\(\int \frac {(d+e x)^{3/2}}{x^2 (a+b x+c x^2)} \, dx\) [72]

Optimal result

Integrand size = 25, antiderivative size = 376 \[ \int \frac {(d+e x)^{3/2}}{x^2 \left (a+b x+c x^2\right )} \, dx=-\frac {d \sqrt {d+e x}}{a x}+\frac {\sqrt {d} (2 b d-3 a e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a^2}-\frac {\sqrt {2} \sqrt {c} \left (b^2 d^2-2 a c d^2+b d \left (\sqrt {b^2-4 a c} d-2 a e\right )-2 a e \left (\sqrt {b^2-4 a c} d-a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} \left (b^2 d^2-2 a c d^2+2 a e \left (\sqrt {b^2-4 a c} d+a e\right )-b d \left (\sqrt {b^2-4 a c} d+2 a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.56 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^{3/2}}{x^2 \left (a+b x+c x^2\right )} \, dx=\frac {-\frac {a d \sqrt {d+e x}}{x}+\frac {\sqrt {2} \sqrt {c} \left (-i b^2 d^2+b d \left (\sqrt {-b^2+4 a c} d+2 i a e\right )-2 i a \left (-c d^2+e \left (-i \sqrt {-b^2+4 a c} d+a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} \left (i b^2 d^2+b d \left (\sqrt {-b^2+4 a c} d-2 i a e\right )+2 i a \left (-c d^2+e \left (i \sqrt {-b^2+4 a c} d+a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}+\sqrt {d} (2 b d-3 a e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a^2} \] Input:

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

Output:

(-((a*d*Sqrt[d + e*x])/x) + (Sqrt[2]*Sqrt[c]*((-I)*b^2*d^2 + b*d*(Sqrt[-b^ 
2 + 4*a*c]*d + (2*I)*a*e) - (2*I)*a*(-(c*d^2) + e*((-I)*Sqrt[-b^2 + 4*a*c] 
*d + a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*S 
qrt[-b^2 + 4*a*c]*e]])/(Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + (b - I*Sqrt[-b^2 
+ 4*a*c])*e]) + (Sqrt[2]*Sqrt[c]*(I*b^2*d^2 + b*d*(Sqrt[-b^2 + 4*a*c]*d - 
(2*I)*a*e) + (2*I)*a*(-(c*d^2) + e*(I*Sqrt[-b^2 + 4*a*c]*d + a*e)))*ArcTan 
[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]* 
e]])/(Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + (b + I*Sqrt[-b^2 + 4*a*c])*e]) + Sq 
rt[d]*(2*b*d - 3*a*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/a^2
 

Rubi [A] (verified)

Time = 1.71 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1199, 2009}

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.

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

\(\Big \downarrow \) 1199

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(2*(-1/2*(d*e*Sqrt[d + e*x])/(a*x) + (Sqrt[d]*e^2*ArcTanh[Sqrt[d + e*x]/Sq 
rt[d]])/(2*a) + (Sqrt[d]*e*(b*d - 2*a*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/a 
^2 - (Sqrt[c]*e*(b^2*d^2 + b*d*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) - 2*a*(c*d^2 
+ e*(Sqrt[b^2 - 4*a*c]*d - a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/ 
Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*a^2*Sqrt[b^2 - 4*a*c]*S 
qrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[c]*e*(b^2*d^2 - 2*a*c*d^2 
+ 2*a*e*(Sqrt[b^2 - 4*a*c]*d + a*e) - b*d*(Sqrt[b^2 - 4*a*c]*d + 2*a*e))*A 
rcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c] 
)*e]])/(Sqrt[2]*a^2*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c]) 
*e])))/e
 

