\(\int \frac {(d+e x)^3}{\sqrt [4]{a+b x+c x^2}} \, dx\) [740]

Optimal result

Integrand size = 22, antiderivative size = 226 \[ \int \frac {(d+e x)^3}{\sqrt [4]{a+b x+c x^2}} \, dx=\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{3/4}}{7 c}+\frac {e \left (360 c^2 d^2+77 b^2 e^2-2 c e (147 b d+40 a e)+66 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/4}}{210 c^3}+\frac {\sqrt {b^2-4 a c} (2 c d-b e) \left (20 c^2 d^2+11 b^2 e^2-4 c e (5 b d+6 a e)\right ) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right |2\right )}{20 \sqrt {2} c^4 \sqrt [4]{a+b x+c x^2}} \] Output:

2/7*e*(e*x+d)^2*(c*x^2+b*x+a)^(3/4)/c+1/210*e*(360*c^2*d^2+77*b^2*e^2-2*c* 
e*(40*a*e+147*b*d)+66*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(3/4)/c^3+1/40*(-4 
*a*c+b^2)^(1/2)*(-b*e+2*c*d)*(20*c^2*d^2+11*b^2*e^2-4*c*e*(6*a*e+5*b*d))*( 
-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/4)*EllipticE(sin(1/2*arcsin((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)^(1/4)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.28 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x)^3}{\sqrt [4]{a+b x+c x^2}} \, dx=\frac {30 e (d+e x)^2 (a+x (b+c x))+\frac {e (a+x (b+c x)) \left (77 b^2 e^2+12 c^2 d (30 d+11 e x)-2 c e (147 b d+40 a e+33 b e x)\right )}{2 c^2}+\frac {21 (2 c d-b e) \left (20 c^2 d^2+11 b^2 e^2-4 c e (5 b d+6 a e)\right ) (b+2 c x) \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{8 \sqrt {2} c^3}}{105 c \sqrt [4]{a+x (b+c x)}} \] Input:

Integrate[(d + e*x)^3/(a + b*x + c*x^2)^(1/4),x]
 

Output:

(30*e*(d + e*x)^2*(a + x*(b + c*x)) + (e*(a + x*(b + c*x))*(77*b^2*e^2 + 1 
2*c^2*d*(30*d + 11*e*x) - 2*c*e*(147*b*d + 40*a*e + 33*b*e*x)))/(2*c^2) + 
(21*(2*c*d - b*e)*(20*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(5*b*d + 6*a*e))*(b + 2 
*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)*Hypergeometric2F1[1/4, 
1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(8*Sqrt[2]*c^3))/(105*c*(a + x*(b 
+ c*x))^(1/4))
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(684\) vs. \(2(226)=452\).

Time = 0.70 (sec) , antiderivative size = 684, normalized size of antiderivative = 3.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1166, 27, 1225, 1094, 834, 761, 1510}

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.

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

\(\Big \downarrow \) 1166

\(\displaystyle \frac {2 \int \frac {(d+e x) \left (14 c d^2-3 b e d-8 a e^2+11 e (2 c d-b e) x\right )}{4 \sqrt [4]{c x^2+b x+a}}dx}{7 c}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{3/4}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(d+e x) \left (14 c d^2-3 b e d-8 a e^2+11 e (2 c d-b e) x\right )}{\sqrt [4]{c x^2+b x+a}}dx}{14 c}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{3/4}}{7 c}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\frac {7 (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right ) \int \frac {1}{\sqrt [4]{c x^2+b x+a}}dx}{20 c^2}+\frac {e \left (a+b x+c x^2\right )^{3/4} \left (-2 c e (40 a e+147 b d)+77 b^2 e^2+66 c e x (2 c d-b e)+360 c^2 d^2\right )}{15 c^2}}{14 c}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{3/4}}{7 c}\)

\(\Big \downarrow \) 1094

\(\displaystyle \frac {\frac {7 \sqrt {(b+2 c x)^2} (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right ) \int \frac {\sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{5 c^2 (b+2 c x)}+\frac {e \left (a+b x+c x^2\right )^{3/4} \left (-2 c e (40 a e+147 b d)+77 b^2 e^2+66 c e x (2 c d-b e)+360 c^2 d^2\right )}{15 c^2}}{14 c}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{3/4}}{7 c}\)

