\(\int \frac {x^4 (d+e x)^{3/2}}{a+b x+c x^2} \, dx\) [66]

Optimal result

Integrand size = 25, antiderivative size = 650 \[ \int \frac {x^4 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=-\frac {2 \left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e\right ) \sqrt {d+e x}}{c^5}-\frac {2 b \left (b^2-2 a c\right ) (d+e x)^{3/2}}{3 c^4}+\frac {2 \left (c^2 d^2+b^2 e^2+c e (b d-a e)\right ) (d+e x)^{5/2}}{5 c^3 e^3}-\frac {2 (2 c d+b e) (d+e x)^{7/2}}{7 c^2 e^3}+\frac {2 (d+e x)^{9/2}}{9 c e^3}+\frac {\sqrt {2} \left (\left (b c d-b^2 e+a c e\right ) \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right )+\frac {2 b^5 c d e-10 a b^3 c^2 d e+10 a^2 b c^3 d e-b^6 e^2+a b^2 c^2 \left (4 c d^2-9 a e^2\right )-b^4 c \left (c d^2-6 a e^2\right )-2 a^2 c^3 \left (c d^2-a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{11/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left (\left (b c d-b^2 e+a c e\right ) \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right )-\frac {2 b^5 c d e-10 a b^3 c^2 d e+10 a^2 b c^3 d e-b^6 e^2+a b^2 c^2 \left (4 c d^2-9 a e^2\right )-b^4 c \left (c d^2-6 a e^2\right )-2 a^2 c^3 \left (c d^2-a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{11/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

-2*(-a^2*c^2*e+3*a*b^2*c*e-2*a*b*c^2*d-b^4*e+b^3*c*d)*(e*x+d)^(1/2)/c^5-2/ 
3*b*(-2*a*c+b^2)*(e*x+d)^(3/2)/c^4+2/5*(c^2*d^2+b^2*e^2+c*e*(-a*e+b*d))*(e 
*x+d)^(5/2)/c^3/e^3-2/7*(b*e+2*c*d)*(e*x+d)^(7/2)/c^2/e^3+2/9*(e*x+d)^(9/2 
)/c/e^3+2^(1/2)*((a*c*e-b^2*e+b*c*d)*(3*a*b*c*e-2*a*c^2*d-b^3*e+b^2*c*d)+( 
2*b^5*c*d*e-10*a*b^3*c^2*d*e+10*a^2*b*c^3*d*e-b^6*e^2+a*b^2*c^2*(-9*a*e^2+ 
4*c*d^2)-b^4*c*(-6*a*e^2+c*d^2)-2*a^2*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^(11/2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)+2^(1/2)*((a*c*e-b^ 
2*e+b*c*d)*(3*a*b*c*e-2*a*c^2*d-b^3*e+b^2*c*d)-(2*b^5*c*d*e-10*a*b^3*c^2*d 
*e+10*a^2*b*c^3*d*e-b^6*e^2+a*b^2*c^2*(-9*a*e^2+4*c*d^2)-b^4*c*(-6*a*e^2+c 
*d^2)-2*a^2*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^(11/2)/(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 = 5.20 (sec) , antiderivative size = 901, normalized size of antiderivative = 1.39 \[ \int \frac {x^4 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\frac {2 \sqrt {d+e x} \left (315 b^4 e^4-105 b^2 c e^3 (4 b d+9 a e+b e x)-9 c^3 e (d+e x)^2 (-2 b d+7 a e+5 b e x)+c^4 (d+e x)^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+21 c^2 e^2 \left (15 a^2 e^2+3 b^2 (d+e x)^2+10 a b e (4 d+e x)\right )\right )}{315 c^5 e^3}-\frac {\left (i b^6 e^2+b^5 e \left (-2 i c d+\sqrt {-b^2+4 a c} e\right )+i b^4 c \left (c d^2+2 i \sqrt {-b^2+4 a c} d e-6 a e^2\right )+a b^2 c^2 \left (-4 i c d^2+6 \sqrt {-b^2+4 a c} d e+9 i a e^2\right )+a b c^2 \left (3 a \sqrt {-b^2+4 a c} e^2-2 c d \left (\sqrt {-b^2+4 a c} d+5 i a e\right )\right )+b^3 c \left (-4 a \sqrt {-b^2+4 a c} e^2+c d \left (\sqrt {-b^2+4 a c} d+10 i a e\right )\right )-2 i a^2 c^3 \left (-c d^2+e \left (-i \sqrt {-b^2+4 a c} d+a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{c^{11/2} \sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}-\frac {\left (-i b^6 e^2+b^5 e \left (2 i c d+\sqrt {-b^2+4 a c} e\right )+b^4 c \left (-i c d^2-2 \sqrt {-b^2+4 a c} d e+6 i a e^2\right )+a b^2 c^2 \left (4 i c d^2+6 \sqrt {-b^2+4 a c} d e-9 i a e^2\right )+a b c^2 \left (3 a \sqrt {-b^2+4 a c} e^2-2 c d \left (\sqrt {-b^2+4 a c} d-5 i a e\right )\right )+b^3 c \left (-4 a \sqrt {-b^2+4 a c} e^2+c d \left (\sqrt {-b^2+4 a c} d-10 i a e\right )\right )+2 i a^2 c^3 \left (-c d^2+e \left (i \sqrt {-b^2+4 a c} d+a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{c^{11/2} \sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}} \] Input:

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

Output:

(2*Sqrt[d + e*x]*(315*b^4*e^4 - 105*b^2*c*e^3*(4*b*d + 9*a*e + b*e*x) - 9* 
c^3*e*(d + e*x)^2*(-2*b*d + 7*a*e + 5*b*e*x) + c^4*(d + e*x)^2*(8*d^2 - 20 
*d*e*x + 35*e^2*x^2) + 21*c^2*e^2*(15*a^2*e^2 + 3*b^2*(d + e*x)^2 + 10*a*b 
*e*(4*d + e*x))))/(315*c^5*e^3) - ((I*b^6*e^2 + b^5*e*((-2*I)*c*d + Sqrt[- 
b^2 + 4*a*c]*e) + I*b^4*c*(c*d^2 + (2*I)*Sqrt[-b^2 + 4*a*c]*d*e - 6*a*e^2) 
 + a*b^2*c^2*((-4*I)*c*d^2 + 6*Sqrt[-b^2 + 4*a*c]*d*e + (9*I)*a*e^2) + a*b 
*c^2*(3*a*Sqrt[-b^2 + 4*a*c]*e^2 - 2*c*d*(Sqrt[-b^2 + 4*a*c]*d + (5*I)*a*e 
)) + b^3*c*(-4*a*Sqrt[-b^2 + 4*a*c]*e^2 + c*d*(Sqrt[-b^2 + 4*a*c]*d + (10* 
I)*a*e)) - (2*I)*a^2*c^3*(-(c*d^2) + e*((-I)*Sqrt[-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]])/(c^(11/2)*Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[-2*c*d + (b - I*Sqrt[-b^ 
2 + 4*a*c])*e]) - (((-I)*b^6*e^2 + b^5*e*((2*I)*c*d + Sqrt[-b^2 + 4*a*c]*e 
) + b^4*c*((-I)*c*d^2 - 2*Sqrt[-b^2 + 4*a*c]*d*e + (6*I)*a*e^2) + a*b^2*c^ 
2*((4*I)*c*d^2 + 6*Sqrt[-b^2 + 4*a*c]*d*e - (9*I)*a*e^2) + a*b*c^2*(3*a*Sq 
rt[-b^2 + 4*a*c]*e^2 - 2*c*d*(Sqrt[-b^2 + 4*a*c]*d - (5*I)*a*e)) + b^3*c*( 
-4*a*Sqrt[-b^2 + 4*a*c]*e^2 + c*d*(Sqrt[-b^2 + 4*a*c]*d - (10*I)*a*e)) + ( 
2*I)*a^2*c^3*(-(c*d^2) + e*(I*Sqrt[-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]])/(c^ 
(11/2)*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 = 2.11 (sec) , antiderivative size = 659, 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.

\(\displaystyle \int \frac {x^4 (d+e x)^{3/2}}{a+b x+c x^2} \, dx\)

\(\Big \downarrow \) 1199

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (\frac {e \left (\frac {10 a^2 b c^3 d e-2 a^2 c^3 \left (c d^2-a e^2\right )-b^4 c \left (c d^2-6 a e^2\right )-10 a b^3 c^2 d e+a b^2 c^2 \left (4 c d^2-9 a e^2\right )+b^6 \left (-e^2\right )+2 b^5 c d e}{\sqrt {b^2-4 a c}}+\left (a c e+b^2 (-e)+b c d\right ) \left (3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{11/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {e \left (\left (a c e+b^2 (-e)+b c d\right ) \left (3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right )-\frac {10 a^2 b c^3 d e-2 a^2 c^3 \left (c d^2-a e^2\right )-b^4 c \left (c d^2-6 a e^2\right )-10 a b^3 c^2 d e+a b^2 c^2 \left (4 c d^2-9 a e^2\right )+b^6 \left (-e^2\right )+2 b^5 c d e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{11/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {e \sqrt {d+e x} \left (-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right )}{c^5}-\frac {b e \left (b^2-2 a c\right ) (d+e x)^{3/2}}{3 c^4}+\frac {(d+e x)^{5/2} \left (c e (b d-a e)+b^2 e^2+c^2 d^2\right )}{5 c^3 e^2}-\frac {(d+e x)^{7/2} (b e+2 c d)}{7 c^2 e^2}+\frac {(d+e x)^{9/2}}{9 c e^2}\right )}{e}\)