Defintions of rubi rules used

rule 1199

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Denominator[m]}, Simp[q/e   Subs 
t[Int[ExpandIntegrand[x^(q*(m + 1) - 1)*(((e*f - d*g)/e + g*(x^q/e))^n/((c* 
d^2 - b*d*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))), x], 
 x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Integer 
Q[n] && FractionQ[m]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.66 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(2 e^{3} \left (\frac {4 c \left (-\frac {\left (2 a^{2} e^{3}-2 a b d \,e^{2}-2 d^{2} e a c +d^{2} e \,b^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a^{2} e^{3}+2 a b d \,e^{2}+2 d^{2} e a c -d^{2} e \,b^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a^{2} e^{3}}-\frac {d \left (\frac {a \sqrt {e x +d}}{2 x}+\frac {\left (3 a e -2 b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{a^{2} e^{3}}\right )\)	\(409\)
default	\(2 e^{3} \left (\frac {4 c \left (-\frac {\left (2 a^{2} e^{3}-2 a b d \,e^{2}-2 d^{2} e a c +d^{2} e \,b^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a^{2} e^{3}+2 a b d \,e^{2}+2 d^{2} e a c -d^{2} e \,b^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a^{2} e^{3}}-\frac {d \left (\frac {a \sqrt {e x +d}}{2 x}+\frac {\left (3 a e -2 b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{a^{2} e^{3}}\right )\)	\(409\)
risch	\(-\frac {d \sqrt {e x +d}}{a x}-\frac {e \left (\frac {\sqrt {d}\, \left (3 a e -2 b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{e a}+\frac {8 c \left (-\frac {\left (-2 a^{2} e^{3}+2 a b d \,e^{2}+2 d^{2} e a c -d^{2} e \,b^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (2 a^{2} e^{3}-2 a b d \,e^{2}-2 d^{2} e a c +d^{2} e \,b^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e a}\right )}{a}\)	\(411\)
pseudoelliptic	\(\frac {2 x \left (d \left (a e -\frac {b d}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-\left (e^{2} a^{2}-a b d e -a \,d^{2} c +\frac {1}{2} b^{2} d^{2}\right ) e \right ) \sqrt {d}\, \sqrt {2}\, c \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+2 \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (x \sqrt {d}\, \sqrt {2}\, \left (\left (-a d e +\frac {1}{2} b \,d^{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-\left (e^{2} a^{2}-a b d e -a \,d^{2} c +\frac {1}{2} b^{2} d^{2}\right ) e \right ) c \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )-\frac {\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (x \left (3 a d e -2 b \,d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )+\sqrt {e x +d}\, d^{\frac {3}{2}} a \right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}{2}\right )}{\sqrt {d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, a^{2} x}\)	\(452\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4321 vs. \(2 (322) = 644\).

Time = 18.95 (sec) , antiderivative size = 8650, normalized size of antiderivative = 23.01 \[ \int \frac {(d+e x)^{3/2}}{x^2 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 890 vs. \(2 (322) = 644\).

Time = 0.36 (sec) , antiderivative size = 890, normalized size of antiderivative = 2.37 \[ \int \frac {(d+e x)^{3/2}}{x^2 \left (a+b x+c x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

