\(\int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx\) [58]

Optimal result

Integrand size = 25, antiderivative size = 490 \[ \int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx=-\frac {2 b \left (b^2-2 a c\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (c^2 d^2+b^2 e^2+c e (b d-a e)\right ) (d+e x)^{3/2}}{3 c^3 e^3}-\frac {2 (2 c d+b e) (d+e x)^{5/2}}{5 c^2 e^3}+\frac {2 (d+e x)^{7/2}}{7 c e^3}+\frac {\sqrt {2} \left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e-\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{9/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e+\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{9/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

-2*b*(-2*a*c+b^2)*(e*x+d)^(1/2)/c^4+2/3*(c^2*d^2+b^2*e^2+c*e*(-a*e+b*d))*( 
e*x+d)^(3/2)/c^3/e^3-2/5*(b*e+2*c*d)*(e*x+d)^(5/2)/c^2/e^3+2/7*(e*x+d)^(7/ 
2)/c/e^3+2^(1/2)*(b^3*c*d-2*a*b*c^2*d-b^4*e+3*a*b^2*c*e-a^2*c^2*e-(-5*a^2* 
b*c^2*e+2*a^2*c^3*d+5*a*b^3*c*e-4*a*b^2*c^2*d-b^5*e+b^4*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^(9/2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)+2^(1/2)*(b^3*c*d 
-2*a*b*c^2*d-b^4*e+3*a*b^2*c*e-a^2*c^2*e+(-5*a^2*b*c^2*e+2*a^2*c^3*d+5*a*b 
^3*c*e-4*a*b^2*c^2*d-b^5*e+b^4*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^(9/2)/(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 = 3.20 (sec) , antiderivative size = 627, normalized size of antiderivative = 1.28 \[ \int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\frac {2 \sqrt {d+e x} \left (-105 b^3 e^3-7 c^2 e (d+e x) (-2 b d+5 a e+3 b e x)+c^3 \left (8 d^3-4 d^2 e x+3 d e^2 x^2+15 e^3 x^3\right )+35 b c e^2 (6 a e+b (d+e x))\right )}{105 c^4 e^3}+\frac {\left (i b^5 e-b^3 c \left (\sqrt {-b^2+4 a c} d+5 i a e\right )+a b c^2 \left (2 \sqrt {-b^2+4 a c} d+5 i a e\right )+a b^2 c \left (4 i c d-3 \sqrt {-b^2+4 a c} e\right )+b^4 \left (-i c d+\sqrt {-b^2+4 a c} e\right )+a^2 c^2 \left (-2 i c d+\sqrt {-b^2+4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{c^{9/2} \sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\left (-i b^5 e+a b c^2 \left (2 \sqrt {-b^2+4 a c} d-5 i a e\right )+b^3 c \left (-\sqrt {-b^2+4 a c} d+5 i a e\right )+a b^2 c \left (-4 i c d-3 \sqrt {-b^2+4 a c} e\right )+b^4 \left (i c d+\sqrt {-b^2+4 a c} e\right )+a^2 c^2 \left (2 i c d+\sqrt {-b^2+4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{c^{9/2} \sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}} \] Input:

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

Output:

(2*Sqrt[d + e*x]*(-105*b^3*e^3 - 7*c^2*e*(d + e*x)*(-2*b*d + 5*a*e + 3*b*e 
*x) + c^3*(8*d^3 - 4*d^2*e*x + 3*d*e^2*x^2 + 15*e^3*x^3) + 35*b*c*e^2*(6*a 
*e + b*(d + e*x))))/(105*c^4*e^3) + ((I*b^5*e - b^3*c*(Sqrt[-b^2 + 4*a*c]* 
d + (5*I)*a*e) + a*b*c^2*(2*Sqrt[-b^2 + 4*a*c]*d + (5*I)*a*e) + a*b^2*c*(( 
4*I)*c*d - 3*Sqrt[-b^2 + 4*a*c]*e) + b^4*((-I)*c*d + Sqrt[-b^2 + 4*a*c]*e) 
 + a^2*c^2*((-2*I)*c*d + Sqrt[-b^2 + 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sq 
rt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(c^(9/2)*Sqrt[- 
1/2*b^2 + 2*a*c]*Sqrt[-2*c*d + (b - I*Sqrt[-b^2 + 4*a*c])*e]) + (((-I)*b^5 
*e + a*b*c^2*(2*Sqrt[-b^2 + 4*a*c]*d - (5*I)*a*e) + b^3*c*(-(Sqrt[-b^2 + 4 
*a*c]*d) + (5*I)*a*e) + a*b^2*c*((-4*I)*c*d - 3*Sqrt[-b^2 + 4*a*c]*e) + b^ 
4*(I*c*d + Sqrt[-b^2 + 4*a*c]*e) + a^2*c^2*((2*I)*c*d + Sqrt[-b^2 + 4*a*c] 
*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^ 
2 + 4*a*c]*e]])/(c^(9/2)*Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[-2*c*d + (b + I*Sqrt[ 
-b^2 + 4*a*c])*e])
 

Rubi [A] (verified)

Time = 7.96 (sec) , antiderivative size = 498, 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.

