\(\int \frac {(a+b x+c x^2)^{3/2}}{x (d+e x)} \, dx\) [24]

Optimal result

Integrand size = 25, antiderivative size = 252 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x (d+e x)} \, dx=-\frac {(4 c d-5 b e) \sqrt {a+b x+c x^2}}{4 e^2}+\frac {c x \sqrt {a+b x+c x^2}}{2 e}-\frac {a^{3/2} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d}+\frac {\left (8 c^2 d^2+3 b^2 e^2-12 c e (b d-a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} e^3}-\frac {\left (c d^2-b d e+a e^2\right )^{3/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{d e^3} \] Output:

-1/4*(-5*b*e+4*c*d)*(c*x^2+b*x+a)^(1/2)/e^2+1/2*c*x*(c*x^2+b*x+a)^(1/2)/e- 
a^(3/2)*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/d+1/8*(8*c^2*d^ 
2+3*b^2*e^2-12*c*e*(-a*e+b*d))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a) 
^(1/2))/c^(1/2)/e^3-(a*e^2-b*d*e+c*d^2)^(3/2)*arctanh(1/2*(b*d-2*a*e+(-b*e 
+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/d/e^3
 

Mathematica [A] (verified)

Time = 1.51 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x (d+e x)} \, dx=\frac {(-4 c d+5 b e+2 c e x) \sqrt {a+x (b+c x)}}{4 e^2}+\frac {2 \left (-c d^2+e (b d-a e)\right )^{3/2} \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{d e^3}+\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{d}-\frac {\left (8 c^2 d^2+3 b^2 e^2+12 c e (-b d+a e)\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{8 \sqrt {c} e^3} \] Input:

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

Output:

((-4*c*d + 5*b*e + 2*c*e*x)*Sqrt[a + x*(b + c*x)])/(4*e^2) + (2*(-(c*d^2) 
+ e*(b*d - a*e))^(3/2)*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + x*(b + c*x)] 
)/Sqrt[-(c*d^2) + e*(b*d - a*e)]])/(d*e^3) + (2*a^(3/2)*ArcTanh[(Sqrt[c]*x 
 - Sqrt[a + x*(b + c*x)])/Sqrt[a]])/d - ((8*c^2*d^2 + 3*b^2*e^2 + 12*c*e*( 
-(b*d) + a*e))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(8*Sqrt[c 
]*e^3)
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.26, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1270, 25, 1162, 25, 1231, 27, 1269, 1092, 219, 1154, 219}

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.