Input:

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

Output:

(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)*Sqrt[d 
+ e*x])/c^5) - (b*(b^2 - 2*a*c)*e*(d + e*x)^(3/2))/(3*c^4) + ((c^2*d^2 + b 
^2*e^2 + c*e*(b*d - a*e))*(d + e*x)^(5/2))/(5*c^3*e^2) - ((2*c*d + b*e)*(d 
 + e*x)^(7/2))/(7*c^2*e^2) + (d + e*x)^(9/2)/(9*c*e^2) + (e*((b*c*d - b^2* 
e + a*c*e)*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e) + (2*b^5*c*d*e - 10*a 
*b^3*c^2*d*e + 10*a^2*b*c^3*d*e - b^6*e^2 + a*b^2*c^2*(4*c*d^2 - 9*a*e^2) 
- b^4*c*(c*d^2 - 6*a*e^2) - 2*a^2*c^3*(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^(11/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (e*(( 
b*c*d - b^2*e + a*c*e)*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e) - (2*b^5* 
c*d*e - 10*a*b^3*c^2*d*e + 10*a^2*b*c^3*d*e - b^6*e^2 + a*b^2*c^2*(4*c*d^2 
 - 9*a*e^2) - b^4*c*(c*d^2 - 6*a*e^2) - 2*a^2*c^3*(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^(11/2)*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 = 1.95 (sec) , antiderivative size = 762, normalized size of antiderivative = 1.17

method	result	size
pseudoelliptic	\(\frac {2 e^{3} \left (\frac {3 \left (\left (a c -b^{2}\right ) e +d b c \right ) \left (\left (a b c -\frac {1}{3} b^{3}\right ) e -\frac {2 d \left (a c -\frac {b^{2}}{2}\right ) c}{3}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+e \left (\left (-\frac {1}{2} b^{6}-\frac {9}{2} a^{2} b^{2} c^{2}+3 a \,b^{4} c +a^{3} c^{3}\right ) e^{2}+5 \left (a^{2} c^{2}-c a \,b^{2}+\frac {1}{5} b^{4}\right ) d b c e -d^{2} \left (a^{2} c^{2}-2 c a \,b^{2}+\frac {1}{2} b^{4}\right ) c^{2}\right )\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 \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (e^{3} \left (-\frac {3 \left (\left (a c -b^{2}\right ) e +d b c \right ) \left (\left (a b c -\frac {1}{3} b^{3}\right ) e -\frac {2 d \left (a c -\frac {b^{2}}{2}\right ) c}{3}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+e \left (\left (-\frac {1}{2} b^{6}-\frac {9}{2} a^{2} b^{2} c^{2}+3 a \,b^{4} c +a^{3} c^{3}\right ) e^{2}+5 \left (a^{2} c^{2}-c a \,b^{2}+\frac {1}{5} b^{4}\right ) d b c e -d^{2} \left (a^{2} c^{2}-2 c a \,b^{2}+\frac {1}{2} b^{4}\right ) c^{2}\right )\right ) \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}\, \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^{4} x^{4}}{9}-\frac {\left (\frac {5 b x}{7}+a \right ) x^{2} c^{3}}{5}+\left (\frac {1}{5} b^{2} x^{2}+a^{2}+\frac {2}{3} a b x \right ) c^{2}+\left (-3 a \,b^{2}-\frac {1}{3} b^{3} x \right ) c +b^{4}\right ) e^{4}+\frac {8 d \left (\frac {5 c^{3} x^{3}}{84}-\frac {3 x \left (\frac {4 b x}{7}+a \right ) c^{2}}{20}+b \left (\frac {3 b x}{20}+a \right ) c -\frac {b^{3}}{2}\right ) c \,e^{3}}{3}-\frac {d^{2} \left (-\frac {c^{2} x^{2}}{21}+\left (\frac {b x}{7}+a \right ) c -b^{2}\right ) c^{2} e^{2}}{5}+\frac {2 d^{3} \left (-\frac {2 c x}{9}+b \right ) c^{3} e}{35}+\frac {8 d^{4} c^{4}}{315}\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^{5} e^{3}}\)	\(762\)
derivativedivides	\(\text {Expression too large to display}\)	\(1094\)
default	\(\text {Expression too large to display}\)	\(1094\)
risch	\(\text {Expression too large to display}\)	\(1128\)

