}{3 c}-\frac {2 a \sqrt {a+b x}}{3 c x^{3/2} (c+d x)^2}\right )}{x^{3/2} (a+b x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} \left (\frac {\int \frac {a (4 b c-7 a d)+3 b (b c-2 a d) x}{x^{3/2} \sqrt {a+b x} (c+d x)^3}dx}{3 c}-\frac {2 a \sqrt {a+b x}}{3 c x^{3/2} (c+d x)^2}\right )}{x^{3/2} (a+b x)^{3/2}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} \left (\frac {\frac {\sqrt {a+b x} (3 b c-7 a d)}{2 c \sqrt {x} (c+d x)^2}-\frac {\int -\frac {a (19 b c-35 a d) (b c-a d)+4 b (3 b c-7 a d) x (b c-a d)}{2 x^{3/2} \sqrt {a+b x} (c+d x)^2}dx}{2 c (b c-a d)}}{3 c}-\frac {2 a \sqrt {a+b x}}{3 c x^{3/2} (c+d x)^2}\right )}{x^{3/2} (a+b x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} \left (\frac {\frac {\int \frac {a (19 b c-35 a d) (b c-a d)+4 b (3 b c-7 a d) x (b c-a d)}{x^{3/2} \sqrt {a+b x} (c+d x)^2}dx}{4 c (b c-a d)}+\frac {\sqrt {a+b x} (3 b c-7 a d)}{2 c \sqrt {x} (c+d x)^2}}{3 c}-\frac {2 a \sqrt {a+b x}}{3 c x^{3/2} (c+d x)^2}\right )}{x^{3/2} (a+b x)^{3/2}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} \left (\frac {\frac {\frac {\sqrt {a+b x} (12 b c-35 a d) (b c-a d)}{c \sqrt {x} (c+d x)}-\frac {\int -\frac {(b c-a d)^2 (5 a (10 b c-21 a d)+2 b (12 b c-35 a d) x)}{2 x^{3/2} \sqrt {a+b x} (c+d x)}dx}{c (b c-a d)}}{4 c (b c-a d)}+\frac {\sqrt {a+b x} (3 b c-7 a d)}{2 c \sqrt {x} (c+d x)^2}}{3 c}-\frac {2 a \sqrt {a+b x}}{3 c x^{3/2} (c+d x)^2}\right )}{x^{3/2} (a+b x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} \left (\frac {\frac {\frac {(b c-a d) \int \frac {5 a (10 b c-21 a d)+2 b (12 b c-35 a d) x}{x^{3/2} \sqrt {a+b x} (c+d x)}dx}{2 c}+\frac {\sqrt {a+b x} (12 b c-35 a d) (b c-a d)}{c \sqrt {x} (c+d x)}}{4 c (b c-a d)}+\frac {\sqrt {a+b x} (3 b c-7 a d)}{2 c \sqrt {x} (c+d x)^2}}{3 c}-\frac {2 a \sqrt {a+b x}}{3 c x^{3/2} (c+d x)^2}\right )}{x^{3/2} (a+b x)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} \left (\frac {\frac {\frac {(b c-a d) \left (-\frac {2 \int -\frac {3 a \left (8 b^2 c^2-40 a b d c+35 a^2 d^2\right )}{2 \sqrt {x} \sqrt {a+b x} (c+d x)}dx}{a c}-\frac {10 \sqrt {a+b x} (10 b c-21 a d)}{c \sqrt {x}}\right )}{2 c}+\frac {\sqrt {a+b x} (12 b c-35 a d) (b c-a d)}{c \sqrt {x} (c+d x)}}{4 c (b c-a d)}+\frac {\sqrt {a+b x} (3 b c-7 a d)}{2 c \sqrt {x} (c+d x)^2}}{3 c}-\frac {2 a \sqrt {a+b x}}{3 c x^{3/2} (c+d x)^2}\right )}{x^{3/2} (a+b x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} \left (\frac {\frac {\frac {(b c-a d) \left (\frac {3 \left (35 a^2 d^2-40 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x} (c+d x)}dx}{c}-\frac {10 \sqrt {a+b x} (10 b c-21 a d)}{c \sqrt {x}}\right )}{2 c}+\frac {\sqrt {a+b x} (12 b c-35 a d) (b c-a d)}{c \sqrt {x} (c+d x)}}{4 c (b c-a d)}+\frac {\sqrt {a+b x} (3 b c-7 a d)}{2 c \sqrt {x} (c+d x)^2}}{3 c}-\frac {2 a \sqrt {a+b x}}{3 c x^{3/2} (c+d x)^2}\right )}{x^{3/2} (a+b x)^{3/2}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} \left (\frac {\frac {\frac {(b c-a d) \left (\frac {6 \left (35 a^2 d^2-40 a b c d+8 b^2 c^2\right ) \int \frac {1}{c-\frac {(b c-a d) x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{c}-\frac {10 \sqrt {a+b x} (10 b c-21 a d)}{c \sqrt {x}}\right )}{2 c}+\frac {\sqrt {a+b x} (12 b c-35 a d) (b c-a d)}{c \sqrt {x} (c+d x)}}{4 c (b c-a d)}+\frac {\sqrt {a+b x} (3 b c-7 a d)}{2 c \sqrt {x} (c+d x)^2}}{3 c}-\frac {2 a \sqrt {a+b x}}{3 c x^{3/2} (c+d x)^2}\right )}{x^{3/2} (a+b x)^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} \left (\frac {\frac {\frac {(b c-a d) \left (\frac {6 \left (35 a^2 d^2-40 a b c d+8 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {x} \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x}}\right )}{c^{3/2} \sqrt {b c-a d}}-\frac {10 \sqrt {a+b x} (10 b c-21 a d)}{c \sqrt {x}}\right )}{2 c}+\frac {\sqrt {a+b x} (12 b c-35 a d) (b c-a d)}{c \sqrt {x} (c+d x)}}{4 c (b c-a d)}+\frac {\sqrt {a+b x} (3 b c-7 a d)}{2 c \sqrt {x} (c+d x)^2}}{3 c}-\frac {2 a \sqrt {a+b x}}{3 c x^{3/2} (c+d x)^2}\right )}{x^{3/2} (a+b x)^{3/2}}\)

