\(\int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx\) [59]

Optimal result

Integrand size = 25, antiderivative size = 397 \[ \int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\frac {2 \left (b^2-a c\right ) \sqrt {d+e x}}{c^3}-\frac {2 (c d+b e) (d+e x)^{3/2}}{3 c^2 e^2}+\frac {2 (d+e x)^{5/2}}{5 c e^2}-\frac {\sqrt {2} \left (b^2 c d-a c^2 d-b^3 e+2 a b c e-\frac {b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{7/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {2} \left (b^2 c d-a c^2 d-b^3 e+2 a b c e+\frac {b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{7/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

2*(-a*c+b^2)*(e*x+d)^(1/2)/c^3-2/3*(b*e+c*d)*(e*x+d)^(3/2)/c^2/e^2+2/5*(e* 
x+d)^(5/2)/c/e^2-2^(1/2)*(b^2*c*d-a*c^2*d-b^3*e+2*a*b*c*e-(-2*a^2*c^2*e+4* 
a*b^2*c*e-3*a*b*c^2*d-b^4*e+b^3*c*d)/(-4*a*c+b^2)^(1/2))*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))/c^(7/2)/(2*c* 
d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)-2^(1/2)*(b^2*c*d-a*c^2*d-b^3*e+2*a*b*c*e 
+(-2*a^2*c^2*e+4*a*b^2*c*e-3*a*b*c^2*d-b^4*e+b^3*c*d)/(-4*a*c+b^2)^(1/2))* 
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))/c^(7/2)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)^(1/2)
 

Mathematica [A] (verified)

Time = 2.47 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.17 \[ \int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\frac {\frac {2 \sqrt {c} \sqrt {d+e x} \left (15 b^2 e^2+c^2 \left (-2 d^2+d e x+3 e^2 x^2\right )-5 c e (3 a e+b (d+e x))\right )}{e^2}-\frac {15 \sqrt {2} \left (-b^4 e+a c^2 \left (\sqrt {b^2-4 a c} d-2 a e\right )+b^2 c \left (-\sqrt {b^2-4 a c} d+4 a e\right )+b^3 \left (c d+\sqrt {b^2-4 a c} e\right )-a b c \left (3 c d+2 \sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {15 \sqrt {2} \left (b^4 e+a c^2 \left (\sqrt {b^2-4 a c} d+2 a e\right )-b^2 c \left (\sqrt {b^2-4 a c} d+4 a e\right )+a b c \left (3 c d-2 \sqrt {b^2-4 a c} e\right )+b^3 \left (-c d+\sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}}{15 c^{7/2}} \] Input:

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

Output:

((2*Sqrt[c]*Sqrt[d + e*x]*(15*b^2*e^2 + c^2*(-2*d^2 + d*e*x + 3*e^2*x^2) - 
 5*c*e*(3*a*e + b*(d + e*x))))/e^2 - (15*Sqrt[2]*(-(b^4*e) + a*c^2*(Sqrt[b 
^2 - 4*a*c]*d - 2*a*e) + b^2*c*(-(Sqrt[b^2 - 4*a*c]*d) + 4*a*e) + b^3*(c*d 
 + Sqrt[b^2 - 4*a*c]*e) - a*b*c*(3*c*d + 2*Sqrt[b^2 - 4*a*c]*e))*ArcTan[(S 
qrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/( 
Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e]) - (15*Sqrt[2]* 
(b^4*e + a*c^2*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) - b^2*c*(Sqrt[b^2 - 4*a*c]*d 
+ 4*a*e) + a*b*c*(3*c*d - 2*Sqrt[b^2 - 4*a*c]*e) + b^3*(-(c*d) + Sqrt[b^2 
- 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + (b + Sqr 
t[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a* 
c])*e]))/(15*c^(7/2))
 

Rubi [A] (verified)

Time = 4.12 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.02, 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.

