\(\int \frac {(d+e x)^{3/2}}{x^3 (a+b x+c x^2)} \, dx\) [73]

Optimal result

Integrand size = 25, antiderivative size = 529 \[ \int \frac {(d+e x)^{3/2}}{x^3 \left (a+b x+c x^2\right )} \, dx=-\frac {d \sqrt {d+e x}}{2 a x^2}+\frac {(4 b d-5 a e) \sqrt {d+e x}}{4 a^2 x}-\frac {\left (8 b^2 d^2-12 a b d e-a \left (8 c d^2-3 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 a^3 \sqrt {d}}+\frac {\sqrt {2} \sqrt {c} \left (b^3 d^2+b^2 d \left (\sqrt {b^2-4 a c} d-2 a e\right )+a \left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d-4 a e\right )\right )-a b \left (3 c d^2+e \left (2 \sqrt {b^2-4 a c} d-a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{a^3 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {2} \sqrt {c} \left (b^3 d^2-b^2 d \left (\sqrt {b^2-4 a c} d+2 a e\right )-a b \left (3 c d^2-e \left (2 \sqrt {b^2-4 a c} d+a e\right )\right )-a \left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d+4 a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{a^3 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

-1/2*d*(e*x+d)^(1/2)/a/x^2+1/4*(-5*a*e+4*b*d)*(e*x+d)^(1/2)/a^2/x-1/4*(8*b 
^2*d^2-12*a*b*d*e-a*(-3*a*e^2+8*c*d^2))*arctanh((e*x+d)^(1/2)/d^(1/2))/a^3 
/d^(1/2)+2^(1/2)*c^(1/2)*(b^3*d^2+b^2*d*((-4*a*c+b^2)^(1/2)*d-2*a*e)+a*(a* 
(-4*a*c+b^2)^(1/2)*e^2-c*d*((-4*a*c+b^2)^(1/2)*d-4*a*e))-a*b*(3*c*d^2+e*(2 
*(-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^3/(-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^3*d^2-b^2*d*((-4*a*c+b^2)^(1/2)*d+ 
2*a*e)-a*b*(3*c*d^2-e*(2*(-4*a*c+b^2)^(1/2)*d+a*e))-a*(a*(-4*a*c+b^2)^(1/2 
)*e^2-c*d*((-4*a*c+b^2)^(1/2)*d+4*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^3/(-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 = 4.27 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.06 \[ \int \frac {(d+e x)^{3/2}}{x^3 \left (a+b x+c x^2\right )} \, dx=\frac {\frac {a \sqrt {d+e x} (-2 a d+4 b d x-5 a e x)}{x^2}+\frac {4 \sqrt {2} \sqrt {c} \left (i b^3 d^2-b^2 d \left (\sqrt {-b^2+4 a c} d+2 i a e\right )+a b \left (-3 i c d^2+e \left (2 \sqrt {-b^2+4 a c} d+i a e\right )\right )+a \left (-a \sqrt {-b^2+4 a c} e^2+c d \left (\sqrt {-b^2+4 a c} d+4 i a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}-\frac {4 \sqrt {2} \sqrt {c} \left (i b^3 d^2+b^2 d \left (\sqrt {-b^2+4 a c} d-2 i a e\right )+a \left (a \sqrt {-b^2+4 a c} e^2+c d \left (-\sqrt {-b^2+4 a c} d+4 i a e\right )\right )+i a b \left (-3 c d^2+e \left (2 i \sqrt {-b^2+4 a c} d+a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\left (-8 b^2 d^2+12 a b d e+a \left (8 c d^2-3 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}}{4 a^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.54 (sec) , antiderivative size = 626, normalized size of antiderivative = 1.18, 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.

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

\(\Big \downarrow \) 1199

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(2*(-1/4*(d*e*Sqrt[d + e*x])/(a*x^2) + (3*e^2*Sqrt[d + e*x])/(8*a*x) + (e* 
(b*d - 2*a*e)*Sqrt[d + e*x])/(2*a^2*x) - (3*e^3*ArcTanh[Sqrt[d + e*x]/Sqrt 
[d]])/(8*a*Sqrt[d]) - (e^2*(b*d - 2*a*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/( 
2*a^2*Sqrt[d]) - (e*(b^2*d^2 - 2*a*b*d*e - a*(c*d^2 - a*e^2))*ArcTanh[Sqrt 
[d + e*x]/Sqrt[d]])/(a^3*Sqrt[d]) + (Sqrt[c]*e*(b^3*d^2 + b^2*d*(Sqrt[b^2 
- 4*a*c]*d - 2*a*e) + a*(a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]* 
d - 4*a*e)) - a*b*(3*c*d^2 + e*(2*Sqrt[b^2 - 4*a*c]*d - a*e)))*ArcTanh[(Sq 
rt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sq 
rt[2]*a^3*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (Sq 
rt[c]*e*(b^3*d^2 - b^2*d*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) - a*b*(3*c*d^2 - e* 
(2*Sqrt[b^2 - 4*a*c]*d + a*e)) - a*(a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^ 
2 - 4*a*c]*d + 4*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^3*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 = 1.86 (sec) , antiderivative size = 509, normalized size of antiderivative = 0.96

