\(\int (d+e x) (f+g x) \sqrt [4]{a+b x+c x^2} \, dx\) [1098]

Optimal result

Integrand size = 25, antiderivative size = 230 \[ \int (d+e x) (f+g x) \sqrt [4]{a+b x+c x^2} \, dx=\frac {\left (28 c^2 d f+9 b^2 e g-8 a c e g-14 b c (e f+d g)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{84 c^3}-\frac {(9 b e g-14 c (e f+d g)-10 c e g x) \left (a+b x+c x^2\right )^{5/4}}{35 c^2}+\frac {\left (-b^2+4 a c\right )^{3/2} \left (28 c^2 d f+9 b^2 e g-8 a c e g-14 b c (e f+d g)\right ) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right ),2\right )}{84 \sqrt {2} c^4 \left (a+b x+c x^2\right )^{3/4}} \] Output:

1/84*(28*c^2*d*f+9*b^2*e*g-8*a*c*e*g-14*b*c*(d*g+e*f))*(2*c*x+b)*(c*x^2+b* 
x+a)^(1/4)/c^3-1/35*(9*b*e*g-14*c*(d*g+e*f)-10*c*e*g*x)*(c*x^2+b*x+a)^(5/4 
)/c^2+1/168*(4*a*c-b^2)^(3/2)*(28*c^2*d*f+9*b^2*e*g-8*a*c*e*g-14*b*c*(d*g+ 
e*f))*(-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 = 10.33 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.83 \[ \int (d+e x) (f+g x) \sqrt [4]{a+b x+c x^2} \, dx=\frac {(a+x (b+c x))^{5/4} (-9 b e g+2 c (7 e f+7 d g+5 e g x))+\frac {5 \left (28 c^2 d f+9 b^2 e g-2 c (4 a e g+7 b (e f+d g))\right ) \left (2 c (b+2 c x) (a+x (b+c x))-\sqrt {2} \left (b^2-4 a c\right )^{3/2} \left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ),2\right )\right )}{24 c^2 (a+x (b+c x))^{3/4}}}{35 c^2} \] Input:

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

Output:

((a + x*(b + c*x))^(5/4)*(-9*b*e*g + 2*c*(7*e*f + 7*d*g + 5*e*g*x)) + (5*( 
28*c^2*d*f + 9*b^2*e*g - 2*c*(4*a*e*g + 7*b*(e*f + 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]) 
)/(24*c^2*(a + x*(b + c*x))^(3/4)))/(35*c^2)
 

Rubi [A] (warning: unable to verify)

Time = 0.66 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.46, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {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.

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

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\left (-8 a c e g+9 b^2 e g-14 b c d g-14 b c e f+28 c^2 d f\right ) \int \sqrt [4]{c x^2+b x+a}dx}{28 c^2}-\frac {\left (a+b x+c x^2\right )^{5/4} (9 b e g-14 c (d g+e f)-10 c e g x)}{35 c^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (-8 a c e g+9 b^2 e g-14 b c d g-14 b c e f+28 c^2 d f\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{3/4}}dx}{12 c}\right )}{28 c^2}-\frac {\left (a+b x+c x^2\right )^{5/4} (9 b e g-14 c (d g+e f)-10 c e g x)}{35 c^2}\)

\(\Big \downarrow \) 1094

\(\displaystyle \frac {\left (-8 a c e g+9 b^2 e g-14 b c d g-14 b c e f+28 c^2 d f\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{3 c (b+2 c x)}\right )}{28 c^2}-\frac {\left (a+b x+c x^2\right )^{5/4} (9 b e g-14 c (d g+e f)-10 c e g x)}{35 c^2}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {\left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right )^{5/4} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{6 \sqrt {2} c^{5/4} (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}\right ) \left (-8 a c e g+9 b^2 e g-14 b c d g-14 b c e f+28 c^2 d f\right )}{28 c^2}-\frac {\left (a+b x+c x^2\right )^{5/4} (9 b e g-14 c (d g+e f)-10 c e g x)}{35 c^2}\)