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

\(\Big \downarrow \) 1270

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1162

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1231

\(\displaystyle \frac {-\frac {\int -\frac {c \left (4 b c d^3-5 b^2 e d^2-4 a c e d^2+12 a b e^2 d-8 a^2 e^3+\left (8 c^2 d^3-12 c e (b d-a e) d+b e^2 (3 b d-4 a e)\right ) x\right )}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-e (5 b d-4 a e)+4 c d^2-2 c d e x\right )}{4 e^2}}{d}+\frac {a \left (\frac {1}{2} \int \frac {2 a+b x}{x \sqrt {c x^2+b x+a}}dx+\sqrt {a+b x+c x^2}\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {4 b c d^3-5 b^2 e d^2-4 a c e d^2+12 a b e^2 d-8 a^2 e^3+\left (8 c^2 d^3-12 c e (b d-a e) d+b e^2 (3 b d-4 a e)\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-e (5 b d-4 a e)+4 c d^2-2 c d e x\right )}{4 e^2}}{d}+\frac {a \left (\frac {1}{2} \int \frac {2 a+b x}{x \sqrt {c x^2+b x+a}}dx+\sqrt {a+b x+c x^2}\right )}{d}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\frac {\left (-12 c d e (b d-a e)+b e^2 (3 b d-4 a e)+8 c^2 d^3\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {8 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-e (5 b d-4 a e)+4 c d^2-2 c d e x\right )}{4 e^2}}{d}+\frac {a \left (\frac {1}{2} \left (b \int \frac {1}{\sqrt {c x^2+b x+a}}dx+2 a \int \frac {1}{x \sqrt {c x^2+b x+a}}dx\right )+\sqrt {a+b x+c x^2}\right )}{d}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {2 \left (-12 c d e (b d-a e)+b e^2 (3 b d-4 a e)+8 c^2 d^3\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}-\frac {8 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-e (5 b d-4 a e)+4 c d^2-2 c d e x\right )}{4 e^2}}{d}+\frac {a \left (\frac {1}{2} \left (2 a \int \frac {1}{x \sqrt {c x^2+b x+a}}dx+2 b \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}\right )+\sqrt {a+b x+c x^2}\right )}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-12 c d e (b d-a e)+b e^2 (3 b d-4 a e)+8 c^2 d^3\right )}{\sqrt {c} e}-\frac {8 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-e (5 b d-4 a e)+4 c d^2-2 c d e x\right )}{4 e^2}}{d}+\frac {a \left (\frac {1}{2} \left (2 a \int \frac {1}{x \sqrt {c x^2+b x+a}}dx+\frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}\right )+\sqrt {a+b x+c x^2}\right )}{d}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {\frac {16 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}+\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-12 c d e (b d-a e)+b e^2 (3 b d-4 a e)+8 c^2 d^3\right )}{\sqrt {c} e}}{8 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-e (5 b d-4 a e)+4 c d^2-2 c d e x\right )}{4 e^2}}{d}+\frac {a \left (\frac {1}{2} \left (\frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}-4 a \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}\right )+\sqrt {a+b x+c x^2}\right )}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-12 c d e (b d-a e)+b e^2 (3 b d-4 a e)+8 c^2 d^3\right )}{\sqrt {c} e}-\frac {8 \left (a e^2-b d e+c d^2\right )^{3/2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e}}{8 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-e (5 b d-4 a e)+4 c d^2-2 c d e x\right )}{4 e^2}}{d}+\frac {a \left (\frac {1}{2} \left (\frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}-2 \sqrt {a} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )\right )+\sqrt {a+b x+c x^2}\right )}{d}\)

Input:

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

Output:

(a*(Sqrt[a + b*x + c*x^2] + (-2*Sqrt[a]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqr 
t[a + b*x + c*x^2])] + (b*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c* 
x^2])])/Sqrt[c])/2))/d + (-1/4*((4*c*d^2 - e*(5*b*d - 4*a*e) - 2*c*d*e*x)* 
Sqrt[a + b*x + c*x^2])/e^2 + (((8*c^2*d^3 + b*e^2*(3*b*d - 4*a*e) - 12*c*d 
*e*(b*d - a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(S 
qrt[c]*e) - (8*(c*d^2 - b*d*e + a*e^2)^(3/2)*ArcTanh[(b*d - 2*a*e + (2*c*d 
 - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/e)/(8*e 
^2))/d
 

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 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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1162

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x 
] - Simp[p/(e*(m + 2*p + 1))   Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - 
b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x 
] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && 
!ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1270

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)/(((d_.) + (e_.)*(x_))*((f_.) + 
 (g_.)*(x_))), x_Symbol] :> Simp[(c*d^2 - b*d*e + a*e^2)/(e*(e*f - d*g)) 
Int[(a + b*x + c*x^2)^(p - 1)/(d + e*x), x], x] - Simp[1/(e*(e*f - d*g)) 
Int[Simp[c*d*f - b*e*f + a*e*g - c*(e*f - d*g)*x, x]*((a + b*x + c*x^2)^(p 
- 1)/(f + g*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[p] 
&& GtQ[p, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(792\) vs. \(2(218)=436\).