method	result	size
pseudoelliptic	\(-\frac {\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, x^{2} \sqrt {d}\, \sqrt {2}\, \left (\left (a \,d^{2} c -\left (a e -b d \right )^{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-e \left (\left (4 d e \,a^{2}-3 a b \,d^{2}\right ) c +b \left (a e -b d \right )^{2}\right )\right ) c \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )-\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (x^{2} \sqrt {d}\, \sqrt {2}\, c \left (\left (a \,d^{2} c -\left (a e -b d \right )^{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (\left (4 d e \,a^{2}-3 a b \,d^{2}\right ) c +b \left (a e -b d \right )^{2}\right )\right ) \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )-\frac {\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\frac {3 x^{2} \left (e^{2} a^{2}-4 a b d e -\frac {8}{3} a \,d^{2} c +\frac {8}{3} b^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2}+\left (\left (-2 b x +a \right ) d +\frac {5 a e x}{2}\right ) \sqrt {d}\, a \sqrt {e x +d}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}\right )}{\sqrt {d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, x^{2} a^{3}}\)	\(509\)
risch	\(-\frac {\sqrt {e x +d}\, \left (5 a e x -4 b d x +2 a d \right )}{4 a^{2} x^{2}}-\frac {e \left (-\frac {\left (-3 e^{2} a^{2}+12 a b d e +8 a \,d^{2} c -8 b^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{a e \sqrt {d}}+\frac {32 c \left (-\frac {\left (a^{2} b \,e^{3}+4 a^{2} c d \,e^{2}-2 a \,b^{2} d \,e^{2}-3 a b c \,d^{2} e +b^{3} d^{2} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-a^{2} b \,e^{3}-4 a^{2} c d \,e^{2}+2 a \,b^{2} d \,e^{2}+3 a b c \,d^{2} e -b^{3} d^{2} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a e}\right )}{4 a^{2}}\)	\(574\)
derivativedivides	\(2 e^{4} \left (\frac {4 c \left (\frac {\left (a^{2} b \,e^{3}+4 a^{2} c d \,e^{2}-2 a \,b^{2} d \,e^{2}-3 a b c \,d^{2} e +b^{3} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-a^{2} b \,e^{3}-4 a^{2} c d \,e^{2}+2 a \,b^{2} d \,e^{2}+3 a b c \,d^{2} e -b^{3} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e^{4} a^{3}}-\frac {\frac {\frac {a e \left (5 a e -4 b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {1}{2} a b \,d^{2} e -\frac {3}{8} a^{2} d \,e^{2}\right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {\left (3 e^{2} a^{2}-12 a b d e -8 a \,d^{2} c +8 b^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}}{e^{4} a^{3}}\right )\)	\(601\)
default	\(2 e^{4} \left (\frac {4 c \left (\frac {\left (a^{2} b \,e^{3}+4 a^{2} c d \,e^{2}-2 a \,b^{2} d \,e^{2}-3 a b c \,d^{2} e +b^{3} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-a^{2} b \,e^{3}-4 a^{2} c d \,e^{2}+2 a \,b^{2} d \,e^{2}+3 a b c \,d^{2} e -b^{3} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e^{4} a^{3}}-\frac {\frac {\frac {a e \left (5 a e -4 b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {1}{2} a b \,d^{2} e -\frac {3}{8} a^{2} d \,e^{2}\right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {\left (3 e^{2} a^{2}-12 a b d e -8 a \,d^{2} c +8 b^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}}{e^{4} a^{3}}\right )\)	\(601\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7202 vs. \(2 (459) = 918\).

Time = 169.42 (sec) , antiderivative size = 14414, normalized size of antiderivative = 27.25 \[ \int \frac {(d+e x)^{3/2}}{x^3 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1129 vs. \(2 (459) = 918\).