Input:

Int[(a*x + b*x^2)^(3/2)/(x^4*(c + d*x)^3),x]
 

Output:

((a*x + b*x^2)^(3/2)*((-2*a*Sqrt[a + b*x])/(3*c*x^(3/2)*(c + d*x)^2) + ((( 
3*b*c - 7*a*d)*Sqrt[a + b*x])/(2*c*Sqrt[x]*(c + d*x)^2) + (((12*b*c - 35*a 
*d)*(b*c - a*d)*Sqrt[a + b*x])/(c*Sqrt[x]*(c + d*x)) + ((b*c - a*d)*((-10* 
(10*b*c - 21*a*d)*Sqrt[a + b*x])/(c*Sqrt[x]) + (6*(8*b^2*c^2 - 40*a*b*c*d 
+ 35*a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*Sqrt[x])/(Sqrt[c]*Sqrt[a + b*x])])/ 
(c^(3/2)*Sqrt[b*c - a*d])))/(2*c))/(4*c*(b*c - a*d)))/(3*c)))/(x^(3/2)*(a 
+ b*x)^(3/2))
 

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 109

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

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 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 = 0.80 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.95

method	result	size
pseudoelliptic	\(-\frac {2 \left (\frac {105 \left (d x +c \right )^{2} \left (a^{2} d^{2}-\frac {8}{7} a b c d +\frac {8}{35} b^{2} c^{2}\right ) x^{2} a \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )}{8}+\sqrt {c \left (a d -b c \right )}\, \left (3 x \left (-\frac {13}{8} a \,d^{2} c +b \,c^{2} d \right ) \left (x \left (b x +a \right )\right )^{\frac {3}{2}}+\sqrt {x \left (b x +a \right )}\, \left (a \left (4 b x +a \right ) c^{3}-7 d \left (\frac {3}{7} b^{2} x^{2}-\frac {8}{7} a b x +a^{2}\right ) x \,c^{2}-17 d^{2} x^{2} \left (-\frac {89 b x}{136}+a \right ) a c -\frac {105 a^{2} d^{3} x^{3}}{8}\right )\right )\right )}{3 \sqrt {c \left (a d -b c \right )}\, c^{4} x^{2} \left (d x +c \right )^{2} a}\)	\(207\)
risch	\(-\frac {2 \left (b x +a \right ) \left (-9 a d x +4 c b x +a c \right )}{3 c^{4} \sqrt {x \left (b x +a \right )}\, x}+\frac {\frac {c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \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}}}}{2 c \left (a d -b c \right ) \left (x +\frac {c}{d}\right )^{2}}+\frac {3 \left (a d -2 b c \right ) 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 )}{4 c \left (a d -b c \right )}-\frac {b \,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 )}{2 c \left (a d -b c \right ) \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{d^{3}}-\frac {a \left (3 a d -2 b c \right ) \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 )}{\sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}+\frac {2 a c \left (a d -b c \right ) \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}}{c^{4}}\)	\(916\)