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

\(\Big \downarrow \) 1199

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (-\frac {e \left (-\frac {-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d}{\sqrt {b^2-4 a c}}+2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{7/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {e \left (\frac {-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d}{\sqrt {b^2-4 a c}}+2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{7/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {e \left (b^2-a c\right ) \sqrt {d+e x}}{c^3}-\frac {(d+e x)^{3/2} (b e+c d)}{3 c^2 e}+\frac {(d+e x)^{5/2}}{5 c e}\right )}{e}\)

Input:

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

Output:

(2*(((b^2 - a*c)*e*Sqrt[d + e*x])/c^3 - ((c*d + b*e)*(d + e*x)^(3/2))/(3*c 
^2*e) + (d + e*x)^(5/2)/(5*c*e) - (e*(b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c* 
e - (b^3*c*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)/Sqrt[b^2 - 
 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^ 
2 - 4*a*c])*e]])/(Sqrt[2]*c^(7/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) 
 - (e*(b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e + (b^3*c*d - 3*a*b*c^2*d - b^ 
4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[ 
c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(7/ 
2)*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 = 1.68 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.25

method	result	size
pseudoelliptic	\(-\frac {2 \left (\left (\left (-\frac {a \,c^{2} d}{2}+b \left (a e +\frac {b d}{2}\right ) c -\frac {e \,b^{3}}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (a \left (\frac {3 b d}{2}+a e \right ) c^{2}+\left (-2 e a \,b^{2}-\frac {1}{2} b^{3} d \right ) c +\frac {b^{4} e}{2}\right )\right ) e^{2} \sqrt {2}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (\left (\left (\frac {a \,c^{2} d}{2}+\left (-a b e -\frac {1}{2} b^{2} d \right ) c +\frac {e \,b^{3}}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (a \left (\frac {3 b d}{2}+a e \right ) c^{2}+\left (-2 e a \,b^{2}-\frac {1}{2} b^{3} d \right ) c +\frac {b^{4} e}{2}\right )\right ) e^{2} \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {e x +d}\, \left (\frac {2 \left (-\frac {3 e x}{2}+d \right ) \left (e x +d \right ) c^{2}}{15}+e \left (\left (\frac {b x}{3}+a \right ) e +\frac {b d}{3}\right ) c -b^{2} e^{2}\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\right )}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, e^{2} c^{3}}\)	\(497\)
risch	\(-\frac {2 \left (-3 c^{2} e^{2} x^{2}+5 e^{2} x b c -c^{2} d e x +15 a c \,e^{2}-15 b^{2} e^{2}+5 b c d e +2 c^{2} d^{2}\right ) \sqrt {e x +d}}{15 e^{2} c^{3}}+\frac {-\frac {\left (2 a^{2} c^{2} e^{2}-4 a \,b^{2} c \,e^{2}+3 a b \,c^{2} d e +b^{4} e^{2}-b^{3} c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a^{2} c^{2} e^{2}+4 a \,b^{2} c \,e^{2}-3 a b \,c^{2} d e -b^{4} e^{2}+b^{3} c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c^{2}}\)	\(552\)
derivativedivides	\(\frac {-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {5}{2}} c^{2}}{5}+\frac {b c e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+a c \,e^{2} \sqrt {e x +d}-b^{2} e^{2} \sqrt {e x +d}\right )}{c^{3}}+\frac {8 e^{2} \left (-\frac {\left (2 a^{2} c^{2} e^{2}-4 a \,b^{2} c \,e^{2}+3 a b \,c^{2} d e +b^{4} e^{2}-b^{3} c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a^{2} c^{2} e^{2}+4 a \,b^{2} c \,e^{2}-3 a b \,c^{2} d e -b^{4} e^{2}+b^{3} c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c^{2}}}{e^{2}}\)	\(559\)
default	\(\frac {-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {5}{2}} c^{2}}{5}+\frac {b c e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+a c \,e^{2} \sqrt {e x +d}-b^{2} e^{2} \sqrt {e x +d}\right )}{c^{3}}+\frac {8 e^{2} \left (-\frac {\left (2 a^{2} c^{2} e^{2}-4 a \,b^{2} c \,e^{2}+3 a b \,c^{2} d e +b^{4} e^{2}-b^{3} c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a^{2} c^{2} e^{2}+4 a \,b^{2} c \,e^{2}-3 a b \,c^{2} d e -b^{4} e^{2}+b^{3} c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c^{2}}}{e^{2}}\)	\(559\)

