\(\int (d+e x)^2 (f+g x) \sqrt [4]{a+b x+c x^2} \, dx\) [1097]

Optimal result

Integrand size = 27, antiderivative size = 380 \[ \int (d+e x)^2 (f+g x) \sqrt [4]{a+b x+c x^2} \, dx=\frac {\left (56 c^3 d^2 f-13 b^3 e^2 g+6 b c e (3 b e f+6 b d g+4 a e g)-4 c^2 (7 b d (2 e f+d g)+4 a e (e f+2 d g))\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{168 c^4}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{5/4}}{9 c}+\frac {\left (117 b^2 e^2 g+56 c^2 d (9 e f+2 d g)-2 c e (56 a e g+81 b (e f+2 d g))+10 c e (18 c e f+8 c d g-13 b e g) x\right ) \left (a+b x+c x^2\right )^{5/4}}{630 c^3}+\frac {\left (-b^2+4 a c\right )^{3/2} \left (56 c^3 d^2 f-13 b^3 e^2 g+6 b c e (3 b e f+6 b d g+4 a e g)-4 c^2 (7 b d (2 e f+d g)+4 a e (e f+2 d g))\right ) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right ),2\right )}{168 \sqrt {2} c^5 \left (a+b x+c x^2\right )^{3/4}} \] Output:

1/168*(56*c^3*d^2*f-13*b^3*e^2*g+6*b*c*e*(4*a*e*g+6*b*d*g+3*b*e*f)-4*c^2*( 
7*b*d*(d*g+2*e*f)+4*a*e*(2*d*g+e*f)))*(2*c*x+b)*(c*x^2+b*x+a)^(1/4)/c^4+2/ 
9*g*(e*x+d)^2*(c*x^2+b*x+a)^(5/4)/c+1/630*(117*b^2*e^2*g+56*c^2*d*(2*d*g+9 
*e*f)-2*c*e*(56*a*e*g+81*b*(2*d*g+e*f))+10*c*e*(-13*b*e*g+8*c*d*g+18*c*e*f 
)*x)*(c*x^2+b*x+a)^(5/4)/c^3+1/336*(4*a*c-b^2)^(3/2)*(56*c^3*d^2*f-13*b^3* 
e^2*g+6*b*c*e*(4*a*e*g+6*b*d*g+3*b*e*f)-4*c^2*(7*b*d*(d*g+2*e*f)+4*a*e*(2* 
d*g+e*f)))*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(3/4)*InverseJacobiAM(1/2*arcta 
n((2*c*x+b)/(4*a*c-b^2)^(1/2)),2^(1/2))*2^(1/2)/c^5/(c*x^2+b*x+a)^(3/4)
 

Mathematica [A] (warning: unable to verify)

Time = 11.26 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.77 \[ \int (d+e x)^2 (f+g x) \sqrt [4]{a+b x+c x^2} \, dx=\frac {1120 c^4 g (d+e x)^2 (a+x (b+c x))^2+8 c^2 (a+x (b+c x))^2 \left (117 b^2 e^2 g+4 c^2 \left (28 d^2 g+45 e^2 f x+2 d e (63 f+10 g x)\right )-2 c e (56 a e g+b (81 e f+162 d g+65 e g x))\right )+15 \left (56 c^3 d^2 f-13 b^3 e^2 g+6 b c e (3 b e f+6 b d g+4 a e g)-4 c^2 (7 b d (2 e f+d g)+4 a e (e f+2 d g))\right ) \left (2 c (b+2 c x) (a+x (b+c x))-\sqrt {2} \left (b^2-4 a c\right )^{3/2} \left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ),2\right )\right )}{5040 c^5 (a+x (b+c x))^{3/4}} \] Input:

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

Output:

(1120*c^4*g*(d + e*x)^2*(a + x*(b + c*x))^2 + 8*c^2*(a + x*(b + c*x))^2*(1 
17*b^2*e^2*g + 4*c^2*(28*d^2*g + 45*e^2*f*x + 2*d*e*(63*f + 10*g*x)) - 2*c 
*e*(56*a*e*g + b*(81*e*f + 162*d*g + 65*e*g*x))) + 15*(56*c^3*d^2*f - 13*b 
^3*e^2*g + 6*b*c*e*(3*b*e*f + 6*b*d*g + 4*a*e*g) - 4*c^2*(7*b*d*(2*e*f + d 
*g) + 4*a*e*(e*f + 2*d*g)))*(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]))/(5040*c^5*(a + x*(b + c*x))^ 
(3/4))
 

