\(\int (d+e x)^2 \sqrt [4]{a+b x+c x^2} \, dx\) [723]

Optimal result

Integrand size = 22, antiderivative size = 213 \[ \int (d+e x)^2 \sqrt [4]{a+b x+c x^2} \, dx=\frac {\left (\left (9 b^2-8 a c\right ) e^2+28 c d (c d-b e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{84 c^3}+\frac {e (28 c d-9 b e+10 c e x) \left (a+b x+c x^2\right )^{5/4}}{35 c^2}+\frac {\left (-b^2+4 a c\right )^{3/2} \left (\left (9 b^2-8 a c\right ) e^2+28 c d (c d-b e)\right ) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right ),2\right )}{84 \sqrt {2} c^4 \left (a+b x+c x^2\right )^{3/4}} \] Output:

1/84*((-8*a*c+9*b^2)*e^2+28*c*d*(-b*e+c*d))*(2*c*x+b)*(c*x^2+b*x+a)^(1/4)/ 
c^3+1/35*e*(10*c*e*x-9*b*e+28*c*d)*(c*x^2+b*x+a)^(5/4)/c^2+1/168*(4*a*c-b^ 
2)^(3/2)*((-8*a*c+9*b^2)*e^2+28*c*d*(-b*e+c*d))*(-c*(c*x^2+b*x+a)/(-4*a*c+ 
b^2))^(3/4)*InverseJacobiAM(1/2*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2)),2^(1/2 
))*2^(1/2)/c^4/(c*x^2+b*x+a)^(3/4)
 

Mathematica [A] (warning: unable to verify)

Time = 9.50 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.90 \[ \int (d+e x)^2 \sqrt [4]{a+b x+c x^2} \, dx=\frac {-54 e (-2 c d+b e) (a+x (b+c x))^2+60 c e (d+e x) (a+x (b+c x))^2+\frac {5 \left (28 c^2 d^2+9 b^2 e^2-4 c e (7 b d+2 a e)\right ) \left (2 c (b+2 c x) (a+x (b+c x))-\sqrt {2} \left (b^2-4 a c\right )^{3/2} \left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ),2\right )\right )}{4 c^2}}{210 c^2 (a+x (b+c x))^{3/4}} \] Input:

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

Output:

(-54*e*(-2*c*d + b*e)*(a + x*(b + c*x))^2 + 60*c*e*(d + e*x)*(a + x*(b + c 
*x))^2 + (5*(28*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(7*b*d + 2*a*e))*(2*c*(b + 2*c 
*x)*(a + x*(b + c*x)) - Sqrt[2]*(b^2 - 4*a*c)^(3/2)*((c*(a + x*(b + c*x))) 
/(-b^2 + 4*a*c))^(3/4)*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 
2]))/(4*c^2))/(210*c^2*(a + x*(b + c*x))^(3/4))
 

Rubi [A] (warning: unable to verify)

Time = 0.44 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.67, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1166, 27, 1160, 1087, 1094, 761}

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.

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

\(\Big \downarrow \) 1166

\(\displaystyle \frac {2 \int \frac {1}{4} \left (14 c d^2-5 b e d-4 a e^2+9 e (2 c d-b e) x\right ) \sqrt [4]{c x^2+b x+a}dx}{7 c}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{5/4}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \left (14 c d^2-5 b e d-4 a e^2+9 e (2 c d-b e) x\right ) \sqrt [4]{c x^2+b x+a}dx}{14 c}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{5/4}}{7 c}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {\left (-4 c e (2 a e+7 b d)+9 b^2 e^2+28 c^2 d^2\right ) \int \sqrt [4]{c x^2+b x+a}dx}{2 c}+\frac {18 e \left (a+b x+c x^2\right )^{5/4} (2 c d-b e)}{5 c}}{14 c}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{5/4}}{7 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {\left (-4 c e (2 a e+7 b d)+9 b^2 e^2+28 c^2 d^2\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{3/4}}dx}{12 c}\right )}{2 c}+\frac {18 e \left (a+b x+c x^2\right )^{5/4} (2 c d-b e)}{5 c}}{14 c}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{5/4}}{7 c}\)

\(\Big \downarrow \) 1094

\(\displaystyle \frac {\frac {\left (-4 c e (2 a e+7 b d)+9 b^2 e^2+28 c^2 d^2\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{3 c (b+2 c x)}\right )}{2 c}+\frac {18 e \left (a+b x+c x^2\right )^{5/4} (2 c d-b e)}{5 c}}{14 c}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{5/4}}{7 c}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {\frac {\left (-4 c e (2 a e+7 b d)+9 b^2 e^2+28 c^2 d^2\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right )^{5/4} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{6 \sqrt {2} c^{5/4} (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}\right )}{2 c}+\frac {18 e \left (a+b x+c x^2\right )^{5/4} (2 c d-b e)}{5 c}}{14 c}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{5/4}}{7 c}\)