Input:

int(x^4*(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)*(e^3*(3/2*((a*c-b^2)*e+d*b*c)*((a*b*c-1/3*b 
^3)*e-2/3*d*(a*c-1/2*b^2)*c)*(-4*e^2*(a*c-1/4*b^2))^(1/2)+e*((-1/2*b^6-9/2 
*a^2*b^2*c^2+3*a*b^4*c+a^3*c^3)*e^2+5*(a^2*c^2-c*a*b^2+1/5*b^4)*d*b*c*e-d^ 
2*(a^2*c^2-2*c*a*b^2+1/2*b^4)*c^2))*((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^3*(-3/2*((a*c-b^2)*e+d*b*c)*((a*b*c-1/3*b^3)*e-2/3*d*(a*c-1/2 
*b^2)*c)*(-4*e^2*(a*c-1/4*b^2))^(1/2)+e*((-1/2*b^6-9/2*a^2*b^2*c^2+3*a*b^4 
*c+a^3*c^3)*e^2+5*(a^2*c^2-c*a*b^2+1/5*b^4)*d*b*c*e-d^2*(a^2*c^2-2*c*a*b^2 
+1/2*b^4)*c^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)*((b*e-2*c*d+(-4*e^2*(a*c-1/ 
4*b^2))^(1/2))*c)^(1/2)*((1/9*c^4*x^4-1/5*(5/7*b*x+a)*x^2*c^3+(1/5*b^2*x^2 
+a^2+2/3*a*b*x)*c^2+(-3*a*b^2-1/3*b^3*x)*c+b^4)*e^4+8/3*d*(5/84*c^3*x^3-3/ 
20*x*(4/7*b*x+a)*c^2+b*(3/20*b*x+a)*c-1/2*b^3)*c*e^3-1/5*d^2*(-1/21*c^2*x^ 
2+(1/7*b*x+a)*c-b^2)*c^2*e^2+2/35*d^3*(-2/9*c*x+b)*c^3*e+8/315*d^4*c^4)*(- 
4*e^2*(a*c-1/4*b^2))^(1/2)))/(-4*e^2*(a*c-1/4*b^2))^(1/2)/c^5/e^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 14340 vs. \(2 (592) = 1184\).

Time = 12.64 (sec) , antiderivative size = 14340, normalized size of antiderivative = 22.06 \[ \int \frac {x^4 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

integrate(x^4*(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Timed out} \] Input:

integrate(x**4*(e*x+d)**(3/2)/(c*x**2+b*x+a),x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^4 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} x^{4}}{c x^{2} + b x + a} \,d x } \] Input:

integrate(x^4*(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="maxima")
 

Output:

integrate((e*x + d)^(3/2)*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. 1612 vs. \(2 (592) = 1184\).

Time = 0.39 (sec) , antiderivative size = 1612, normalized size of antiderivative = 2.48 \[ \int \frac {x^4 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

integrate(x^4*(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^5*c^2 - 6*a*b^3 
*c^3 + 8*a^2*b*c^4)*d^2 - 2*(b^6*c - 7*a*b^4*c^2 + 13*a^2*b^2*c^3 - 4*a^3* 
c^4)*d*e + (b^7 - 8*a*b^5*c + 19*a^2*b^3*c^2 - 12*a^3*b*c^3)*e^2)*c^2*e^2 
- 2*((b^3*c^4 - 2*a*b*c^5)*sqrt(b^2 - 4*a*c)*d^3 - (2*b^4*c^3 - 5*a*b^2*c^ 
4 + a^2*c^5)*sqrt(b^2 - 4*a*c)*d^2*e + (b^5*c^2 - 2*a*b^3*c^3 - a^2*b*c^4) 
*sqrt(b^2 - 4*a*c)*d*e^2 - (a*b^4*c^2 - 3*a^2*b^2*c^3 + a^3*c^4)*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^4*c^5 - 4*a*b^2*c^6 + 2*a^2*c^7)*d^3*e - (5*b^5*c^4 - 24*a*b^3* 
c^5 + 22*a^2*b*c^6)*d^2*e^2 + 2*(2*b^6*c^3 - 11*a*b^4*c^4 + 14*a^2*b^2*c^5 
 - 2*a^3*c^6)*d*e^3 - (b^7*c^2 - 6*a*b^5*c^3 + 9*a^2*b^3*c^4 - 2*a^3*b*c^5 
)*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^10*d*e^30 - b*c^9*e^31 + sqrt(-4*(c^10*d^2*e^30 
 - b*c^9*d*e^31 + a*c^9*e^32)*c^10*e^30 + (2*c^10*d*e^30 - b*c^9*e^31)^2)) 
/(c^10*e^30)))/((sqrt(b^2 - 4*a*c)*c^8*d^2 - sqrt(b^2 - 4*a*c)*b*c^7*d*e + 
 sqrt(b^2 - 4*a*c)*a*c^7*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^2 - 6*a*b^3*c^3 + 8*a^2*b*c^4)*d^2 - 2*(b^ 
6*c - 7*a*b^4*c^2 + 13*a^2*b^2*c^3 - 4*a^3*c^4)*d*e + (b^7 - 8*a*b^5*c + 1 
9*a^2*b^3*c^2 - 12*a^3*b*c^3)*e^2)*c^2*e^2 + 2*((b^3*c^4 - 2*a*b*c^5)*sqrt 
(b^2 - 4*a*c)*d^3 - (2*b^4*c^3 - 5*a*b^2*c^4 + a^2*c^5)*sqrt(b^2 - 4*a*c)* 
d^2*e + (b^5*c^2 - 2*a*b^3*c^3 - a^2*b*c^4)*sqrt(b^2 - 4*a*c)*d*e^2 - (...
 