\(\Big \downarrow \) 834

\(\displaystyle \frac {\frac {7 \sqrt {(b+2 c x)^2} (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right ) \left (\frac {\sqrt {b^2-4 a c} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{5 c^2 (b+2 c x)}+\frac {e \left (a+b x+c x^2\right )^{3/4} \left (-2 c e (40 a e+147 b d)+77 b^2 e^2+66 c e x (2 c d-b e)+360 c^2 d^2\right )}{15 c^2}}{14 c}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{3/4}}{7 c}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {\frac {7 \sqrt {(b+2 c x)^2} (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right ) \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{5 c^2 (b+2 c x)}+\frac {e \left (a+b x+c x^2\right )^{3/4} \left (-2 c e (40 a e+147 b d)+77 b^2 e^2+66 c e x (2 c d-b e)+360 c^2 d^2\right )}{15 c^2}}{14 c}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{3/4}}{7 c}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {\frac {7 \sqrt {(b+2 c x)^2} (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right ) \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \left (\frac {\sqrt [4]{b^2-4 a c} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt [4]{a+b x+c x^2} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )}\right )}{2 \sqrt {c}}\right )}{5 c^2 (b+2 c x)}+\frac {e \left (a+b x+c x^2\right )^{3/4} \left (-2 c e (40 a e+147 b d)+77 b^2 e^2+66 c e x (2 c d-b e)+360 c^2 d^2\right )}{15 c^2}}{14 c}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{3/4}}{7 c}\)

Input:

Int[(d + e*x)^3/(a + b*x + c*x^2)^(1/4),x]
 

Output:

(2*e*(d + e*x)^2*(a + b*x + c*x^2)^(3/4))/(7*c) + ((e*(360*c^2*d^2 + 77*b^ 
2*e^2 - 2*c*e*(147*b*d + 40*a*e) + 66*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^ 
2)^(3/4))/(15*c^2) + (7*(2*c*d - b*e)*(20*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(5* 
b*d + 6*a*e))*Sqrt[(b + 2*c*x)^2]*(-1/2*(Sqrt[b^2 - 4*a*c]*(-(((a + b*x + 
c*x^2)^(1/4)*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]))) + ((b^2 - 4*a*c)^ 
(1/4)*(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)]*EllipticE[2*ArcTan[(Sqrt[2]*c^(1/4)*(a 
 + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(Sqrt[2]*c^(1/4)*Sqrt[b 
^2 - 4*a*c + 4*c*(a + b*x + c*x^2)])))/Sqrt[c] + ((b^2 - 4*a*c)^(3/4)*(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])/(4*Sqrt[2]*c^(3/4)*Sqrt[b^2 - 4*a 
*c + 4*c*(a + b*x + c*x^2)])))/(5*c^2*(b + 2*c*x)))/(14*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 834

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

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 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 1510

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* 
(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E 
llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e 
}, x] && PosQ[c/a]
 

Maple [F]

\[\int \frac {\left (e x +d \right )^{3}}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {(d+e x)^3}{\sqrt [4]{a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate((e*x+d)^3/(c*x^2+b*x+a)^(1/4),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((e*x+d)**3/(c*x**2+b*x+a)**(1/4),x)
 

Output:

Integral((d + e*x)**3/(a + b*x + c*x**2)**(1/4), x)
 

Maxima [F]

\[ \int \frac {(d+e x)^3}{\sqrt [4]{a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate((e*x+d)^3/(c*x^2+b*x+a)^(1/4),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {(d+e x)^3}{\sqrt [4]{a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate((e*x+d)^3/(c*x^2+b*x+a)^(1/4),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3}{\sqrt [4]{a+b x+c x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^3}{{\left (c\,x^2+b\,x+a\right )}^{1/4}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(d+e x)^3}{\sqrt [4]{a+b x+c x^2}} \, dx=\left (\int \frac {x^{3}}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x \right ) e^{3}+3 \left (\int \frac {x^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x \right ) d \,e^{2}+3 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x \right ) d^{2} e +\left (\int \frac {1}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x \right ) d^{3} \] Input:

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

Output:

int(x**3/(a + b*x + c*x**2)**(1/4),x)*e**3 + 3*int(x**2/(a + b*x + c*x**2) 
**(1/4),x)*d*e**2 + 3*int(x/(a + b*x + c*x**2)**(1/4),x)*d**2*e + int(1/(a 
 + b*x + c*x**2)**(1/4),x)*d**3