\(\int \frac {(d+e x)^4}{\sqrt {a+b x+c x^2}} \, dx\) [608]

Optimal result

Integrand size = 22, antiderivative size = 296 \[ \int \frac {(d+e x)^4}{\sqrt {a+b x+c x^2}} \, dx=\frac {7 e (2 c d-b e) (d+e x)^2 \sqrt {a+b x+c x^2}}{24 c^2}+\frac {e (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}+\frac {e \left (608 c^3 d^3-105 b^3 e^3+20 b c e^2 (24 b d+11 a e)-8 c^2 d e (101 b d+64 a e)+2 c e \left (104 c^2 d^2+35 b^2 e^2-4 c e (26 b d+9 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{192 c^4}+\frac {\left (128 c^4 d^4+35 b^4 e^4-128 c^3 d^2 e (2 b d+3 a e)-40 b^2 c e^3 (4 b d+3 a e)+48 c^2 e^2 \left (6 b^2 d^2+8 a b d e+a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{9/2}} \] Output:

7/24*e*(-b*e+2*c*d)*(e*x+d)^2*(c*x^2+b*x+a)^(1/2)/c^2+1/4*e*(e*x+d)^3*(c*x 
^2+b*x+a)^(1/2)/c+1/192*e*(608*c^3*d^3-105*b^3*e^3+20*b*c*e^2*(11*a*e+24*b 
*d)-8*c^2*d*e*(64*a*e+101*b*d)+2*c*e*(104*c^2*d^2+35*b^2*e^2-4*c*e*(9*a*e+ 
26*b*d))*x)*(c*x^2+b*x+a)^(1/2)/c^4+1/128*(128*c^4*d^4+35*b^4*e^4-128*c^3* 
d^2*e*(3*a*e+2*b*d)-40*b^2*c*e^3*(3*a*e+4*b*d)+48*c^2*e^2*(a^2*e^2+8*a*b*d 
*e+6*b^2*d^2))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(9/2)
 

Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.83 \[ \int \frac {(d+e x)^4}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {c} e \sqrt {a+x (b+c x)} \left (-105 b^3 e^3+10 b c e^2 (48 b d+22 a e+7 b e x)+16 c^3 \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )-8 c^2 e \left (a e (64 d+9 e x)+b \left (108 d^2+40 d e x+7 e^2 x^2\right )\right )\right )-3 \left (128 c^4 d^4+35 b^4 e^4-128 c^3 d^2 e (2 b d+3 a e)-40 b^2 c e^3 (4 b d+3 a e)+48 c^2 e^2 \left (6 b^2 d^2+8 a b d e+a^2 e^2\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{384 c^{9/2}} \] Input:

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

Output:

(2*Sqrt[c]*e*Sqrt[a + x*(b + c*x)]*(-105*b^3*e^3 + 10*b*c*e^2*(48*b*d + 22 
*a*e + 7*b*e*x) + 16*c^3*(48*d^3 + 36*d^2*e*x + 16*d*e^2*x^2 + 3*e^3*x^3) 
- 8*c^2*e*(a*e*(64*d + 9*e*x) + b*(108*d^2 + 40*d*e*x + 7*e^2*x^2))) - 3*( 
128*c^4*d^4 + 35*b^4*e^4 - 128*c^3*d^2*e*(2*b*d + 3*a*e) - 40*b^2*c*e^3*(4 
*b*d + 3*a*e) + 48*c^2*e^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2))*Log[b + 2*c* 
x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(384*c^(9/2))
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1166, 27, 1236, 27, 1225, 1092, 219}

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.