default	\(\text {Expression too large to display}\)	\(3621\)

Input:

int((b*x^2+a*x)^(3/2)/x^4/(d*x+c)^3,x,method=_RETURNVERBOSE)
 

Output:

-2/3/(c*(a*d-b*c))^(1/2)*(105/8*(d*x+c)^2*(a^2*d^2-8/7*a*b*c*d+8/35*b^2*c^ 
2)*x^2*a*arctan((x*(b*x+a))^(1/2)/x*c/(c*(a*d-b*c))^(1/2))+(c*(a*d-b*c))^( 
1/2)*(3*x*(-13/8*a*d^2*c+b*c^2*d)*(x*(b*x+a))^(3/2)+(x*(b*x+a))^(1/2)*(a*( 
4*b*x+a)*c^3-7*d*(3/7*b^2*x^2-8/7*a*b*x+a^2)*x*c^2-17*d^2*x^2*(-89/136*b*x 
+a)*a*c-105/8*a^2*d^3*x^3)))/c^4/x^2/(d*x+c)^2/a
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 710, normalized size of antiderivative = 3.24 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^4 (c+d x)^3} \, dx=\left [\frac {3 \, {\left ({\left (8 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (8 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{3} + {\left (8 \, b^{2} c^{4} - 40 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x + 2 \, \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a x}}{d x + c}\right ) - 2 \, {\left (8 \, a b c^{5} - 8 \, a^{2} c^{4} d + 5 \, {\left (10 \, b^{2} c^{3} d^{2} - 31 \, a b c^{2} d^{3} + 21 \, a^{2} c d^{4}\right )} x^{3} + {\left (88 \, b^{2} c^{4} d - 263 \, a b c^{3} d^{2} + 175 \, a^{2} c^{2} d^{3}\right )} x^{2} + 8 \, {\left (4 \, b^{2} c^{5} - 11 \, a b c^{4} d + 7 \, a^{2} c^{3} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{24 \, {\left ({\left (b c^{6} d^{2} - a c^{5} d^{3}\right )} x^{4} + 2 \, {\left (b c^{7} d - a c^{6} d^{2}\right )} x^{3} + {\left (b c^{8} - a c^{7} d\right )} x^{2}\right )}}, -\frac {3 \, {\left ({\left (8 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (8 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{3} + {\left (8 \, b^{2} c^{4} - 40 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} \sqrt {b x^{2} + a x}}{b c x + a c}\right ) + {\left (8 \, a b c^{5} - 8 \, a^{2} c^{4} d + 5 \, {\left (10 \, b^{2} c^{3} d^{2} - 31 \, a b c^{2} d^{3} + 21 \, a^{2} c d^{4}\right )} x^{3} + {\left (88 \, b^{2} c^{4} d - 263 \, a b c^{3} d^{2} + 175 \, a^{2} c^{2} d^{3}\right )} x^{2} + 8 \, {\left (4 \, b^{2} c^{5} - 11 \, a b c^{4} d + 7 \, a^{2} c^{3} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{12 \, {\left ({\left (b c^{6} d^{2} - a c^{5} d^{3}\right )} x^{4} + 2 \, {\left (b c^{7} d - a c^{6} d^{2}\right )} x^{3} + {\left (b c^{8} - a c^{7} d\right )} x^{2}\right )}}\right ] \] Input:

integrate((b*x^2+a*x)^(3/2)/x^4/(d*x+c)^3,x, algorithm="fricas")
 