\(\displaystyle \int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx\)

\(\Big \downarrow \) 1199

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (\frac {e \left (-\frac {-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d}{\sqrt {b^2-4 a c}}-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{9/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {e \left (\frac {-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d}{\sqrt {b^2-4 a c}}-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{9/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {b e \left (b^2-2 a c\right ) \sqrt {d+e x}}{c^4}+\frac {(d+e x)^{3/2} \left (c e (b d-a e)+b^2 e^2+c^2 d^2\right )}{3 c^3 e^2}-\frac {(d+e x)^{5/2} (b e+2 c d)}{5 c^2 e^2}+\frac {(d+e x)^{7/2}}{7 c e^2}\right )}{e}\)

Input:

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

Output:

(2*(-((b*(b^2 - 2*a*c)*e*Sqrt[d + e*x])/c^4) + ((c^2*d^2 + b^2*e^2 + c*e*( 
b*d - a*e))*(d + e*x)^(3/2))/(3*c^3*e^2) - ((2*c*d + b*e)*(d + e*x)^(5/2)) 
/(5*c^2*e^2) + (d + e*x)^(7/2)/(7*c*e^2) + (e*(b^3*c*d - 2*a*b*c^2*d - b^4 
*e + 3*a*b^2*c*e - a^2*c^2*e - (b^4*c*d - 4*a*b^2*c^2*d + 2*a^2*c^3*d - b^ 
5*e + 5*a*b^3*c*e - 5*a^2*b*c^2*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqr 
t[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^( 
9/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (e*(b^3*c*d - 2*a*b*c^2*d 
- b^4*e + 3*a*b^2*c*e - a^2*c^2*e + (b^4*c*d - 4*a*b^2*c^2*d + 2*a^2*c^3*d 
 - b^5*e + 5*a*b^3*c*e - 5*a^2*b*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^(9/2)*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 = 594, normalized size of antiderivative = 1.21