Input:

int(x^3*(e*x+d)^(1/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
 

Output:

-2*(((-1/2*a*c^2*d+b*(a*e+1/2*b*d)*c-1/2*e*b^3)*(-4*e^2*(a*c-1/4*b^2))^(1/ 
2)+e*(a*(3/2*b*d+a*e)*c^2+(-2*e*a*b^2-1/2*b^3*d)*c+1/2*b^4*e))*e^2*2^(1/2) 
*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*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))+(((1/2*a*c^ 
2*d+(-a*b*e-1/2*b^2*d)*c+1/2*e*b^3)*(-4*e^2*(a*c-1/4*b^2))^(1/2)+e*(a*(3/2 
*b*d+a*e)*c^2+(-2*e*a*b^2-1/2*b^3*d)*c+1/2*b^4*e))*e^2*2^(1/2)*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))+(e 
*x+d)^(1/2)*(2/15*(-3/2*e*x+d)*(e*x+d)*c^2+e*((1/3*b*x+a)*e+1/3*b*d)*c-b^2 
*e^2)*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*(-4*e^2*(a*c-1/4* 
b^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)/((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b 
^2))^(1/2))*c)^(1/2)/(-4*e^2*(a*c-1/4*b^2))^(1/2)/e^2/c^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4245 vs. \(2 (347) = 694\).

Time = 0.36 (sec) , antiderivative size = 4245, normalized size of antiderivative = 10.69 \[ \int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

integrate(x^3*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\int \frac {x^{3} \sqrt {d + e x}}{a + b x + c x^{2}}\, dx \] Input:

integrate(x**3*(e*x+d)**(1/2)/(c*x**2+b*x+a),x)
                                                                                    
                                                                                    
 

Output:

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

Maxima [F]

\[ \int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\int { \frac {\sqrt {e x + d} x^{3}}{c x^{2} + b x + a} \,d x } \] Input:

integrate(x^3*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1074 vs. \(2 (347) = 694\).

Time = 0.33 (sec) , antiderivative size = 1074, normalized size of antiderivative = 2.71 \[ \int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

integrate(x^3*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="giac")
 

Output:

1/4*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*((b^4*c - 5*a*b^2*c^ 
2 + 4*a^2*c^3)*d - (b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*e)*c^2*e^2 - 2*((b^2*c^ 
3 - a*c^4)*sqrt(b^2 - 4*a*c)*d^2 - (b^3*c^2 - a*b*c^3)*sqrt(b^2 - 4*a*c)*d 
*e + (a*b^2*c^2 - a^2*c^3)*sqrt(b^2 - 4*a*c)*e^2)*sqrt(-4*c^2*d + 2*(b*c - 
 sqrt(b^2 - 4*a*c)*c)*e)*abs(c)*abs(e) + (2*(b^3*c^4 - 3*a*b*c^5)*d^2*e - 
(3*b^4*c^3 - 11*a*b^2*c^4 + 4*a^2*c^5)*d*e^2 + (b^5*c^2 - 4*a*b^3*c^3 + 2* 
a^2*b*c^4)*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*c^6*d*e^12 - b*c^5*e^13 + sqrt(-4*(c^6*d 
^2*e^12 - b*c^5*d*e^13 + a*c^5*e^14)*c^6*e^12 + (2*c^6*d*e^12 - b*c^5*e^13 
)^2))/(c^6*e^12)))/((sqrt(b^2 - 4*a*c)*c^6*d^2 - sqrt(b^2 - 4*a*c)*b*c^5*d 
*e + sqrt(b^2 - 4*a*c)*a*c^5*e^2)*c^2*abs(e)) - 1/4*(sqrt(-4*c^2*d + 2*(b* 
c + sqrt(b^2 - 4*a*c)*c)*e)*((b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d - (b^5 - 
6*a*b^3*c + 8*a^2*b*c^2)*e)*c^2*e^2 + 2*((b^2*c^3 - a*c^4)*sqrt(b^2 - 4*a* 
c)*d^2 - (b^3*c^2 - a*b*c^3)*sqrt(b^2 - 4*a*c)*d*e + (a*b^2*c^2 - a^2*c^3) 
*sqrt(b^2 - 4*a*c)*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*a 
bs(c)*abs(e) + (2*(b^3*c^4 - 3*a*b*c^5)*d^2*e - (3*b^4*c^3 - 11*a*b^2*c^4 
+ 4*a^2*c^5)*d*e^2 + (b^5*c^2 - 4*a*b^3*c^3 + 2*a^2*b*c^4)*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)/s 
qrt(-(2*c^6*d*e^12 - b*c^5*e^13 - sqrt(-4*(c^6*d^2*e^12 - b*c^5*d*e^13 + a 
*c^5*e^14)*c^6*e^12 + (2*c^6*d*e^12 - b*c^5*e^13)^2))/(c^6*e^12)))/((sq...
 