Input:

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

Output:

-1/35*((9*b*e*g - 14*c*(e*f + d*g) - 10*c*e*g*x)*(a + b*x + c*x^2)^(5/4))/ 
c^2 + ((28*c^2*d*f - 14*b*c*e*f - 14*b*c*d*g + 9*b^2*e*g - 8*a*c*e*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]*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)])))/(28*c^2)
 

Defintions of rubi rules used

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+e x) (f+g x) \sqrt [4]{a+b x+c x^2} \, dx=\frac {-320 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} a^{2} c e g +144 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} a \,b^{2} e g -224 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} a b c d g -224 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} a b c e f +80 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} a b c e g x +1120 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} a \,c^{2} d f -36 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b^{3} e g x +56 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b^{2} c d g x +56 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b^{2} c e f x +24 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b^{2} c e g \,x^{2}+560 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b \,c^{2} d f x +336 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b \,c^{2} d g \,x^{2}+336 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b \,c^{2} e f \,x^{2}+240 \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b \,c^{2} e g \,x^{3}+160 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x \right ) a^{2} c^{2} e g -220 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x \right ) a \,b^{2} c e g +280 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x \right ) a b \,c^{2} d g +280 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x \right ) a b \,c^{2} e f -560 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x \right ) a \,c^{3} d f +45 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x \right ) b^{4} e g -70 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x \right ) b^{3} c d g -70 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x \right ) b^{3} c e f +140 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x \right ) b^{2} c^{2} d f}{840 b \,c^{2}} \] Input:

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

Output:

( - 320*(a + b*x + c*x**2)**(1/4)*a**2*c*e*g + 144*(a + b*x + c*x**2)**(1/ 
4)*a*b**2*e*g - 224*(a + b*x + c*x**2)**(1/4)*a*b*c*d*g - 224*(a + b*x + c 
*x**2)**(1/4)*a*b*c*e*f + 80*(a + b*x + c*x**2)**(1/4)*a*b*c*e*g*x + 1120* 
(a + b*x + c*x**2)**(1/4)*a*c**2*d*f - 36*(a + b*x + c*x**2)**(1/4)*b**3*e 
*g*x + 56*(a + b*x + c*x**2)**(1/4)*b**2*c*d*g*x + 56*(a + b*x + c*x**2)** 
(1/4)*b**2*c*e*f*x + 24*(a + b*x + c*x**2)**(1/4)*b**2*c*e*g*x**2 + 560*(a 
 + b*x + c*x**2)**(1/4)*b*c**2*d*f*x + 336*(a + b*x + c*x**2)**(1/4)*b*c** 
2*d*g*x**2 + 336*(a + b*x + c*x**2)**(1/4)*b*c**2*e*f*x**2 + 240*(a + b*x 
+ c*x**2)**(1/4)*b*c**2*e*g*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*g - 220*int(((a + b*x + c*x**2)**(1/4)*x) 
/(a + b*x + c*x**2),x)*a*b**2*c*e*g + 280*int(((a + b*x + c*x**2)**(1/4)*x 
)/(a + b*x + c*x**2),x)*a*b*c**2*d*g + 280*int(((a + b*x + c*x**2)**(1/4)* 
x)/(a + b*x + c*x**2),x)*a*b*c**2*e*f - 560*int(((a + b*x + c*x**2)**(1/4) 
*x)/(a + b*x + c*x**2),x)*a*c**3*d*f + 45*int(((a + b*x + c*x**2)**(1/4)*x 
)/(a + b*x + c*x**2),x)*b**4*e*g - 70*int(((a + b*x + c*x**2)**(1/4)*x)/(a 
 + b*x + c*x**2),x)*b**3*c*d*g - 70*int(((a + b*x + c*x**2)**(1/4)*x)/(a + 
 b*x + c*x**2),x)*b**3*c*e*f + 140*int(((a + b*x + c*x**2)**(1/4)*x)/(a + 
b*x + c*x**2),x)*b**2*c**2*d*f)/(840*b*c**2)