Input:

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

Output:

(2*e*(d + e*x)*(a + b*x + c*x^2)^(5/4))/(7*c) + ((18*e*(2*c*d - b*e)*(a + 
b*x + c*x^2)^(5/4))/(5*c) + ((28*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(7*b*d + 2*a* 
e))*(((b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(3*c) - ((b^2 - 4*a*c)^(5/4)*Sq 
rt[(b + 2*c*x)^2]*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c] 
)*Sqrt[(b^2 - 4*a*c + 4*c*(a + b*x + c*x^2))/((b^2 - 4*a*c)*(1 + (2*Sqrt[c 
]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]*EllipticF[2*ArcTan[(Sqrt[2 
]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(6*Sqrt[2]* 
c^(5/4)*(b + 2*c*x)*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2)])))/(2*c))/(1 
4*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 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1094

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

rule 1160

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

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]
 

Maple [F]

\[\int \left (e x +d \right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int (d+e x)^2 \sqrt [4]{a+b x+c x^2} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {1}{4}} {\left (e x + d\right )}^{2} \,d x } \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

\[ \int (d+e x)^2 \sqrt [4]{a+b x+c x^2} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {1}{4}} {\left (e x + d\right )}^{2} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int (d+e x)^2 \sqrt [4]{a+b x+c x^2} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {1}{4}} {\left (e x + d\right )}^{2} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^2 \sqrt [4]{a+b x+c x^2} \, dx=\int {\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{1/4} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (d+e x)^2 \sqrt [4]{a+b x+c x^2} \, dx=\frac {-320 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} a^{2} c \,e^{2}+144 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} a \,b^{2} e^{2}-448 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} a b c d e +80 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} a b c \,e^{2} x +1120 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} a \,c^{2} d^{2}-36 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b^{3} e^{2} x +112 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b^{2} c d e x +24 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b^{2} c \,e^{2} x^{2}+560 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b \,c^{2} d^{2} x +672 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b \,c^{2} d e \,x^{2}+240 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b \,c^{2} e^{2} x^{3}+160 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x \right ) a^{2} c^{2} e^{2}-220 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x \right ) a \,b^{2} c \,e^{2}+560 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x \right ) a b \,c^{2} d e -560 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x \right ) a \,c^{3} d^{2}+45 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x \right ) b^{4} e^{2}-140 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x \right ) b^{3} c d e +140 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x \right ) b^{2} c^{2} d^{2}}{840 b \,c^{2}} \] Input:

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

Output:

( - 320*(a + b*x + c*x**2)**(1/4)*a**2*c*e**2 + 144*(a + b*x + c*x**2)**(1 
/4)*a*b**2*e**2 - 448*(a + b*x + c*x**2)**(1/4)*a*b*c*d*e + 80*(a + b*x + 
c*x**2)**(1/4)*a*b*c*e**2*x + 1120*(a + b*x + c*x**2)**(1/4)*a*c**2*d**2 - 
 36*(a + b*x + c*x**2)**(1/4)*b**3*e**2*x + 112*(a + b*x + c*x**2)**(1/4)* 
b**2*c*d*e*x + 24*(a + b*x + c*x**2)**(1/4)*b**2*c*e**2*x**2 + 560*(a + b* 
x + c*x**2)**(1/4)*b*c**2*d**2*x + 672*(a + b*x + c*x**2)**(1/4)*b*c**2*d* 
e*x**2 + 240*(a + b*x + c*x**2)**(1/4)*b*c**2*e**2*x**3 + 160*int(((a + b* 
x + c*x**2)**(1/4)*x)/(a + b*x + c*x**2),x)*a**2*c**2*e**2 - 220*int(((a + 
 b*x + c*x**2)**(1/4)*x)/(a + b*x + c*x**2),x)*a*b**2*c*e**2 + 560*int(((a 
 + b*x + c*x**2)**(1/4)*x)/(a + b*x + c*x**2),x)*a*b*c**2*d*e - 560*int((( 
a + b*x + c*x**2)**(1/4)*x)/(a + b*x + c*x**2),x)*a*c**3*d**2 + 45*int(((a 
 + b*x + c*x**2)**(1/4)*x)/(a + b*x + c*x**2),x)*b**4*e**2 - 140*int(((a + 
 b*x + c*x**2)**(1/4)*x)/(a + b*x + c*x**2),x)*b**3*c*d*e + 140*int(((a + 
b*x + c*x**2)**(1/4)*x)/(a + b*x + c*x**2),x)*b**2*c**2*d**2)/(840*b*c**2)