Mupad [B] (verification not implemented)

Time = 15.29 (sec) , antiderivative size = 31485, normalized size of antiderivative = 48.44 \[ \int \frac {x^4 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(d + e*x)^(1/2)*((2*d^4)/(c*e^3) - ((a*e^5 + c*d^2*e^3 - b*d*e^4)*((12*d^2 
)/(c*e^3) - (2*(a*e^5 + c*d^2*e^3 - b*d*e^4))/(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))/(c*e^3)))/(c*e^3 
) + ((b*e^4 - 2*c*d*e^3)*((8*d^3)/(c*e^3) - (((8*d)/(c*e^3) + (2*(b*e^4 - 
2*c*d*e^3))/(c^2*e^6))*(a*e^5 + c*d^2*e^3 - b*d*e^4))/(c*e^3) + ((b*e^4 - 
2*c*d*e^3)*((12*d^2)/(c*e^3) - (2*(a*e^5 + c*d^2*e^3 - b*d*e^4))/(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 
))/(c*e^3)))/(c*e^3)))/(c*e^3)) - atan(((((8*(4*a^4*c^9*e^5 - a*b^6*c^6*e^ 
5 + b^7*c^6*d*e^4 + 7*a^2*b^4*c^7*e^5 - 13*a^3*b^2*c^8*e^5 + 4*a^3*c^10*d^ 
2*e^3 + b^5*c^8*d^3*e^2 - 2*b^6*c^7*d^2*e^3 - 21*a^2*b^2*c^9*d^2*e^3 - 6*a 
*b^5*c^7*d*e^4 + 4*a^3*b*c^9*d*e^4 - 6*a*b^3*c^9*d^3*e^2 + 13*a*b^4*c^8*d^ 
2*e^3 + 8*a^2*b*c^10*d^3*e^2 + 7*a^2*b^3*c^8*d*e^4))/c^9 - (8*(d + e*x)^(1 
/2)*(-(b^13*e^3 + 8*a^5*c^8*d^3 - b^10*c^3*d^3 - b^10*e^3*(-(4*a*c - b^2)^ 
3)^(1/2) + 12*a*b^8*c^4*d^3 + 44*a^6*b*c^6*e^3 - 24*a^6*c^7*d*e^2 + 3*b^11 
*c^2*d^2*e - 52*a^2*b^6*c^5*d^3 + 96*a^3*b^4*c^6*d^3 - 66*a^4*b^2*c^7*d^3 
+ 88*a^2*b^9*c^2*e^3 - 253*a^3*b^7*c^3*e^3 + 363*a^4*b^5*c^4*e^3 - 231*a^5 
*b^3*c^5*e^3 + a^5*c^5*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^7*c^3*d^3*(-(4*a*c 
 - b^2)^3)^(1/2) - 15*a*b^11*c*e^3 - 3*b^12*c*d*e^2 + 10*a^2*b^3*c^5*d^3*( 
-(4*a*c - b^2)^3)^(1/2) - 28*a^2*b^6*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 35 
*a^3*b^4*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - 15*a^4*b^2*c^4*e^3*(-(4*a*c...
 

Reduce [F]

\[ \int \frac {x^4 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\int \frac {x^{4} \left (e x +d \right )^{\frac {3}{2}}}{c \,x^{2}+b x +a}d x \] Input:

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

Output:

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