Mupad [B] (verification not implemented)

Time = 12.61 (sec) , antiderivative size = 11143, normalized size of antiderivative = 28.07 \[ \int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

int((x^3*(d + e*x)^(1/2))/(a + b*x + c*x^2),x)
 

Output:

atan(((((8*(4*a^3*c^6*e^4 + a*b^4*c^4*e^4 - b^5*c^4*d*e^3 - 5*a^2*b^2*c^5* 
e^4 + 4*a^2*c^7*d^2*e^2 + b^4*c^5*d^2*e^2 + 5*a*b^3*c^5*d*e^3 - 4*a^2*b*c^ 
6*d*e^3 - 5*a*b^2*c^6*d^2*e^2))/c^5 - (8*(d + e*x)^(1/2)*(-(b^9*e - 8*a^4* 
c^5*d - b^6*e*(-(4*a*c - b^2)^3)^(1/2) - b^8*c*d - 33*a^2*b^4*c^3*d + 38*a 
^3*b^2*c^4*d + 42*a^2*b^5*c^2*e - 63*a^3*b^3*c^3*e + a^3*c^3*e*(-(4*a*c - 
b^2)^3)^(1/2) - 11*a*b^7*c*e + 10*a*b^6*c^2*d + 28*a^4*b*c^4*e + b^5*c*d*( 
-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c 
^2*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^3*d*(-(4*a*c - b^2)^3)^(1/2) - 6 
*a^2*b^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^ 
2*c^8)))^(1/2)*(b^3*c^7*e^3 - 2*b^2*c^8*d*e^2 - 4*a*b*c^8*e^3 + 8*a*c^9*d* 
e^2))/c^5)*(-(b^9*e - 8*a^4*c^5*d - b^6*e*(-(4*a*c - b^2)^3)^(1/2) - b^8*c 
*d - 33*a^2*b^4*c^3*d + 38*a^3*b^2*c^4*d + 42*a^2*b^5*c^2*e - 63*a^3*b^3*c 
^3*e + a^3*c^3*e*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e + 10*a*b^6*c^2*d 
+ 28*a^4*b*c^4*e + b^5*c*d*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e*(-(4*a*c 
 - b^2)^3)^(1/2) - 4*a*b^3*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^3*d* 
(-(4*a*c - b^2)^3)^(1/2) - 6*a^2*b^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(1 
6*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) - (8*(d + e*x)^(1/2)*(b^8*e^4 + 
 2*a^4*c^4*e^4 + 20*a^2*b^4*c^2*e^4 - 16*a^3*b^2*c^3*e^4 - 2*a^3*c^5*d^2*e 
^2 + b^6*c^2*d^2*e^2 - 8*a*b^6*c*e^4 - 2*b^7*c*d*e^3 + 9*a^2*b^2*c^4*d^2*e 
^2 + 14*a*b^5*c^2*d*e^3 + 14*a^3*b*c^4*d*e^3 - 6*a*b^4*c^3*d^2*e^2 - 28...
 

Reduce [F]

\[ \int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\int \frac {x^{3} \sqrt {e x +d}}{c \,x^{2}+b x +a}d x \] Input:

int(x^3*(e*x+d)^(1/2)/(c*x^2+b*x+a),x)
 

Output:

int(x^3*(e*x+d)^(1/2)/(c*x^2+b*x+a),x)