Time = 0.33 (sec) , antiderivative size = 1129, normalized size of antiderivative = 2.13 \[ \int \frac {(d+e x)^{3/2}}{x^3 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*(8*b^2*d^2 - 8*a*c*d^2 - 12*a*b*d*e + 3*a^2*e^2)*arctan(sqrt(e*x + d)/ 
sqrt(-d))/(a^3*sqrt(-d)) - 1/4*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c) 
*c)*e)*((b^4 - 5*a*b^2*c + 4*a^2*c^2)*d^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e + (a 
^2*b^2 - 4*a^3*c)*e^2)*a^2*e^2 + 2*(sqrt(b^2 - 4*a*c)*a*b^3*d^2*e + sqrt(b 
^2 - 4*a*c)*a^3*b*e^3 - (a*b^2*c - a^2*c^2)*sqrt(b^2 - 4*a*c)*d^3 - (2*a^2 
*b^2 - a^3*c)*sqrt(b^2 - 4*a*c)*d*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 
 4*a*c)*c)*e)*abs(a)*abs(e) + (a^4*b^2*e^4 - 2*(a^2*b^3*c - 3*a^3*b*c^2)*d 
^3*e + (a^2*b^4 + a^3*b^2*c - 8*a^4*c^2)*d^2*e^2 - 2*(a^3*b^3 - a^4*b*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^3*c*d - a^3*b*e + sqrt(-4*(a^3*c*d^2 - a^3*b*d*e 
 + a^4*e^2)*a^3*c + (2*a^3*c*d - a^3*b*e)^2))/(a^3*c)))/((sqrt(b^2 - 4*a*c 
)*a^4*c*d^2 - sqrt(b^2 - 4*a*c)*a^4*b*d*e + sqrt(b^2 - 4*a*c)*a^5*e^2)*abs 
(a)*abs(c)*abs(e)) + 1/4*(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e) 
*((b^4 - 5*a*b^2*c + 4*a^2*c^2)*d^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e + (a^2*b^2 
 - 4*a^3*c)*e^2)*a^2*e^2 - 2*(sqrt(b^2 - 4*a*c)*a*b^3*d^2*e + sqrt(b^2 - 4 
*a*c)*a^3*b*e^3 - (a*b^2*c - a^2*c^2)*sqrt(b^2 - 4*a*c)*d^3 - (2*a^2*b^2 - 
 a^3*c)*sqrt(b^2 - 4*a*c)*d*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c 
)*c)*e)*abs(a)*abs(e) + (a^4*b^2*e^4 - 2*(a^2*b^3*c - 3*a^3*b*c^2)*d^3*e + 
 (a^2*b^4 + a^3*b^2*c - 8*a^4*c^2)*d^2*e^2 - 2*(a^3*b^3 - a^4*b*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)*sq...
 

Mupad [B] (verification not implemented)

Time = 17.71 (sec) , antiderivative size = 44649, normalized size of antiderivative = 84.40 \[ \int \frac {(d+e x)^{3/2}}{x^3 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

(((3*a*d*e^2 - 4*b*d^2*e)*(d + e*x)^(1/2))/(4*a^2) - ((5*a*e^2 - 4*b*d*e)* 
(d + e*x)^(3/2))/(4*a^2))/((d + e*x)^2 - 2*d*(d + e*x) + d^2) + atan(((((( 
(192*a^11*b^2*c^3*e^12 - 24*a^10*b^4*c^2*e^12 - 384*a^12*c^4*e^12 + 768*a^ 
10*c^6*d^4*e^8 + 384*a^11*c^5*d^2*e^10 + 128*a^8*b^4*c^4*d^4*e^8 - 96*a^8* 
b^5*c^3*d^3*e^9 - 32*a^8*b^6*c^2*d^2*e^10 - 704*a^9*b^2*c^5*d^4*e^8 + 320* 
a^9*b^3*c^4*d^3*e^9 + 488*a^9*b^4*c^3*d^2*e^10 - 1536*a^10*b^2*c^4*d^2*e^1 
0 + 1408*a^11*b*c^4*d*e^11 + 56*a^9*b^5*c^2*d*e^11 + 256*a^10*b*c^5*d^3*e^ 
9 - 576*a^10*b^3*c^3*d*e^11)/(2*a^8) - ((d + e*x)^(1/2)*((b^8*d^3 - a^3*b^ 
5*e^3 + 8*a^4*c^4*d^3 + b^5*d^3*(-(4*a*c - b^2)^3)^(1/2) + 7*a^4*b^3*c*e^3 
 - 12*a^5*b*c^2*e^3 + a^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^6*d*e^2 
 - 24*a^5*c^3*d*e^2 + 33*a^2*b^4*c^2*d^3 - 38*a^3*b^2*c^3*d^3 - a^3*b^2*e^ 
3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d^3 - 3*a*b^7*d^2*e - 4*a*b^3*c*d^ 
3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a 
^2*b^5*c*d^2*e - 24*a^3*b^4*c*d*e^2 + 60*a^4*b*c^3*d^2*e + 3*a^2*b*c^2*d^3 
*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 75* 
a^3*b^3*c^2*d^2*e + 54*a^4*b^2*c^2*d*e^2 - 3*a^3*c^2*d^2*e*(-(4*a*c - b^2) 
^3)^(1/2) + 9*a^2*b^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a^3*b*c*d*e^2*( 
-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2)*( 
1024*a^13*c^4*e^10 + 64*a^11*b^4*c^2*e^10 - 512*a^12*b^2*c^3*e^10 + 1536*a 
^12*c^5*d^2*e^8 + 128*a^10*b^4*c^3*d^2*e^8 - 896*a^11*b^2*c^4*d^2*e^8 -...
 

Reduce [F]

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

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

Output:

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