Rubi [A] (warning: unable to verify)

Time = 0.99 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1236, 27, 1225, 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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2 \int \frac {1}{4} (d+e x) (18 c d f-5 b d g-8 a e g+(18 c e f+8 c d g-13 b e g) x) \sqrt [4]{c x^2+b x+a}dx}{9 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{5/4}}{9 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (d+e x) (18 c d f-5 b d g-8 a e g+(18 c e f+8 c d g-13 b e g) x) \sqrt [4]{c x^2+b x+a}dx}{18 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{5/4}}{9 c}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\frac {9 \left (-4 c^2 (4 a e (2 d g+e f)+7 b d (d g+2 e f))+6 b c e (4 a e g+6 b d g+3 b e f)-13 b^3 e^2 g+56 c^3 d^2 f\right ) \int \sqrt [4]{c x^2+b x+a}dx}{28 c^2}+\frac {\left (a+b x+c x^2\right )^{5/4} \left (-2 c e (56 a e g+81 b (2 d g+e f))+117 b^2 e^2 g+10 c e x (-13 b e g+8 c d g+18 c e f)+56 c^2 d (2 d g+9 e f)\right )}{35 c^2}}{18 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{5/4}}{9 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {9 \left (-4 c^2 (4 a e (2 d g+e f)+7 b d (d g+2 e f))+6 b c e (4 a e g+6 b d g+3 b e f)-13 b^3 e^2 g+56 c^3 d^2 f\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{3/4}}dx}{12 c}\right )}{28 c^2}+\frac {\left (a+b x+c x^2\right )^{5/4} \left (-2 c e (56 a e g+81 b (2 d g+e f))+117 b^2 e^2 g+10 c e x (-13 b e g+8 c d g+18 c e f)+56 c^2 d (2 d g+9 e f)\right )}{35 c^2}}{18 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{5/4}}{9 c}\)

\(\Big \downarrow \) 1094

\(\displaystyle \frac {\frac {9 \left (-4 c^2 (4 a e (2 d g+e f)+7 b d (d g+2 e f))+6 b c e (4 a e g+6 b d g+3 b e f)-13 b^3 e^2 g+56 c^3 d^2 f\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{3 c (b+2 c x)}\right )}{28 c^2}+\frac {\left (a+b x+c x^2\right )^{5/4} \left (-2 c e (56 a e g+81 b (2 d g+e f))+117 b^2 e^2 g+10 c e x (-13 b e g+8 c d g+18 c e f)+56 c^2 d (2 d g+9 e f)\right )}{35 c^2}}{18 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{5/4}}{9 c}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {\frac {9 \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right )^{5/4} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{6 \sqrt {2} c^{5/4} (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}\right ) \left (-4 c^2 (4 a e (2 d g+e f)+7 b d (d g+2 e f))+6 b c e (4 a e g+6 b d g+3 b e f)-13 b^3 e^2 g+56 c^3 d^2 f\right )}{28 c^2}+\frac {\left (a+b x+c x^2\right )^{5/4} \left (-2 c e (56 a e g+81 b (2 d g+e f))+117 b^2 e^2 g+10 c e x (-13 b e g+8 c d g+18 c e f)+56 c^2 d (2 d g+9 e f)\right )}{35 c^2}}{18 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{5/4}}{9 c}\)

Input:

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

Output:

(2*g*(d + e*x)^2*(a + b*x + c*x^2)^(5/4))/(9*c) + (((117*b^2*e^2*g + 56*c^ 
2*d*(9*e*f + 2*d*g) - 2*c*e*(56*a*e*g + 81*b*(e*f + 2*d*g)) + 10*c*e*(18*c 
*e*f + 8*c*d*g - 13*b*e*g)*x)*(a + b*x + c*x^2)^(5/4))/(35*c^2) + (9*(56*c 
^3*d^2*f - 13*b^3*e^2*g + 6*b*c*e*(3*b*e*f + 6*b*d*g + 4*a*e*g) - 4*c^2*(7 
*b*d*(2*e*f + d*g) + 4*a*e*(e*f + 2*d*g)))*(((b + 2*c*x)*(a + b*x + c*x^2) 
^(1/4))/(3*c) - ((b^2 - 4*a*c)^(5/4)*Sqrt[(b + 2*c*x)^2]*(1 + (2*Sqrt[c]*S 
qrt[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)])))/(28*c^2))/(18*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 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 [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((e^2*g*x^3 + d^2*f + (e^2*f + 2*d*e*g)*x^2 + (2*d*e*f + d^2*g)*x) 
*(c*x^2 + b*x + a)^(1/4), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((e*x+d)^2*(g*x+f)*(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*(g*x + f), x)
 

Giac [F]

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

integrate((e*x+d)^2*(g*x+f)*(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*(g*x + f), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+e x)^2 (f+g x) \sqrt [4]{a+b x+c x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(1984*(a + b*x + c*x**2)**(1/4)*a**2*b*c*e**2*g - 3840*(a + b*x + c*x**2)* 
*(1/4)*a**2*c**2*d*e*g - 1920*(a + b*x + c*x**2)**(1/4)*a**2*c**2*e**2*f - 
 624*(a + b*x + c*x**2)**(1/4)*a*b**3*e**2*g + 1728*(a + b*x + c*x**2)**(1 
/4)*a*b**2*c*d*e*g + 864*(a + b*x + c*x**2)**(1/4)*a*b**2*c*e**2*f - 496*( 
a + b*x + c*x**2)**(1/4)*a*b**2*c*e**2*g*x - 1344*(a + b*x + c*x**2)**(1/4 
)*a*b*c**2*d**2*g - 2688*(a + b*x + c*x**2)**(1/4)*a*b*c**2*d*e*f + 960*(a 
 + b*x + c*x**2)**(1/4)*a*b*c**2*d*e*g*x + 480*(a + b*x + c*x**2)**(1/4)*a 
*b*c**2*e**2*f*x + 224*(a + b*x + c*x**2)**(1/4)*a*b*c**2*e**2*g*x**2 + 67 
20*(a + b*x + c*x**2)**(1/4)*a*c**3*d**2*f + 156*(a + b*x + c*x**2)**(1/4) 
*b**4*e**2*g*x - 432*(a + b*x + c*x**2)**(1/4)*b**3*c*d*e*g*x - 216*(a + b 
*x + c*x**2)**(1/4)*b**3*c*e**2*f*x - 104*(a + b*x + c*x**2)**(1/4)*b**3*c 
*e**2*g*x**2 + 336*(a + b*x + c*x**2)**(1/4)*b**2*c**2*d**2*g*x + 672*(a + 
 b*x + c*x**2)**(1/4)*b**2*c**2*d*e*f*x + 288*(a + b*x + c*x**2)**(1/4)*b* 
*2*c**2*d*e*g*x**2 + 144*(a + b*x + c*x**2)**(1/4)*b**2*c**2*e**2*f*x**2 + 
 80*(a + b*x + c*x**2)**(1/4)*b**2*c**2*e**2*g*x**3 + 3360*(a + b*x + c*x* 
*2)**(1/4)*b*c**3*d**2*f*x + 2016*(a + b*x + c*x**2)**(1/4)*b*c**3*d**2*g* 
x**2 + 4032*(a + b*x + c*x**2)**(1/4)*b*c**3*d*e*f*x**2 + 2880*(a + b*x + 
c*x**2)**(1/4)*b*c**3*d*e*g*x**3 + 1440*(a + b*x + c*x**2)**(1/4)*b*c**3*e 
**2*f*x**3 + 1120*(a + b*x + c*x**2)**(1/4)*b*c**3*e**2*g*x**4 - 1440*int( 
((a + b*x + c*x**2)**(1/4)*x)/(a + b*x + c*x**2),x)*a**2*b*c**2*e**2*g ...