\(\int \frac {x^2 (d+e x)^{3/2}}{a+b x+c x^2} \, dx\) [68]

Optimal result

Integrand size = 25, antiderivative size = 441 \[ \int \frac {x^2 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=-\frac {2 \left (b c d-b^2 e+a c e\right ) \sqrt {d+e x}}{c^3}-\frac {2 b (d+e x)^{3/2}}{3 c^2}+\frac {2 (d+e x)^{5/2}}{5 c e}+\frac {\sqrt {2} \left ((c d-b e) \left (b c d-b^2 e+2 a c e\right )+\frac {2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{7/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left ((c d-b e) \left (b c d-b^2 e+2 a c e\right )-\frac {2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{7/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

-2*(a*c*e-b^2*e+b*c*d)*(e*x+d)^(1/2)/c^3-2/3*b*(e*x+d)^(3/2)/c^2+2/5*(e*x+ 
d)^(5/2)/c/e+2^(1/2)*((-b*e+c*d)*(2*a*c*e-b^2*e+b*c*d)+(2*b^3*c*d*e-6*a*b* 
c^2*d*e-b^4*e^2-b^2*c*(-4*a*e^2+c*d^2)+2*a*c^2*(-a*e^2+c*d^2))/(-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*e 
+c*d)*(2*a*c*e-b^2*e+b*c*d)-(2*b^3*c*d*e-6*a*b*c^2*d*e-b^4*e^2-b^2*c*(-4*a 
*e^2+c*d^2)+2*a*c^2*(-a*e^2+c*d^2))/(-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 = 2.88 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.22 \[ \int \frac {x^2 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\frac {\frac {2 \sqrt {c} \sqrt {d+e x} \left (15 b^2 e^2+3 c^2 (d+e x)^2-5 c e (4 b d+3 a e+b e x)\right )}{e}-\frac {15 \sqrt {2} \left (-b^4 e^2+b^3 e \left (2 c d+\sqrt {b^2-4 a c} e\right )+b c \left (-2 a \sqrt {b^2-4 a c} e^2+c d \left (\sqrt {b^2-4 a c} d-6 a e\right )\right )-b^2 c \left (c d^2+2 e \left (\sqrt {b^2-4 a c} d-2 a e\right )\right )+2 a c^2 \left (c d^2+e \left (\sqrt {b^2-4 a c} d-a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {15 \sqrt {2} \left (b^4 e^2+b^3 e \left (-2 c d+\sqrt {b^2-4 a c} e\right )+2 a c^2 \left (-c d^2+e \left (\sqrt {b^2-4 a c} d+a e\right )\right )+b^2 c \left (c d^2-2 e \left (\sqrt {b^2-4 a c} d+2 a e\right )\right )+b c \left (-2 a \sqrt {b^2-4 a c} e^2+c d \left (\sqrt {b^2-4 a c} d+6 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}}{15 c^{7/2}} \] Input:

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

Output:

((2*Sqrt[c]*Sqrt[d + e*x]*(15*b^2*e^2 + 3*c^2*(d + e*x)^2 - 5*c*e*(4*b*d + 
 3*a*e + b*e*x)))/e - (15*Sqrt[2]*(-(b^4*e^2) + b^3*e*(2*c*d + Sqrt[b^2 - 
4*a*c]*e) + b*c*(-2*a*Sqrt[b^2 - 4*a*c]*e^2 + c*d*(Sqrt[b^2 - 4*a*c]*d - 6 
*a*e)) - b^2*c*(c*d^2 + 2*e*(Sqrt[b^2 - 4*a*c]*d - 2*a*e)) + 2*a*c^2*(c*d^ 
2 + e*(Sqrt[b^2 - 4*a*c]*d - a*e)))*ArcTan[(Sqrt[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^2 + b^3*e*(-2*c*d + Sq 
rt[b^2 - 4*a*c]*e) + 2*a*c^2*(-(c*d^2) + e*(Sqrt[b^2 - 4*a*c]*d + a*e)) + 
b^2*c*(c*d^2 - 2*e*(Sqrt[b^2 - 4*a*c]*d + 2*a*e)) + b*c*(-2*a*Sqrt[b^2 - 4 
*a*c]*e^2 + c*d*(Sqrt[b^2 - 4*a*c]*d + 6*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*Sq 
rt[d + e*x])/Sqrt[-2*c*d + (b + 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*c^(7/2))
 

Rubi [A] (verified)

Time = 1.60 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.01, 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.