Time = 1.43 (sec) , antiderivative size = 793, normalized size of antiderivative = 3.15

method	result	size
default	\(\frac {\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3}+\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2}+a \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )}{d}-\frac {\frac {\left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}{3}+\frac {\left (b e -2 c d \right ) \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{4 c}+\frac {\left (\frac {4 c \left (a \,e^{2}-b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{2 e}+\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\right )}{2 e \sqrt {c}}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{2}}}{d}\)	\(793\)

Input:

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

Output:

1/d*(1/3*(c*x^2+b*x+a)^(3/2)+1/2*b*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/ 
8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+a*((c*x 
^2+b*x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)- 
a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)))-1/d*(1/3*(c*(x+d/e 
)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+1/2*(b*e-2*c*d)/e 
*(1/4*(2*c*(x+d/e)+(b*e-2*c*d)/e)/c*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a* 
e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2 
/e^2)/c^(3/2)*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+(b*e-2 
*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)))+(a*e^2-b*d*e+c*d^2)/e^2*( 
(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2*(b*e 
-2*c*d)/e*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+(b*e-2*c*d 
)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^2)/e^ 
2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d 
)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+(b*e-2*c*d)/e*( 
x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Degree mismatch inside factorisatio 
n over extensionNot implemented, e.g. for multivariate mod/approx polynomi 
alsError:
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 562, normalized size of antiderivative = 2.23 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x (d+e x)} \, dx=\frac {8 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (-2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a \,e^{2}-b d e +c \,d^{2}}-2 a e +b d -b e x +2 c d x \right ) a c \,e^{2}-8 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (-2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a \,e^{2}-b d e +c \,d^{2}}-2 a e +b d -b e x +2 c d x \right ) b c d e +8 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (-2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a \,e^{2}-b d e +c \,d^{2}}-2 a e +b d -b e x +2 c d x \right ) c^{2} d^{2}-8 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (e x +d \right ) a c \,e^{2}+8 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (e x +d \right ) b c d e -8 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (e x +d \right ) c^{2} d^{2}+10 \sqrt {c \,x^{2}+b x +a}\, b c d \,e^{2}-8 \sqrt {c \,x^{2}+b x +a}\, c^{2} d^{2} e +4 \sqrt {c \,x^{2}+b x +a}\, c^{2} d \,e^{2} x +8 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a c \,e^{3}-8 \sqrt {a}\, \mathrm {log}\left (x \right ) a c \,e^{3}+12 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) a c d \,e^{2}+3 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) b^{2} d \,e^{2}-12 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) b c \,d^{2} e +8 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) c^{2} d^{3}}{8 c d \,e^{3}} \] Input:

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

Output:

(8*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e* 
*2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*c*e**2 - 8*sqrt(a* 
e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e 
 + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*b*c*d*e + 8*sqrt(a*e**2 - b*d* 
e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) 
- 2*a*e + b*d - b*e*x + 2*c*d*x)*c**2*d**2 - 8*sqrt(a*e**2 - b*d*e + c*d** 
2)*log(d + e*x)*a*c*e**2 + 8*sqrt(a*e**2 - b*d*e + c*d**2)*log(d + e*x)*b* 
c*d*e - 8*sqrt(a*e**2 - b*d*e + c*d**2)*log(d + e*x)*c**2*d**2 + 10*sqrt(a 
 + b*x + c*x**2)*b*c*d*e**2 - 8*sqrt(a + b*x + c*x**2)*c**2*d**2*e + 4*sqr 
t(a + b*x + c*x**2)*c**2*d*e**2*x + 8*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + 
 c*x**2) - 2*a - b*x)*a*c*e**3 - 8*sqrt(a)*log(x)*a*c*e**3 + 12*sqrt(c)*lo 
g( - 2*sqrt(c)*sqrt(a + b*x + c*x**2) - b - 2*c*x)*a*c*d*e**2 + 3*sqrt(c)* 
log( - 2*sqrt(c)*sqrt(a + b*x + c*x**2) - b - 2*c*x)*b**2*d*e**2 - 12*sqrt 
(c)*log( - 2*sqrt(c)*sqrt(a + b*x + c*x**2) - b - 2*c*x)*b*c*d**2*e + 8*sq 
rt(c)*log( - 2*sqrt(c)*sqrt(a + b*x + c*x**2) - b - 2*c*x)*c**2*d**3)/(8*c 
*d*e**3)