\(\int \frac {(a x+b x^2)^{3/2}}{x^2 (c+d x)} \, dx\) [95]

Optimal result

Integrand size = 24, antiderivative size = 114 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^2 (c+d x)} \, dx=\frac {b \sqrt {a x+b x^2}}{d}-\frac {\sqrt {b} (2 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{d^2}+\frac {2 (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{\sqrt {c} d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^2 (c+d x)} \, dx=\frac {\sqrt {x} \sqrt {a+b x} \left (b \sqrt {c} d \sqrt {x} \sqrt {a+b x}-2 (-b c+a d)^{3/2} \arctan \left (\frac {-d \sqrt {x} \sqrt {a+b x}+\sqrt {b} (c+d x)}{\sqrt {c} \sqrt {-b c+a d}}\right )+\sqrt {b} \sqrt {c} (2 b c-3 a d) \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )\right )}{\sqrt {c} d^2 \sqrt {x (a+b x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.32, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1261, 113, 27, 175, 65, 104, 219, 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.

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

\(\Big \downarrow \) 1261

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

\(\Big \downarrow \) 113

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 175

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

\(\Big \downarrow \) 65

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

((a*x + b*x^2)^(3/2)*((b*Sqrt[x]*Sqrt[a + b*x])/d - ((2*Sqrt[b]*(2*b*c - 3 
*a*d)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/d - (4*(b*c - a*d)^(3/2)*A 
rcTanh[(Sqrt[b*c - a*d]*Sqrt[x])/(Sqrt[c]*Sqrt[a + b*x])])/(Sqrt[c]*d))/(2 
*d)))/(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 65

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2   Sub 
st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d 
}, x] &&  !GtQ[c, 0]
 

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 113

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 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
 

rule 175

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

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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 = 0.63 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.91

method	result	size
pseudoelliptic	\(-\frac {b \left (-d \sqrt {x \left (b x +a \right )}-\frac {\left (3 a d -2 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )}{\sqrt {b}}\right )+\frac {2 \left (a d -b c \right )^{2} \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )}{\sqrt {c \left (a d -b c \right )}}}{d^{2}}\)	\(104\)
risch	\(\frac {x \left (b x +a \right ) b}{d \sqrt {x \left (b x +a \right )}}+\frac {\frac {\sqrt {b}\, \left (3 a d -2 b c \right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{d}-\frac {2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}}{2 d}\)	\(220\)
default	\(\frac {\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{a \,x^{2}}-\frac {6 b \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3}+\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2}\right )}{a}}{c}+\frac {d \left (\frac {\left (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}}\right )^{\frac {3}{2}}}{3}+\frac {\left (a d -2 b c \right ) \left (\frac {\left (2 b \left (x +\frac {c}{d}\right )+\frac {a d -2 b c}{d}\right ) \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}}}}{4 b}+\frac {\left (-\frac {4 b c \left (a d -b c \right )}{d^{2}}-\frac {\left (a d -2 b c \right )^{2}}{d^{2}}\right ) \ln \left (\frac {\frac {a d -2 b c}{2 d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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}}}\right )}{8 b^{\frac {3}{2}}}\right )}{2 d}-\frac {c \left (a d -b c \right ) \left (\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}}}+\frac {\left (a d -2 b c \right ) \ln \left (\frac {\frac {a d -2 b c}{2 d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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}}}\right )}{2 d \sqrt {b}}+\frac {c \left (a d -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 )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{d^{2}}\right )}{c^{2}}-\frac {d \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3}+\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2}\right )}{c^{2}}\)	\(731\)

Input:

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

Output:

-1/d^2*(b*(-d*(x*(b*x+a))^(1/2)-(3*a*d-2*b*c)/b^(1/2)*arctanh((x*(b*x+a))^ 
(1/2)/x/b^(1/2)))+2*(a*d-b*c)^2/(c*(a*d-b*c))^(1/2)*arctan((x*(b*x+a))^(1/ 
2)/x*c/(c*(a*d-b*c))^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 523, normalized size of antiderivative = 4.59 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^2 (c+d x)} \, dx=\left [\frac {2 \, \sqrt {b x^{2} + a x} b d - {\left (2 \, b c - 3 \, a d\right )} \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - 2 \, {\left (b c - a d\right )} \sqrt {\frac {b c - a d}{c}} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x - 2 \, \sqrt {b x^{2} + a x} c \sqrt {\frac {b c - a d}{c}}}{d x + c}\right )}{2 \, d^{2}}, \frac {2 \, \sqrt {b x^{2} + a x} b d + 4 \, {\left (b c - a d\right )} \sqrt {-\frac {b c - a d}{c}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} c \sqrt {-\frac {b c - a d}{c}}}{{\left (b c - a d\right )} x}\right ) - {\left (2 \, b c - 3 \, a d\right )} \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{2 \, d^{2}}, \frac {\sqrt {b x^{2} + a x} b d + {\left (2 \, b c - 3 \, a d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) - {\left (b c - a d\right )} \sqrt {\frac {b c - a d}{c}} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x - 2 \, \sqrt {b x^{2} + a x} c \sqrt {\frac {b c - a d}{c}}}{d x + c}\right )}{d^{2}}, \frac {\sqrt {b x^{2} + a x} b d + {\left (2 \, b c - 3 \, a d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) + 2 \, {\left (b c - a d\right )} \sqrt {-\frac {b c - a d}{c}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} c \sqrt {-\frac {b c - a d}{c}}}{{\left (b c - a d\right )} x}\right )}{d^{2}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.45 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^2 (c+d x)} \, dx=\frac {-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a d +2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b c -2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a d +2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b c +\sqrt {x}\, \sqrt {b x +a}\, b c d +3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a c d -2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) b \,c^{2}}{c \,d^{2}} \] Input:

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

Output:

( - 2*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*d + 2*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)))*b*c - 2*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*d + 2*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)))*b*c + sqrt(x)*sqrt(a + b 
*x)*b*c*d + 3*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a*c*d 
 - 2*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*b*c**2)/(c*d** 
2)