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

\(\Big \downarrow \) 1166

\(\displaystyle \frac {\int \frac {(d+e x)^2 \left (8 c d^2-e (b d+6 a e)+7 e (2 c d-b e) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{4 c}+\frac {e (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(d+e x)^2 \left (8 c d^2-e (b d+6 a e)+7 e (2 c d-b e) x\right )}{\sqrt {c x^2+b x+a}}dx}{8 c}+\frac {e (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\frac {\int \frac {(d+e x) \left (48 c^2 d^3-4 c e (5 b d+23 a e) d+7 b e^2 (b d+4 a e)+e \left (104 c^2 d^2+35 b^2 e^2-4 c e (26 b d+9 a e)\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {7 e (d+e x)^2 \sqrt {a+b x+c x^2} (2 c d-b e)}{3 c}}{8 c}+\frac {e (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {(d+e x) \left (48 c^2 d^3-4 c e (5 b d+23 a e) d+7 b e^2 (b d+4 a e)+e \left (104 c^2 d^2+35 b^2 e^2-4 c e (26 b d+9 a e)\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{6 c}+\frac {7 e (d+e x)^2 \sqrt {a+b x+c x^2} (2 c d-b e)}{3 c}}{8 c}+\frac {e (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\frac {\frac {3 \left (48 c^2 e^2 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )-40 b^2 c e^3 (3 a e+4 b d)-128 c^3 d^2 e (3 a e+2 b d)+35 b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c^2}+\frac {e \sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (9 a e+26 b d)+35 b^2 e^2+104 c^2 d^2\right )-8 c^2 d e (64 a e+101 b d)+20 b c e^2 (11 a e+24 b d)-105 b^3 e^3+608 c^3 d^3\right )}{4 c^2}}{6 c}+\frac {7 e (d+e x)^2 \sqrt {a+b x+c x^2} (2 c d-b e)}{3 c}}{8 c}+\frac {e (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {3 \left (48 c^2 e^2 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )-40 b^2 c e^3 (3 a e+4 b d)-128 c^3 d^2 e (3 a e+2 b d)+35 b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c^2}+\frac {e \sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (9 a e+26 b d)+35 b^2 e^2+104 c^2 d^2\right )-8 c^2 d e (64 a e+101 b d)+20 b c e^2 (11 a e+24 b d)-105 b^3 e^3+608 c^3 d^3\right )}{4 c^2}}{6 c}+\frac {7 e (d+e x)^2 \sqrt {a+b x+c x^2} (2 c d-b e)}{3 c}}{8 c}+\frac {e (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {3 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (48 c^2 e^2 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )-40 b^2 c e^3 (3 a e+4 b d)-128 c^3 d^2 e (3 a e+2 b d)+35 b^4 e^4+128 c^4 d^4\right )}{8 c^{5/2}}+\frac {e \sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (9 a e+26 b d)+35 b^2 e^2+104 c^2 d^2\right )-8 c^2 d e (64 a e+101 b d)+20 b c e^2 (11 a e+24 b d)-105 b^3 e^3+608 c^3 d^3\right )}{4 c^2}}{6 c}+\frac {7 e (d+e x)^2 \sqrt {a+b x+c x^2} (2 c d-b e)}{3 c}}{8 c}+\frac {e (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}\)

Input:

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

Output:

(e*(d + e*x)^3*Sqrt[a + b*x + c*x^2])/(4*c) + ((7*e*(2*c*d - b*e)*(d + e*x 
)^2*Sqrt[a + b*x + c*x^2])/(3*c) + ((e*(608*c^3*d^3 - 105*b^3*e^3 + 20*b*c 
*e^2*(24*b*d + 11*a*e) - 8*c^2*d*e*(101*b*d + 64*a*e) + 2*c*e*(104*c^2*d^2 
 + 35*b^2*e^2 - 4*c*e*(26*b*d + 9*a*e))*x)*Sqrt[a + b*x + c*x^2])/(4*c^2) 
+ (3*(128*c^4*d^4 + 35*b^4*e^4 - 128*c^3*d^2*e*(2*b*d + 3*a*e) - 40*b^2*c* 
e^3*(4*b*d + 3*a*e) + 48*c^2*e^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2))*ArcTan 
h[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(5/2)))/(6*c))/(8*c 
)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1166

