\(\int \frac {x (d+e x)^{3/2}}{a+b x+c x^2} \, dx\) [69]

Optimal result

Integrand size = 23, antiderivative size = 376 \[ \int \frac {x (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\frac {2 (c d-b e) \sqrt {d+e x}}{c^2}+\frac {2 (d+e x)^{3/2}}{3 c}-\frac {\sqrt {2} \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)+\frac {2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {2} \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)-\frac {2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

2*(-b*e+c*d)*(e*x+d)^(1/2)/c^2+2/3*(e*x+d)^(3/2)/c-2^(1/2)*(c^2*d^2+b^2*e^ 
2-c*e*(a*e+2*b*d)+(2*b^2*c*d*e-4*a*c^2*d*e-b^3*e^2-b*c*(-3*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^(5/2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)-2^(1 
/2)*(c^2*d^2+b^2*e^2-c*e*(a*e+2*b*d)-(2*b^2*c*d*e-4*a*c^2*d*e-b^3*e^2-b*c* 
(-3*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^(5/2)/(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 = 3.10 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.31 \[ \int \frac {x (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\frac {2 \sqrt {c} \sqrt {d+e x} (4 c d-3 b e+c e x)+\frac {3 \left (i b^3 e^2+b^2 e \left (-2 i c d+\sqrt {-b^2+4 a c} e\right )+i b c \left (c d^2+e \left (2 i \sqrt {-b^2+4 a c} d-3 a e\right )\right )+c \left (-a \sqrt {-b^2+4 a c} e^2+c d \left (\sqrt {-b^2+4 a c} d+4 i a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {3 \left (-i b^3 e^2+b^2 e \left (2 i c d+\sqrt {-b^2+4 a c} e\right )+b c \left (-i c d^2+e \left (-2 \sqrt {-b^2+4 a c} d+3 i a e\right )\right )+c \left (-a \sqrt {-b^2+4 a c} e^2+c d \left (\sqrt {-b^2+4 a c} d-4 i a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{3 c^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1196, 25, 1196, 1197, 1480, 221}

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.

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

\(\Big \downarrow \) 1196

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1196

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

\(\Big \downarrow \) 1197

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(2*(d + e*x)^(3/2))/(3*c) - ((-2*(c*d - b*e)*Sqrt[d + e*x])/c + (2*(((c^2* 
d^2 + b^2*e^2 - c*e*(2*b*d + a*e) + (2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - 
 b*c*(c*d^2 - 3*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]*Sqrt[c]*Sqrt[2 
*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + ((c^2*d^2 + b^2*e^2 - c*e*(2*b*d + a* 
e) - (2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*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]*Sqrt[c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])* 
e])))/c)/c
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 1196

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

rule 1197

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)), x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - 
b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr 
eeQ[{a, b, c, d, e, f, g}, x]
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 1.61 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.29

method	result	size
pseudoelliptic	\(-\frac {2 \left (-\frac {\left (\left (-c^{2} d^{2}+\left (a \,e^{2}+2 b d e \right ) c -b^{2} e^{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (4 a d \,e^{2}+b \,d^{2} e \right ) c^{2}+\left (-3 a \,e^{3} b -2 d \,e^{2} b^{2}\right ) c +b^{3} e^{3}\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )}{2}+\left (\frac {\sqrt {2}\, \left (\left (-c^{2} d^{2}+\left (a \,e^{2}+2 b d e \right ) c -b^{2} e^{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (-4 a d \,e^{2}-b \,d^{2} e \right ) c^{2}+\left (3 a \,e^{3} b +2 d \,e^{2} b^{2}\right ) c -b^{3} e^{3}\right ) \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )}{2}+\left (\frac {\left (-e x -4 d \right ) c}{3}+b e \right ) \sqrt {e x +d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\right )}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, c^{2}}\)	\(485\)
risch	\(-\frac {2 \left (-c e x +3 b e -4 c d \right ) \sqrt {e x +d}}{3 c^{2}}-\frac {8 \left (-\frac {\left (-3 a b c \,e^{3}+4 d \,e^{2} a \,c^{2}+b^{3} e^{3}-2 d \,e^{2} b^{2} c +d^{2} e b \,c^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (3 a b c \,e^{3}-4 d \,e^{2} a \,c^{2}-b^{3} e^{3}+2 d \,e^{2} b^{2} c -d^{2} e b \,c^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c}\)	\(512\)
derivativedivides	\(-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {3}{2}} c}{3}+\sqrt {e x +d}\, b e -c d \sqrt {e x +d}\right )}{c^{2}}+\frac {\frac {\left (-3 a b c \,e^{3}+4 d \,e^{2} a \,c^{2}+b^{3} e^{3}-2 d \,e^{2} b^{2} c +d^{2} e b \,c^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (3 a b c \,e^{3}-4 d \,e^{2} a \,c^{2}-b^{3} e^{3}+2 d \,e^{2} b^{2} c -d^{2} e b \,c^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c}\)	\(521\)
default	\(-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {3}{2}} c}{3}+\sqrt {e x +d}\, b e -c d \sqrt {e x +d}\right )}{c^{2}}+\frac {\frac {\left (-3 a b c \,e^{3}+4 d \,e^{2} a \,c^{2}+b^{3} e^{3}-2 d \,e^{2} b^{2} c +d^{2} e b \,c^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (3 a b c \,e^{3}-4 d \,e^{2} a \,c^{2}-b^{3} e^{3}+2 d \,e^{2} b^{2} c -d^{2} e b \,c^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c}\)	\(521\)

