\(\int (d+e x) (a+b x+c x^2)^{4/3} \, dx\) [681]

Optimal result

Integrand size = 20, antiderivative size = 120 \[ \int (d+e x) \left (a+b x+c x^2\right )^{4/3} \, dx=\frac {3 e \left (a+b x+c x^2\right )^{7/3}}{14 c}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{4/3} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{16\ 2^{2/3} c^2 \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{4/3}} \] Output:

3/14*e*(c*x^2+b*x+a)^(7/3)/c+1/32*(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(4/ 
3)*hypergeom([-4/3, 1/2],[3/2],(2*c*x+b)^2/(-4*a*c+b^2))*2^(1/3)/c^2/(-c*( 
c*x^2+b*x+a)/(-4*a*c+b^2))^(4/3)
 

Mathematica [A] (verified)

Time = 10.19 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.94 \[ \int (d+e x) \left (a+b x+c x^2\right )^{4/3} \, dx=\frac {(a+x (b+c x))^{4/3} \left (48 c^2 e (a+x (b+c x))-\frac {7 \sqrt [3]{2} c (-2 c d+b e) (b+2 c x) \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (-\frac {c (a+x (b+c x))}{b^2-4 a c}\right )^{4/3}}\right )}{224 c^3} \] Input:

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

Output:

((a + x*(b + c*x))^(4/3)*(48*c^2*e*(a + x*(b + c*x)) - (7*2^(1/3)*c*(-2*c* 
d + b*e)*(b + 2*c*x)*Hypergeometric2F1[-4/3, 1/2, 3/2, (b + 2*c*x)^2/(b^2 
- 4*a*c)])/(-((c*(a + x*(b + c*x)))/(b^2 - 4*a*c)))^(4/3)))/(224*c^3)
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(572\) vs. \(2(120)=240\).

Time = 0.45 (sec) , antiderivative size = 572, normalized size of antiderivative = 4.77, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1160, 1087, 1087, 1095, 759}

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.

\(\displaystyle \int (d+e x) \left (a+b x+c x^2\right )^{4/3} \, dx\)

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(2 c d-b e) \left (\frac {3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{22 c}-\frac {2 \left (b^2-4 a c\right ) \int \sqrt [3]{c x^2+b x+a}dx}{11 c}\right )}{2 c}+\frac {3 e \left (a+b x+c x^2\right )^{7/3}}{14 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(2 c d-b e) \left (\frac {3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{22 c}-\frac {2 \left (b^2-4 a c\right ) \left (\frac {3 (b+2 c x) \sqrt [3]{a+b x+c x^2}}{10 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{2/3}}dx}{10 c}\right )}{11 c}\right )}{2 c}+\frac {3 e \left (a+b x+c x^2\right )^{7/3}}{14 c}\)

\(\Big \downarrow \) 1095

\(\displaystyle \frac {(2 c d-b e) \left (\frac {3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{22 c}-\frac {2 \left (b^2-4 a c\right ) \left (\frac {3 (b+2 c x) \sqrt [3]{a+b x+c x^2}}{10 c}-\frac {3 \left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{10 c (b+2 c x)}\right )}{11 c}\right )}{2 c}+\frac {3 e \left (a+b x+c x^2\right )^{7/3}}{14 c}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {(2 c d-b e) \left (\frac {3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{22 c}-\frac {2 \left (b^2-4 a c\right ) \left (\frac {3 (b+2 c x) \sqrt [3]{a+b x+c x^2}}{10 c}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} \left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+\left (b^2-4 a c\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right ),-7-4 \sqrt {3}\right )}{5\ 2^{2/3} c^{4/3} (b+2 c x) \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}\right )}{11 c}\right )}{2 c}+\frac {3 e \left (a+b x+c x^2\right )^{7/3}}{14 c}\)

Input:

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

Output:

(3*e*(a + b*x + c*x^2)^(7/3))/(14*c) + ((2*c*d - b*e)*((3*(b + 2*c*x)*(a + 
 b*x + c*x^2)^(4/3))/(22*c) - (2*(b^2 - 4*a*c)*((3*(b + 2*c*x)*(a + b*x + 
c*x^2)^(1/3))/(10*c) - (3^(3/4)*Sqrt[2 + Sqrt[3]]*(b^2 - 4*a*c)*Sqrt[(b + 
2*c*x)^2]*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))* 
Sqrt[((b^2 - 4*a*c)^(2/3) - 2^(2/3)*c^(1/3)*(b^2 - 4*a*c)^(1/3)*(a + b*x + 
 c*x^2)^(1/3) + 2*2^(1/3)*c^(2/3)*(a + b*x + c*x^2)^(2/3))/((1 + Sqrt[3])* 
(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))^2]*Elliptic 
F[ArcSin[((1 - Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c 
*x^2)^(1/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x 
 + c*x^2)^(1/3))], -7 - 4*Sqrt[3]])/(5*2^(2/3)*c^(4/3)*(b + 2*c*x)*Sqrt[(( 
b^2 - 4*a*c)^(1/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2 
)^(1/3)))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + 
c*x^2)^(1/3))^2]*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2)])))/(11*c)))/(2* 
c)
 

Defintions of rubi rules used

rule 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[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 1095

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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