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, x]
 

rule 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

rule 1236

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 
1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1 
)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m 
*(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])
 

Maple [A] (verified)

Time = 1.18 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.93

method	result	size
risch	\(\frac {e \left (48 c^{3} x^{3} e^{3}-56 b \,c^{2} e^{3} x^{2}+256 c^{3} d \,e^{2} x^{2}-72 a \,c^{2} e^{3} x +70 x \,b^{2} c \,e^{3}-320 b \,c^{2} d \,e^{2} x +576 d^{2} e \,c^{3} x +220 a b c \,e^{3}-512 d \,e^{2} a \,c^{2}-105 b^{3} e^{3}+480 d \,e^{2} b^{2} c -864 d^{2} e b \,c^{2}+768 d^{3} c^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{192 c^{4}}+\frac {\left (48 e^{4} a^{2} c^{2}-120 a \,b^{2} c \,e^{4}+384 a b \,c^{2} d \,e^{3}-384 d^{2} e^{2} a \,c^{3}+35 b^{4} e^{4}-160 d \,e^{3} b^{3} c +288 d^{2} e^{2} b^{2} c^{2}-256 d^{3} e b \,c^{3}+128 d^{4} c^{4}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {9}{2}}}\)	\(276\)
default	\(\frac {d^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+e^{4} \left (\frac {x^{3} \sqrt {c \,x^{2}+b x +a}}{4 c}-\frac {7 b \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )}{8 c}-\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )+4 d \,e^{3} \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )+6 d^{2} e^{2} \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )+4 d^{3} e \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )\)	\(717\)

Input:

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

Output:

1/192*e*(48*c^3*e^3*x^3-56*b*c^2*e^3*x^2+256*c^3*d*e^2*x^2-72*a*c^2*e^3*x+ 
70*b^2*c*e^3*x-320*b*c^2*d*e^2*x+576*c^3*d^2*e*x+220*a*b*c*e^3-512*a*c^2*d 
*e^2-105*b^3*e^3+480*b^2*c*d*e^2-864*b*c^2*d^2*e+768*c^3*d^3)/c^4*(c*x^2+b 
*x+a)^(1/2)+1/128*(48*a^2*c^2*e^4-120*a*b^2*c*e^4+384*a*b*c^2*d*e^3-384*a* 
c^3*d^2*e^2+35*b^4*e^4-160*b^3*c*d*e^3+288*b^2*c^2*d^2*e^2-256*b*c^3*d^3*e 
+128*c^4*d^4)/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 599, normalized size of antiderivative = 2.02 \[ \int \frac {(d+e x)^4}{\sqrt {a+b x+c x^2}} \, dx=\left [\frac {3 \, {\left (128 \, c^{4} d^{4} - 256 \, b c^{3} d^{3} e + 96 \, {\left (3 \, b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e^{2} - 32 \, {\left (5 \, b^{3} c - 12 \, a b c^{2}\right )} d e^{3} + {\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} e^{4}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (48 \, c^{4} e^{4} x^{3} + 768 \, c^{4} d^{3} e - 864 \, b c^{3} d^{2} e^{2} + 32 \, {\left (15 \, b^{2} c^{2} - 16 \, a c^{3}\right )} d e^{3} - 5 \, {\left (21 \, b^{3} c - 44 \, a b c^{2}\right )} e^{4} + 8 \, {\left (32 \, c^{4} d e^{3} - 7 \, b c^{3} e^{4}\right )} x^{2} + 2 \, {\left (288 \, c^{4} d^{2} e^{2} - 160 \, b c^{3} d e^{3} + {\left (35 \, b^{2} c^{2} - 36 \, a c^{3}\right )} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, c^{5}}, -\frac {3 \, {\left (128 \, c^{4} d^{4} - 256 \, b c^{3} d^{3} e + 96 \, {\left (3 \, b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e^{2} - 32 \, {\left (5 \, b^{3} c - 12 \, a b c^{2}\right )} d e^{3} + {\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (48 \, c^{4} e^{4} x^{3} + 768 \, c^{4} d^{3} e - 864 \, b c^{3} d^{2} e^{2} + 32 \, {\left (15 \, b^{2} c^{2} - 16 \, a c^{3}\right )} d e^{3} - 5 \, {\left (21 \, b^{3} c - 44 \, a b c^{2}\right )} e^{4} + 8 \, {\left (32 \, c^{4} d e^{3} - 7 \, b c^{3} e^{4}\right )} x^{2} + 2 \, {\left (288 \, c^{4} d^{2} e^{2} - 160 \, b c^{3} d e^{3} + {\left (35 \, b^{2} c^{2} - 36 \, a c^{3}\right )} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, c^{5}}\right ] \] Input:

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

Output:

[1/768*(3*(128*c^4*d^4 - 256*b*c^3*d^3*e + 96*(3*b^2*c^2 - 4*a*c^3)*d^2*e^ 
2 - 32*(5*b^3*c - 12*a*b*c^2)*d*e^3 + (35*b^4 - 120*a*b^2*c + 48*a^2*c^2)* 
e^4)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c 
*x + b)*sqrt(c) - 4*a*c) + 4*(48*c^4*e^4*x^3 + 768*c^4*d^3*e - 864*b*c^3*d 
^2*e^2 + 32*(15*b^2*c^2 - 16*a*c^3)*d*e^3 - 5*(21*b^3*c - 44*a*b*c^2)*e^4 
+ 8*(32*c^4*d*e^3 - 7*b*c^3*e^4)*x^2 + 2*(288*c^4*d^2*e^2 - 160*b*c^3*d*e^ 
3 + (35*b^2*c^2 - 36*a*c^3)*e^4)*x)*sqrt(c*x^2 + b*x + a))/c^5, -1/384*(3* 
(128*c^4*d^4 - 256*b*c^3*d^3*e + 96*(3*b^2*c^2 - 4*a*c^3)*d^2*e^2 - 32*(5* 
b^3*c - 12*a*b*c^2)*d*e^3 + (35*b^4 - 120*a*b^2*c + 48*a^2*c^2)*e^4)*sqrt( 
-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x 
 + a*c)) - 2*(48*c^4*e^4*x^3 + 768*c^4*d^3*e - 864*b*c^3*d^2*e^2 + 32*(15* 
b^2*c^2 - 16*a*c^3)*d*e^3 - 5*(21*b^3*c - 44*a*b*c^2)*e^4 + 8*(32*c^4*d*e^ 
3 - 7*b*c^3*e^4)*x^2 + 2*(288*c^4*d^2*e^2 - 160*b*c^3*d*e^3 + (35*b^2*c^2 
- 36*a*c^3)*e^4)*x)*sqrt(c*x^2 + b*x + a))/c^5]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (299) = 598\).