Output:

[1/24*(3*((8*b^2*c^2*d^2 - 40*a*b*c*d^3 + 35*a^2*d^4)*x^4 + 2*(8*b^2*c^3*d 
 - 40*a*b*c^2*d^2 + 35*a^2*c*d^3)*x^3 + (8*b^2*c^4 - 40*a*b*c^3*d + 35*a^2 
*c^2*d^2)*x^2)*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*(8*a*b*c^5 - 8*a^2*c^4*d + 5 
*(10*b^2*c^3*d^2 - 31*a*b*c^2*d^3 + 21*a^2*c*d^4)*x^3 + (88*b^2*c^4*d - 26 
3*a*b*c^3*d^2 + 175*a^2*c^2*d^3)*x^2 + 8*(4*b^2*c^5 - 11*a*b*c^4*d + 7*a^2 
*c^3*d^2)*x)*sqrt(b*x^2 + a*x))/((b*c^6*d^2 - a*c^5*d^3)*x^4 + 2*(b*c^7*d 
- a*c^6*d^2)*x^3 + (b*c^8 - a*c^7*d)*x^2), -1/12*(3*((8*b^2*c^2*d^2 - 40*a 
*b*c*d^3 + 35*a^2*d^4)*x^4 + 2*(8*b^2*c^3*d - 40*a*b*c^2*d^2 + 35*a^2*c*d^ 
3)*x^3 + (8*b^2*c^4 - 40*a*b*c^3*d + 35*a^2*c^2*d^2)*x^2)*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)) + (8*a*b 
*c^5 - 8*a^2*c^4*d + 5*(10*b^2*c^3*d^2 - 31*a*b*c^2*d^3 + 21*a^2*c*d^4)*x^ 
3 + (88*b^2*c^4*d - 263*a*b*c^3*d^2 + 175*a^2*c^2*d^3)*x^2 + 8*(4*b^2*c^5 
- 11*a*b*c^4*d + 7*a^2*c^3*d^2)*x)*sqrt(b*x^2 + a*x))/((b*c^6*d^2 - a*c^5* 
d^3)*x^4 + 2*(b*c^7*d - a*c^6*d^2)*x^3 + (b*c^8 - a*c^7*d)*x^2)]
 

Sympy [F]

\[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^4 (c+d x)^3} \, dx=\int \frac {\left (x \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{4} \left (c + d x\right )^{3}}\, dx \] Input:

integrate((b*x**2+a*x)**(3/2)/x**4/(d*x+c)**3,x)
 

Output:

Integral((x*(a + b*x))**(3/2)/(x**4*(c + d*x)**3), x)
 

Maxima [F]

\[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^4 (c+d x)^3} \, dx=\int { \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{3} x^{4}} \,d x } \] Input:

integrate((b*x^2+a*x)^(3/2)/x^4/(d*x+c)^3,x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a*x)^(3/2)/((d*x + c)^3*x^4), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^4 (c+d x)^3} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((b*x^2+a*x)^(3/2)/x^4/(d*x+c)^3,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^4 (c+d x)^3} \, dx=\int \frac {{\left (b\,x^2+a\,x\right )}^{3/2}}{x^4\,{\left (c+d\,x\right )}^3} \,d x \] Input:

int((a*x + b*x^2)^(3/2)/(x^4*(c + d*x)^3),x)
 

Output:

int((a*x + b*x^2)^(3/2)/(x^4*(c + d*x)^3), x)
 

Reduce [F]

\[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^4 (c+d x)^3} \, dx=\int \frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{x^{4} \left (d x +c \right )^{3}}d x \] Input:

int((b*x^2+a*x)^(3/2)/x^4/(d*x+c)^3,x)
 

Output:

int((b*x^2+a*x)^(3/2)/x^4/(d*x+c)^3,x)