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

\(\Big \downarrow \) 1199

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (\frac {e \left ((c d-b e) \left (2 a c e+b^2 (-e)+b c d\right )+\frac {-b^2 c \left (c d^2-4 a e^2\right )-6 a b c^2 d e+2 a c^2 \left (c d^2-a e^2\right )+b^4 \left (-e^2\right )+2 b^3 c d e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{7/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {e \left ((c d-b e) \left (2 a c e+b^2 (-e)+b c d\right )-\frac {-b^2 c \left (c d^2-4 a e^2\right )-6 a b c^2 d e+2 a c^2 \left (c d^2-a e^2\right )+b^4 \left (-e^2\right )+2 b^3 c d e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{7/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {e \sqrt {d+e x} \left (a c e+b^2 (-e)+b c d\right )}{c^3}-\frac {b e (d+e x)^{3/2}}{3 c^2}+\frac {(d+e x)^{5/2}}{5 c}\right )}{e}\)

Input:

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

Output:

(2*(-((e*(b*c*d - b^2*e + a*c*e)*Sqrt[d + e*x])/c^3) - (b*e*(d + e*x)^(3/2 
))/(3*c^2) + (d + e*x)^(5/2)/(5*c) + (e*((c*d - b*e)*(b*c*d - b^2*e + 2*a* 
c*e) + (2*b^3*c*d*e - 6*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 4*a*e^2) + 
2*a*c^2*(c*d^2 - a*e^2))/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*((c*d - b*e)*(b*c*d - b^2*e + 2*a 
*c*e) - (2*b^3*c*d*e - 6*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 4*a*e^2) + 
 2*a*c^2*(c*d^2 - a*e^2))/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 = 1.76 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.23

method	result	size
pseudoelliptic	\(-\frac {2 \left (e \left (\left (b e -c d \right ) \left (\left (a e +\frac {b d}{2}\right ) c -\frac {e \,b^{2}}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (-a \,c^{3} d^{2}+\left (e^{2} a^{2}+3 a b d e +\frac {1}{2} b^{2} d^{2}\right ) c^{2}+\left (-2 a \,b^{2} e^{2}-b^{3} d e \right ) c +\frac {b^{4} e^{2}}{2}\right ) e \right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (e \sqrt {2}\, \left (-\left (b e -c d \right ) \left (\left (a e +\frac {b d}{2}\right ) c -\frac {e \,b^{2}}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (-a \,c^{3} d^{2}+\left (e^{2} a^{2}+3 a b d e +\frac {1}{2} b^{2} d^{2}\right ) c^{2}+\left (-2 a \,b^{2} e^{2}-b^{3} d e \right ) c +\frac {b^{4} e^{2}}{2}\right ) e \right ) \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {e x +d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (-\frac {\left (e x +d \right )^{2} c^{2}}{5}+e \left (\left (\frac {b x}{3}+a \right ) e +\frac {4 b d}{3}\right ) c -b^{2} e^{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right )\right )}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, e \,c^{3}}\)	\(541\)
risch	\(-\frac {2 \left (-3 c^{2} e^{2} x^{2}+5 e^{2} x b c -6 c^{2} d e x +15 a c \,e^{2}-15 b^{2} e^{2}+20 b c d e -3 c^{2} d^{2}\right ) \sqrt {e x +d}}{15 e \,c^{3}}+\frac {-\frac {\left (2 a^{2} c^{2} e^{3}-4 a \,b^{2} c \,e^{3}+6 a b \,c^{2} d \,e^{2}-2 a \,c^{3} d^{2} e +e^{3} b^{4}-2 b^{3} c d \,e^{2}+b^{2} c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a^{2} c^{2} e^{3}+4 a \,b^{2} c \,e^{3}-6 a b \,c^{2} d \,e^{2}+2 a \,c^{3} d^{2} e -e^{3} b^{4}+2 b^{3} c d \,e^{2}-b^{2} c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c^{2}}\)	\(670\)
derivativedivides	\(\frac {-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {5}{2}} c^{2}}{5}+\frac {b c e \left (e x +d \right )^{\frac {3}{2}}}{3}+a c \,e^{2} \sqrt {e x +d}-b^{2} e^{2} \sqrt {e x +d}+b c d e \sqrt {e x +d}\right )}{c^{3}}+\frac {8 e \left (-\frac {\left (2 a^{2} c^{2} e^{3}-4 a \,b^{2} c \,e^{3}+6 a b \,c^{2} d \,e^{2}-2 a \,c^{3} d^{2} e +e^{3} b^{4}-2 b^{3} c d \,e^{2}+b^{2} c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a^{2} c^{2} e^{3}+4 a \,b^{2} c \,e^{3}-6 a b \,c^{2} d \,e^{2}+2 a \,c^{3} d^{2} e -e^{3} b^{4}+2 b^{3} c d \,e^{2}-b^{2} c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c^{2}}}{e}\)	\(674\)
default	\(\frac {-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {5}{2}} c^{2}}{5}+\frac {b c e \left (e x +d \right )^{\frac {3}{2}}}{3}+a c \,e^{2} \sqrt {e x +d}-b^{2} e^{2} \sqrt {e x +d}+b c d e \sqrt {e x +d}\right )}{c^{3}}+\frac {8 e \left (-\frac {\left (2 a^{2} c^{2} e^{3}-4 a \,b^{2} c \,e^{3}+6 a b \,c^{2} d \,e^{2}-2 a \,c^{3} d^{2} e +e^{3} b^{4}-2 b^{3} c d \,e^{2}+b^{2} c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a^{2} c^{2} e^{3}+4 a \,b^{2} c \,e^{3}-6 a b \,c^{2} d \,e^{2}+2 a \,c^{3} d^{2} e -e^{3} b^{4}+2 b^{3} c d \,e^{2}-b^{2} c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c^{2}}}{e}\)	\(674\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8530 vs. \(2 (391) = 782\).