-sqrt(e*x + d)*d/(a*x) - (2*b*d^2 - 3*a*d*e)*arctan(sqrt(e*x + d)/sqrt(-d) 
)/(a^2*sqrt(-d)) + 1/4*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*( 
(b^3 - 4*a*b*c)*d^2 - 2*(a*b^2 - 4*a^2*c)*d*e)*e^2 - 2*(sqrt(b^2 - 4*a*c)* 
b*c*d^3 + 2*sqrt(b^2 - 4*a*c)*a*b*d*e^2 - sqrt(b^2 - 4*a*c)*a^2*e^3 - (b^2 
 + a*c)*sqrt(b^2 - 4*a*c)*d^2*e)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c 
)*c)*e)*abs(e) + (2*a^2*b*e^4 - 2*(b^2*c - 2*a*c^2)*d^3*e + (b^3 + 2*a*b*c 
)*d^2*e^2 - 2*(a*b^2 + 2*a^2*c)*d*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 
 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*a^2*c*d - a^2*b*e 
 + sqrt(-4*(a^2*c*d^2 - a^2*b*d*e + a^3*e^2)*a^2*c + (2*a^2*c*d - a^2*b*e) 
^2))/(a^2*c)))/((sqrt(b^2 - 4*a*c)*a^2*c*d^2 - sqrt(b^2 - 4*a*c)*a^2*b*d*e 
 + sqrt(b^2 - 4*a*c)*a^3*e^2)*abs(c)*abs(e)) - 1/4*(sqrt(-4*c^2*d + 2*(b*c 
 + sqrt(b^2 - 4*a*c)*c)*e)*((b^3 - 4*a*b*c)*d^2 - 2*(a*b^2 - 4*a^2*c)*d*e) 
*e^2 + 2*(sqrt(b^2 - 4*a*c)*b*c*d^3 + 2*sqrt(b^2 - 4*a*c)*a*b*d*e^2 - sqrt 
(b^2 - 4*a*c)*a^2*e^3 - (b^2 + a*c)*sqrt(b^2 - 4*a*c)*d^2*e)*sqrt(-4*c^2*d 
 + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(e) + (2*a^2*b*e^4 - 2*(b^2*c - 2*a 
*c^2)*d^3*e + (b^3 + 2*a*b*c)*d^2*e^2 - 2*(a*b^2 + 2*a^2*c)*d*e^3)*sqrt(-4 
*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d 
)/sqrt(-(2*a^2*c*d - a^2*b*e - sqrt(-4*(a^2*c*d^2 - a^2*b*d*e + a^3*e^2)*a 
^2*c + (2*a^2*c*d - a^2*b*e)^2))/(a^2*c)))/((sqrt(b^2 - 4*a*c)*a^2*c*d^2 - 
 sqrt(b^2 - 4*a*c)*a^2*b*d*e + sqrt(b^2 - 4*a*c)*a^3*e^2)*abs(c)*abs(e)...
 

Mupad [B] (verification not implemented)

Time = 17.57 (sec) , antiderivative size = 29890, normalized size of antiderivative = 79.49 \[ \int \frac {(d+e x)^{3/2}}{x^2 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

(d^(1/2)*atan(((d^(1/2)*((8*(d + e*x)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*d^ 
8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14 + 4 
*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b^3*c^4*d^5*e^11 + 8* 
a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^13 + 
 33*a^4*b^2*c^3*d^2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8 - 28* 
a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3*b*c^5*d^5*e^11 - 60*a^4* 
b*c^4*d^3*e^13))/a^4 - (d^(1/2)*((8*(56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e 
^11 - 100*a^6*c^4*d^2*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c^3*d^5*e 
^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^9 + 96*a^3*b^3*c^4*d 
^5*e^10 + 111*a^3*b^4*c^3*d^4*e^11 + 22*a^3*b^5*c^2*d^3*e^12 - 237*a^4*b^2 
*c^4*d^4*e^11 - 161*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^13 + 111*a 
^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*c^4*d^7*e^8 + 6*a*b^6* 
c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^10 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d 
^5*e^10 + 252*a^5*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14))/a^4 + (d^(1/2)*( 
(8*(d + e*x)^(1/2)*(16*a^7*b*c^3*e^13 + 88*a^7*c^4*d*e^12 - 4*a^6*b^3*c^2* 
e^13 - 40*a^5*c^6*d^5*e^8 + 184*a^6*c^5*d^3*e^10 + 8*a^2*b^6*c^3*d^5*e^8 - 
 8*a^2*b^7*c^2*d^4*e^9 - 56*a^3*b^4*c^4*d^5*e^8 + 36*a^3*b^5*c^3*d^4*e^9 + 
 28*a^3*b^6*c^2*d^3*e^10 + 108*a^4*b^2*c^5*d^5*e^8 + 36*a^4*b^3*c^4*d^4*e^ 
9 - 179*a^4*b^4*c^3*d^3*e^10 - 33*a^4*b^5*c^2*d^2*e^11 + 234*a^5*b^2*c^4*d 
^3*e^10 + 215*a^5*b^3*c^3*d^2*e^11 - 224*a^5*b*c^5*d^4*e^9 + 16*a^5*b^4...
 

Reduce [F]

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

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

Output:

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