Input:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5572 vs. \(2 (330) = 660\).

Time = 0.88 (sec) , antiderivative size = 5572, normalized size of antiderivative = 14.82 \[ \int \frac {x (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate((e*x + d)^(3/2)*x/(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. 986 vs. \(2 (330) = 660\).

Time = 0.31 (sec) , antiderivative size = 986, normalized size of antiderivative = 2.62 \[ \int \frac {x (d+e x)^{3/2}}{a+b x+c x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 13.29 (sec) , antiderivative size = 13841, normalized size of antiderivative = 36.81 \[ \int \frac {x (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(2*(d + e*x)^(3/2))/(3*c) - ((2*d)/c + (2*(b*e - 2*c*d))/c^2)*(d + e*x)^(1 
/2) - atan(((((8*(a*b^3*c^3*e^5 - 4*a^2*b*c^4*e^5 + 4*a*c^6*d^3*e^2 + 4*a^ 
2*c^5*d*e^4 - b^4*c^3*d*e^4 - b^2*c^5*d^3*e^2 + 2*b^3*c^4*d^2*e^3 - 8*a*b* 
c^5*d^2*e^3 + 3*a*b^2*c^4*d*e^4))/c^3 - (8*(d + e*x)^(1/2)*(-(b^7*e^3 - 8* 
a^2*c^5*d^3 - b^4*c^3*d^3 + b^4*e^3*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^2*c^4 
*d^3 - 20*a^3*b*c^3*e^3 - b*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 24*a^3*c^4* 
d*e^2 + 3*b^5*c^2*d^2*e + 25*a^2*b^3*c^2*e^3 + a^2*c^2*e^3*(-(4*a*c - b^2) 
^3)^(1/2) - 9*a*b^5*c*e^3 - 3*b^6*c*d*e^2 - 3*a*b^2*c*e^3*(-(4*a*c - b^2)^ 
3)^(1/2) - 21*a*b^3*c^3*d^2*e + 24*a*b^4*c^2*d*e^2 + 36*a^2*b*c^4*d^2*e - 
3*a*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*b^3*c*d*e^2*(-(4*a*c - b^2)^3)^ 
(1/2) - 54*a^2*b^2*c^3*d*e^2 + 3*b^2*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 
6*a*b*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^7 + b^4*c^5 - 8*a*b 
^2*c^6)))^(1/2)*(b^3*c^5*e^3 - 2*b^2*c^6*d*e^2 - 4*a*b*c^6*e^3 + 8*a*c^7*d 
*e^2))/c^3)*(-(b^7*e^3 - 8*a^2*c^5*d^3 - b^4*c^3*d^3 + b^4*e^3*(-(4*a*c - 
b^2)^3)^(1/2) + 6*a*b^2*c^4*d^3 - 20*a^3*b*c^3*e^3 - b*c^3*d^3*(-(4*a*c - 
b^2)^3)^(1/2) + 24*a^3*c^4*d*e^2 + 3*b^5*c^2*d^2*e + 25*a^2*b^3*c^2*e^3 + 
a^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e^3 - 3*b^6*c*d*e^2 - 3*a 
*b^2*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 21*a*b^3*c^3*d^2*e + 24*a*b^4*c^2*d* 
e^2 + 36*a^2*b*c^4*d^2*e - 3*a*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*b^3* 
c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 54*a^2*b^2*c^3*d*e^2 + 3*b^2*c^2*d^2...
 

Reduce [F]

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

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

Output:

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