Time = 2.45 (sec) , antiderivative size = 8530, normalized size of antiderivative = 19.34 \[ \int \frac {x^2 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate((e*x + d)^(3/2)*x^2/(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. 1195 vs. \(2 (391) = 782\).

Time = 0.37 (sec) , antiderivative size = 1195, normalized size of antiderivative = 2.71 \[ \int \frac {x^2 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

integrate(x^2*(e*x+d)^(3/2)/(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^3*c^2 - 4*a*b*c 
^3)*d^2 - 2*(b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d*e + (b^5 - 6*a*b^3*c + 8*a 
^2*b*c^2)*e^2)*c^2*e^2 - 2*(sqrt(b^2 - 4*a*c)*b*c^4*d^3 + sqrt(b^2 - 4*a*c 
)*b^3*c^2*d*e^2 - (2*b^2*c^3 - a*c^4)*sqrt(b^2 - 4*a*c)*d^2*e - (a*b^2*c^2 
 - a^2*c^3)*sqrt(b^2 - 4*a*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a 
*c)*c)*e)*abs(c)*abs(e) + (2*(b^2*c^5 - 2*a*c^6)*d^3*e - (5*b^3*c^4 - 14*a 
*b*c^5)*d^2*e^2 + 2*(2*b^4*c^3 - 7*a*b^2*c^4 + 2*a^2*c^5)*d*e^3 - (b^5*c^2 
 - 4*a*b^3*c^3 + 2*a^2*b*c^4)*e^4)*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^6 - b*c^5*e^7 
 + sqrt(-4*(c^6*d^2*e^6 - b*c^5*d*e^7 + a*c^5*e^8)*c^6*e^6 + (2*c^6*d*e^6 
- b*c^5*e^7)^2))/(c^6*e^6)))/((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^3*c^2 - 4*a*b*c^3)*d^2 - 2*(b^4* 
c - 5*a*b^2*c^2 + 4*a^2*c^3)*d*e + (b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*e^2)*c^ 
2*e^2 + 2*(sqrt(b^2 - 4*a*c)*b*c^4*d^3 + sqrt(b^2 - 4*a*c)*b^3*c^2*d*e^2 - 
 (2*b^2*c^3 - a*c^4)*sqrt(b^2 - 4*a*c)*d^2*e - (a*b^2*c^2 - a^2*c^3)*sqrt( 
b^2 - 4*a*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(c)* 
abs(e) + (2*(b^2*c^5 - 2*a*c^6)*d^3*e - (5*b^3*c^4 - 14*a*b*c^5)*d^2*e^2 + 
 2*(2*b^4*c^3 - 7*a*b^2*c^4 + 2*a^2*c^5)*d*e^3 - (b^5*c^2 - 4*a*b^3*c^3 + 
2*a^2*b*c^4)*e^4)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arc...
 

Mupad [B] (verification not implemented)

Time = 13.79 (sec) , antiderivative size = 19465, normalized size of antiderivative = 44.14 \[ \int \frac {x^2 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((8*(4*a^3*c^6*e^5 + a*b^4*c^4*e^5 - b^5*c^4*d*e^4 - 5*a^2*b^2*c^5* 
e^5 + 4*a^2*c^7*d^2*e^3 - b^3*c^6*d^3*e^2 + 2*b^4*c^5*d^2*e^3 + 4*a*b*c^7* 
d^3*e^2 + 4*a*b^3*c^5*d*e^4 - 9*a*b^2*c^6*d^2*e^3))/c^5 - (8*(d + e*x)^(1/ 
2)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 - b^6*e^3*(-(4*a*c - b^2)^3)^( 
1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d 
^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 + a^3* 
c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 
11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 - 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2 
) - 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e^3*(-(4*a*c - b^2) 
^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e + 
 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^ 
4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 - 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^( 
1/2) - 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^2*c^3*d^2*e*(-(4*a 
*c - b^2)^3)^(1/2) - 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b 
*c^3*d*e^2*(-(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^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 - b^6*e^3*(-(4*a*c - b^2)^ 
3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c 
^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 + 
a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(...
 

Reduce [F]

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

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

Output:

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