\(\int (d+e x)^3 (a+b x+c x^2)^{4/3} \, dx\) [679]

Optimal result

Integrand size = 22, antiderivative size = 223 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{4/3} \, dx=\frac {3 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/3}}{20 c}+\frac {3 e \left (391 c^2 d^2+65 b^2 e^2-3 c e (100 b d+17 a e)+91 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/3}}{2380 c^3}+\frac {(2 c d-b e) \left (34 c^2 d^2+13 b^2 e^2-2 c e (17 b d+9 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{4/3} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{544\ 2^{2/3} c^4 \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{4/3}} \] Output:

3/20*e*(e*x+d)^2*(c*x^2+b*x+a)^(7/3)/c+3/2380*e*(391*c^2*d^2+65*b^2*e^2-3* 
c*e*(17*a*e+100*b*d)+91*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(7/3)/c^3+1/1088 
*(-b*e+2*c*d)*(34*c^2*d^2+13*b^2*e^2-2*c*e*(9*a*e+17*b*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^4/(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(4/3)
 

Mathematica [A] (verified)

Time = 10.47 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.92 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{4/3} \, dx=\frac {3 (a+x (b+c x))^{4/3} \left (3808 e (d+e x)^2 (a+x (b+c x))+\frac {32 e (a+x (b+c x)) \left (65 b^2 e^2+c^2 d (391 d+182 e x)-c e (300 b d+51 a e+91 b e x)\right )}{c^2}-\frac {70 \sqrt [3]{2} (-2 c d+b e) \left (34 c^2 d^2+13 b^2 e^2-2 c e (17 b d+9 a e)\right ) (b+2 c x) \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{3 c^3 \left (-\frac {c (a+x (b+c x))}{b^2-4 a c}\right )^{4/3}}\right )}{76160 c} \] Input:

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

Output:

(3*(a + x*(b + c*x))^(4/3)*(3808*e*(d + e*x)^2*(a + x*(b + c*x)) + (32*e*( 
a + x*(b + c*x))*(65*b^2*e^2 + c^2*d*(391*d + 182*e*x) - c*e*(300*b*d + 51 
*a*e + 91*b*e*x)))/c^2 - (70*2^(1/3)*(-2*c*d + b*e)*(34*c^2*d^2 + 13*b^2*e 
^2 - 2*c*e*(17*b*d + 9*a*e))*(b + 2*c*x)*Hypergeometric2F1[-4/3, 1/2, 3/2, 
 (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*c^3*(-((c*(a + x*(b + c*x)))/(b^2 - 4*a* 
c)))^(4/3))))/(76160*c)
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(683\) vs. \(2(223)=446\).

Time = 0.67 (sec) , antiderivative size = 683, normalized size of antiderivative = 3.06, 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, 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.

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

\(\Big \downarrow \) 1166

\(\displaystyle \frac {3 \int \frac {1}{3} (d+e x) \left (20 c d^2-7 b e d-6 a e^2+13 e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{4/3}dx}{20 c}+\frac {3 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/3}}{20 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (d+e x) \left (20 c d^2-7 b e d-6 a e^2+13 e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{4/3}dx}{20 c}+\frac {3 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/3}}{20 c}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\frac {5 (2 c d-b e) \left (-2 c e (9 a e+17 b d)+13 b^2 e^2+34 c^2 d^2\right ) \int \left (c x^2+b x+a\right )^{4/3}dx}{17 c^2}+\frac {3 e \left (a+b x+c x^2\right )^{7/3} \left (-3 c e (17 a e+100 b d)+65 b^2 e^2+91 c e x (2 c d-b e)+391 c^2 d^2\right )}{119 c^2}}{20 c}+\frac {3 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/3}}{20 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {5 (2 c d-b e) \left (-2 c e (9 a e+17 b d)+13 b^2 e^2+34 c^2 d^2\right ) \left (\frac {3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{22 c}-\frac {2 \left (b^2-4 a c\right ) \int \sqrt [3]{c x^2+b x+a}dx}{11 c}\right )}{17 c^2}+\frac {3 e \left (a+b x+c x^2\right )^{7/3} \left (-3 c e (17 a e+100 b d)+65 b^2 e^2+91 c e x (2 c d-b e)+391 c^2 d^2\right )}{119 c^2}}{20 c}+\frac {3 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/3}}{20 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {5 (2 c d-b e) \left (-2 c e (9 a e+17 b d)+13 b^2 e^2+34 c^2 d^2\right ) \left (\frac {3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{22 c}-\frac {2 \left (b^2-4 a c\right ) \left (\frac {3 (b+2 c x) \sqrt [3]{a+b x+c x^2}}{10 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{2/3}}dx}{10 c}\right )}{11 c}\right )}{17 c^2}+\frac {3 e \left (a+b x+c x^2\right )^{7/3} \left (-3 c e (17 a e+100 b d)+65 b^2 e^2+91 c e x (2 c d-b e)+391 c^2 d^2\right )}{119 c^2}}{20 c}+\frac {3 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/3}}{20 c}\)

\(\Big \downarrow \) 1095

\(\displaystyle \frac {\frac {5 (2 c d-b e) \left (-2 c e (9 a e+17 b d)+13 b^2 e^2+34 c^2 d^2\right ) \left (\frac {3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{22 c}-\frac {2 \left (b^2-4 a c\right ) \left (\frac {3 (b+2 c x) \sqrt [3]{a+b x+c x^2}}{10 c}-\frac {3 \left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{10 c (b+2 c x)}\right )}{11 c}\right )}{17 c^2}+\frac {3 e \left (a+b x+c x^2\right )^{7/3} \left (-3 c e (17 a e+100 b d)+65 b^2 e^2+91 c e x (2 c d-b e)+391 c^2 d^2\right )}{119 c^2}}{20 c}+\frac {3 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/3}}{20 c}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {\frac {5 (2 c d-b e) \left (-2 c e (9 a e+17 b d)+13 b^2 e^2+34 c^2 d^2\right ) \left (\frac {3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{22 c}-\frac {2 \left (b^2-4 a c\right ) \left (\frac {3 (b+2 c x) \sqrt [3]{a+b x+c x^2}}{10 c}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} \left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+\left (b^2-4 a c\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right ),-7-4 \sqrt {3}\right )}{5\ 2^{2/3} c^{4/3} (b+2 c x) \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}\right )}{11 c}\right )}{17 c^2}+\frac {3 e \left (a+b x+c x^2\right )^{7/3} \left (-3 c e (17 a e+100 b d)+65 b^2 e^2+91 c e x (2 c d-b e)+391 c^2 d^2\right )}{119 c^2}}{20 c}+\frac {3 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/3}}{20 c}\)

Input:

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

Output:

(3*e*(d + e*x)^2*(a + b*x + c*x^2)^(7/3))/(20*c) + ((3*e*(391*c^2*d^2 + 65 
*b^2*e^2 - 3*c*e*(100*b*d + 17*a*e) + 91*c*e*(2*c*d - b*e)*x)*(a + b*x + c 
*x^2)^(7/3))/(119*c^2) + (5*(2*c*d - b*e)*(34*c^2*d^2 + 13*b^2*e^2 - 2*c*e 
*(17*b*d + 9*a*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)*Sqr 
t[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]*EllipticF[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)))/(17*c^2))/(20*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 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 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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((c*e^3*x^5 + (3*c*d*e^2 + b*e^3)*x^4 + a*d^3 + (3*c*d^2*e + 3*b*d 
*e^2 + a*e^3)*x^3 + (c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*x^2 + (b*d^3 + 3*a*d^2 
*e)*x)*(c*x^2 + b*x + a)^(1/3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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