method	result	size
pseudoelliptic	\(\frac {-e^{3} \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {2}\, \left (\left (\left (a^{2} c^{2}-3 c a \,b^{2}+b^{4}\right ) e +2 a b \,c^{2} d -b^{3} c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-5 e \left (b \left (a^{2} c^{2}-c a \,b^{2}+\frac {1}{5} b^{4}\right ) e -\frac {2 d \left (a^{2} c^{2}-2 c a \,b^{2}+\frac {1}{2} b^{4}\right ) c}{5}\right )\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (e^{3} \sqrt {2}\, \left (\left (\left (a^{2} c^{2}-3 c a \,b^{2}+b^{4}\right ) e +2 a b \,c^{2} d -b^{3} c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+5 e \left (b \left (a^{2} c^{2}-c a \,b^{2}+\frac {1}{5} b^{4}\right ) e -\frac {2 d \left (a^{2} c^{2}-2 c a \,b^{2}+\frac {1}{2} b^{4}\right ) c}{5}\right )\right ) \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+4 \sqrt {e x +d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\left (\frac {c^{3} x^{3}}{14}-\frac {x \left (\frac {3 b x}{5}+a \right ) c^{2}}{6}+b \left (\frac {b x}{6}+a \right ) c -\frac {b^{3}}{2}\right ) e^{3}-\frac {\left (-\frac {3 c^{2} x^{2}}{35}+\left (\frac {b x}{5}+a \right ) c -b^{2}\right ) d c \,e^{2}}{6}+\frac {d^{2} \left (-\frac {2 c x}{7}+b \right ) c^{2} e}{15}+\frac {4 d^{3} c^{3}}{105}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right )}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, c^{4} e^{3}}\)	\(594\)
derivativedivides	\(\frac {\frac {2 \left (\frac {\left (e x +d \right )^{\frac {7}{2}} c^{3}}{7}-\frac {b \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {a \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {b^{2} c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {b \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a b c \,e^{3} \sqrt {e x +d}-b^{3} e^{3} \sqrt {e x +d}\right )}{c^{4}}-\frac {8 e^{3} \left (-\frac {\left (5 a^{2} b \,c^{2} e^{2}-2 a^{2} c^{3} d e -5 a \,b^{3} c \,e^{2}+4 a \,b^{2} c^{2} d e +b^{5} e^{2}-b^{4} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} 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 (-5 a^{2} b \,c^{2} e^{2}+2 a^{2} c^{3} d e +5 a \,b^{3} c \,e^{2}-4 a \,b^{2} c^{2} d e -b^{5} e^{2}+b^{4} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} 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^{3}}}{e^{3}}\)	\(708\)
default	\(\frac {\frac {2 \left (\frac {\left (e x +d \right )^{\frac {7}{2}} c^{3}}{7}-\frac {b \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {a \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {b^{2} c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {b \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a b c \,e^{3} \sqrt {e x +d}-b^{3} e^{3} \sqrt {e x +d}\right )}{c^{4}}-\frac {8 e^{3} \left (-\frac {\left (5 a^{2} b \,c^{2} e^{2}-2 a^{2} c^{3} d e -5 a \,b^{3} c \,e^{2}+4 a \,b^{2} c^{2} d e +b^{5} e^{2}-b^{4} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} 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 (-5 a^{2} b \,c^{2} e^{2}+2 a^{2} c^{3} d e +5 a \,b^{3} c \,e^{2}-4 a \,b^{2} c^{2} d e -b^{5} e^{2}+b^{4} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} 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^{3}}}{e^{3}}\)	\(708\)
risch	\(\frac {2 \left (15 c^{3} x^{3} e^{3}-21 b \,c^{2} e^{3} x^{2}+3 c^{3} d \,e^{2} x^{2}-35 a \,c^{2} e^{3} x +35 x \,b^{2} c \,e^{3}-7 b \,c^{2} d \,e^{2} x -4 d^{2} e \,c^{3} x +210 a b c \,e^{3}-35 d \,e^{2} a \,c^{2}-105 b^{3} e^{3}+35 d \,e^{2} b^{2} c +14 d^{2} e b \,c^{2}+8 d^{3} c^{3}\right ) \sqrt {e x +d}}{105 e^{3} c^{4}}+\frac {-\frac {\left (-5 a^{2} b \,c^{2} e^{2}+2 a^{2} c^{3} d e +5 a \,b^{3} c \,e^{2}-4 a \,b^{2} c^{2} d e -b^{5} e^{2}+b^{4} c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e -3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (5 a^{2} b \,c^{2} e^{2}-2 a^{2} c^{3} d e -5 a \,b^{3} c \,e^{2}+4 a \,b^{2} c^{2} d e +b^{5} e^{2}-b^{4} c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e -3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c^{3}}\)	\(708\)

Input:

int(x^4*(e*x+d)^(1/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
 

Output:

(-e^3*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*2^(1/2)*(((a^2*c^ 
2-3*a*b^2*c+b^4)*e+2*a*b*c^2*d-b^3*c*d)*(-4*e^2*(a*c-1/4*b^2))^(1/2)-5*e*( 
b*(a^2*c^2-c*a*b^2+1/5*b^4)*e-2/5*d*(a^2*c^2-2*c*a*b^2+1/2*b^4)*c))*arctan 
h((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^3*2^(1/2)*(((a 
^2*c^2-3*a*b^2*c+b^4)*e+2*a*b*c^2*d-b^3*c*d)*(-4*e^2*(a*c-1/4*b^2))^(1/2)+ 
5*e*(b*(a^2*c^2-c*a*b^2+1/5*b^4)*e-2/5*d*(a^2*c^2-2*c*a*b^2+1/2*b^4)*c))*a 
rctan((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))+4*(e*x+d)^(1/2)*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2) 
*((1/14*c^3*x^3-1/6*x*(3/5*b*x+a)*c^2+b*(1/6*b*x+a)*c-1/2*b^3)*e^3-1/6*(-3 
/35*c^2*x^2+(1/5*b*x+a)*c-b^2)*d*c*e^2+1/15*d^2*(-2/7*c*x+b)*c^2*e+4/105*d 
^3*c^3)*(-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^4/e^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5507 vs. \(2 (436) = 872\).