Time = 1.02 (sec) , antiderivative size = 622, normalized size of antiderivative = 2.10 \[ \int \frac {(d+e x)^4}{\sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(a + b*x + c*x**2)*(e**4*x**3/(4*c) + x**2*(-7*b*e**4/(8*c) 
 + 4*d*e**3)/(3*c) + x*(-3*a*e**4/(4*c) - 5*b*(-7*b*e**4/(8*c) + 4*d*e**3) 
/(6*c) + 6*d**2*e**2)/(2*c) + (-2*a*(-7*b*e**4/(8*c) + 4*d*e**3)/(3*c) - 3 
*b*(-3*a*e**4/(4*c) - 5*b*(-7*b*e**4/(8*c) + 4*d*e**3)/(6*c) + 6*d**2*e**2 
)/(4*c) + 4*d**3*e)/c) + (-a*(-3*a*e**4/(4*c) - 5*b*(-7*b*e**4/(8*c) + 4*d 
*e**3)/(6*c) + 6*d**2*e**2)/(2*c) - b*(-2*a*(-7*b*e**4/(8*c) + 4*d*e**3)/( 
3*c) - 3*b*(-3*a*e**4/(4*c) - 5*b*(-7*b*e**4/(8*c) + 4*d*e**3)/(6*c) + 6*d 
**2*e**2)/(4*c) + 4*d**3*e)/(2*c) + d**4)*Piecewise((log(b + 2*sqrt(c)*sqr 
t(a + b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(a - b**2/(4*c), 0)), ((b/(2*c) + 
x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), True)), Ne(c, 0)), (2*(e**4* 
(a + b*x)**(9/2)/(9*b**4) + (a + b*x)**(7/2)*(-4*a*e**4 + 4*b*d*e**3)/(7*b 
**4) + (a + b*x)**(5/2)*(6*a**2*e**4 - 12*a*b*d*e**3 + 6*b**2*d**2*e**2)/( 
5*b**4) + (a + b*x)**(3/2)*(-4*a**3*e**4 + 12*a**2*b*d*e**3 - 12*a*b**2*d* 
*2*e**2 + 4*b**3*d**3*e)/(3*b**4) + sqrt(a + b*x)*(a**4*e**4 - 4*a**3*b*d* 
e**3 + 6*a**2*b**2*d**2*e**2 - 4*a*b**3*d**3*e + b**4*d**4)/b**4)/b, Ne(b, 
 0)), (Piecewise((d**4*x, Eq(e, 0)), ((d + e*x)**5/(5*e), True))/sqrt(a), 
True))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^4}{\sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^4}{\sqrt {a+b x+c x^2}} \, dx=\frac {1}{192} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (\frac {6 \, e^{4} x}{c} + \frac {32 \, c^{3} d e^{3} - 7 \, b c^{2} e^{4}}{c^{4}}\right )} x + \frac {288 \, c^{3} d^{2} e^{2} - 160 \, b c^{2} d e^{3} + 35 \, b^{2} c e^{4} - 36 \, a c^{2} e^{4}}{c^{4}}\right )} x + \frac {768 \, c^{3} d^{3} e - 864 \, b c^{2} d^{2} e^{2} + 480 \, b^{2} c d e^{3} - 512 \, a c^{2} d e^{3} - 105 \, b^{3} e^{4} + 220 \, a b c e^{4}}{c^{4}}\right )} - \frac {{\left (128 \, c^{4} d^{4} - 256 \, b c^{3} d^{3} e + 288 \, b^{2} c^{2} d^{2} e^{2} - 384 \, a c^{3} d^{2} e^{2} - 160 \, b^{3} c d e^{3} + 384 \, a b c^{2} d e^{3} + 35 \, b^{4} e^{4} - 120 \, a b^{2} c e^{4} + 48 \, a^{2} c^{2} e^{4}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{128 \, c^{\frac {9}{2}}} \] Input:

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

Output:

1/192*sqrt(c*x^2 + b*x + a)*(2*(4*(6*e^4*x/c + (32*c^3*d*e^3 - 7*b*c^2*e^4 
)/c^4)*x + (288*c^3*d^2*e^2 - 160*b*c^2*d*e^3 + 35*b^2*c*e^4 - 36*a*c^2*e^ 
4)/c^4)*x + (768*c^3*d^3*e - 864*b*c^2*d^2*e^2 + 480*b^2*c*d*e^3 - 512*a*c 
^2*d*e^3 - 105*b^3*e^4 + 220*a*b*c*e^4)/c^4) - 1/128*(128*c^4*d^4 - 256*b* 
c^3*d^3*e + 288*b^2*c^2*d^2*e^2 - 384*a*c^3*d^2*e^2 - 160*b^3*c*d*e^3 + 38 
4*a*b*c^2*d*e^3 + 35*b^4*e^4 - 120*a*b^2*c*e^4 + 48*a^2*c^2*e^4)*log(abs(2 
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(9/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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