Time = 0.65 (sec) , antiderivative size = 5507, normalized size of antiderivative = 11.24 \[ \int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

integrate(x^4*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\int \frac {x^{4} \sqrt {d + e x}}{a + b x + c x^{2}}\, dx \] Input:

integrate(x**4*(e*x+d)**(1/2)/(c*x**2+b*x+a),x)
                                                                                    
                                                                                    
 

Output:

Integral(x**4*sqrt(d + e*x)/(a + b*x + c*x**2), x)
 

Maxima [F]

\[ \int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\int { \frac {\sqrt {e x + d} x^{4}}{c x^{2} + b x + a} \,d x } \] Input:

integrate(x^4*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="maxima")
 

Output:

integrate(sqrt(e*x + d)*x^4/(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. 1200 vs. \(2 (436) = 872\).

Time = 0.33 (sec) , antiderivative size = 1200, normalized size of antiderivative = 2.45 \[ \int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

integrate(x^4*(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^5*c - 6*a*b^3*c 
^2 + 8*a^2*b*c^3)*d - (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e)*c^ 
2*e^2 - 2*((b^3*c^3 - 2*a*b*c^4)*sqrt(b^2 - 4*a*c)*d^2 - (b^4*c^2 - 2*a*b^ 
2*c^3)*sqrt(b^2 - 4*a*c)*d*e + (a*b^3*c^2 - 2*a^2*b*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^4*c^4 - 4*a*b^2*c^5 + 2*a^2*c^6)*d^2*e - (3*b^5*c^3 - 14*a*b^3*c^4 + 12 
*a^2*b*c^5)*d*e^2 + (b^6*c^2 - 5*a*b^4*c^3 + 5*a^2*b^2*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^8*d*e^24 - b*c^7*e^25 + sqrt(-4*(c^8*d^2*e^24 - b*c^7*d*e^25 + 
a*c^7*e^26)*c^8*e^24 + (2*c^8*d*e^24 - b*c^7*e^25)^2))/(c^8*e^24)))/((sqrt 
(b^2 - 4*a*c)*c^7*d^2 - sqrt(b^2 - 4*a*c)*b*c^6*d*e + sqrt(b^2 - 4*a*c)*a* 
c^6*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^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d - (b^6 - 7*a*b^4*c + 13*a^2*b^2* 
c^2 - 4*a^3*c^3)*e)*c^2*e^2 + 2*((b^3*c^3 - 2*a*b*c^4)*sqrt(b^2 - 4*a*c)*d 
^2 - (b^4*c^2 - 2*a*b^2*c^3)*sqrt(b^2 - 4*a*c)*d*e + (a*b^3*c^2 - 2*a^2*b* 
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^4*c^4 - 4*a*b^2*c^5 + 2*a^2*c^6)*d^2*e - (3*b^5*c 
^3 - 14*a*b^3*c^4 + 12*a^2*b*c^5)*d*e^2 + (b^6*c^2 - 5*a*b^4*c^3 + 5*a^2*b 
^2*c^4)*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sq 
rt(1/2)*sqrt(e*x + d)/sqrt(-(2*c^8*d*e^24 - b*c^7*e^25 - sqrt(-4*(c^8*d...
 

Mupad [B] (verification not implemented)

Time = 13.55 (sec) , antiderivative size = 13879, normalized size of antiderivative = 28.32 \[ \int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(d + e*x)^(3/2)*((4*d^2)/(c*e^3) - (2*(a*e^5 + c*d^2*e^3 - b*d*e^4))/(3*c^ 
2*e^6) + (((8*d)/(c*e^3) + (2*(b*e^4 - 2*c*d*e^3))/(c^2*e^6))*(b*e^4 - 2*c 
*d*e^3))/(3*c*e^3)) - atan(((((8*(a*b^5*c^5*e^4 + 8*a^3*b*c^7*e^4 - b^6*c^ 
5*d*e^3 - 6*a^2*b^3*c^6*e^4 + b^5*c^6*d^2*e^2 + 6*a*b^4*c^6*d*e^3 - 6*a*b^ 
3*c^7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^2 - 8*a^2*b^2*c^7*d*e^3))/c^7 - (8*(d + 
e*x)^(1/2)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10 
*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7 
*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2) 
^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4* 
a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d 
*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^ 
2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3) 
^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4* 
c^9 - 8*a*b^2*c^10)))^(1/2)*(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2 - 4*a*b*c^10*e 
^3 + 8*a*c^11*d*e^2))/c^7)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2) 
^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^ 
5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e 
*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e 
 - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) 
 + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^...
 

Reduce [F]

\[ \int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\int \frac {x^{4} \sqrt {e x +d}}{c \,x^{2}+b x +a}d x \] Input:

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

Output:

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