\(\int \frac {(d+e x^3) (a+b x^3+c x^6)^p}{x^3} \, dx\) [84]

Optimal result

Integrand size = 25, antiderivative size = 274 \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^3} \, dx=-\frac {d \left (1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (-\frac {2}{3},-p,-p,\frac {1}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{2 x^2}+e x \left (1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (\frac {1}{3},-p,-p,\frac {4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right ) \] Output:

-1/2*d*(c*x^6+b*x^3+a)^p*AppellF1(-2/3,-p,-p,1/3,-2*c*x^3/(b-(-4*a*c+b^2)^ 
(1/2)),-2*c*x^3/(b+(-4*a*c+b^2)^(1/2)))/x^2/((1+2*c*x^3/(b-(-4*a*c+b^2)^(1 
/2)))^p)/((1+2*c*x^3/(b+(-4*a*c+b^2)^(1/2)))^p)+e*x*(c*x^6+b*x^3+a)^p*Appe 
llF1(1/3,-p,-p,4/3,-2*c*x^3/(b-(-4*a*c+b^2)^(1/2)),-2*c*x^3/(b+(-4*a*c+b^2 
)^(1/2)))/((1+2*c*x^3/(b-(-4*a*c+b^2)^(1/2)))^p)/((1+2*c*x^3/(b+(-4*a*c+b^ 
2)^(1/2)))^p)
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.86 \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^3} \, dx=\frac {\left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \left (-d \operatorname {AppellF1}\left (-\frac {2}{3},-p,-p,\frac {1}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+2 e x^3 \operatorname {AppellF1}\left (\frac {1}{3},-p,-p,\frac {4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right )}{2 x^2} \] Input:

Integrate[((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x^3,x]
 

Output:

((a + b*x^3 + c*x^6)^p*(-(d*AppellF1[-2/3, -p, -p, 1/3, (-2*c*x^3)/(b + Sq 
rt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])]) + 2*e*x^3*AppellF1[ 
1/3, -p, -p, 4/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt 
[b^2 - 4*a*c])]))/(2*x^2*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 
- 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^p 
)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1864, 2009}

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.

\(\displaystyle \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^3} \, dx\)

\(\Big \downarrow \) 1864

\(\displaystyle \int \left (\frac {d \left (a+b x^3+c x^6\right )^p}{x^3}+e \left (a+b x^3+c x^6\right )^p\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle e x \left (\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (\frac {1}{3},-p,-p,\frac {4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )-\frac {d \left (\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (-\frac {2}{3},-p,-p,\frac {1}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{2 x^2}\)

Input:

Int[((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x^3,x]
 

Output:

-1/2*(d*(a + b*x^3 + c*x^6)^p*AppellF1[-2/3, -p, -p, 1/3, (-2*c*x^3)/(b - 
Sqrt[b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/(x^2*(1 + (2*c*x^ 
3)/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^p) + 
 (e*x*(a + b*x^3 + c*x^6)^p*AppellF1[1/3, -p, -p, 4/3, (-2*c*x^3)/(b - Sqr 
t[b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/((1 + (2*c*x^3)/(b - 
 Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^p)
 

Defintions of rubi rules used

rule 1864

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*( 
(d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n 
+ c*x^(2*n))^p, (f*x)^m*(d + e*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m 
, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[q, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

\[\int \frac {\left (e \,x^{3}+d \right ) \left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{x^{3}}d x\]

Input:

int((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^3,x)
 

Output:

int((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^3,x)
 

Fricas [F]

\[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^3} \, dx=\int { \frac {{\left (e x^{3} + d\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{3}} \,d x } \] Input:

integrate((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^3,x, algorithm="fricas")
 

Output:

integral((e*x^3 + d)*(c*x^6 + b*x^3 + a)^p/x^3, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^3} \, dx=\text {Timed out} \] Input:

integrate((e*x**3+d)*(c*x**6+b*x**3+a)**p/x**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^3} \, dx=\int { \frac {{\left (e x^{3} + d\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{3}} \,d x } \] Input:

integrate((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^3,x, algorithm="maxima")
 

Output:

integrate((e*x^3 + d)*(c*x^6 + b*x^3 + a)^p/x^3, x)
 

Giac [F]

\[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^3} \, dx=\int { \frac {{\left (e x^{3} + d\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{3}} \,d x } \] Input:

integrate((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^3,x, algorithm="giac")
 

Output:

integrate((e*x^3 + d)*(c*x^6 + b*x^3 + a)^p/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^3} \, dx=\int \frac {\left (e\,x^3+d\right )\,{\left (c\,x^6+b\,x^3+a\right )}^p}{x^3} \,d x \] Input:

int(((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x^3,x)
 

Output:

int(((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x^3, x)
 

Reduce [F]

\[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^3} \, dx=\text {too large to display} \] Input:

int((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^3,x)
 

Output:

(6*(a + b*x**3 + c*x**6)**p*a*e*p + 6*(a + b*x**3 + c*x**6)**p*b*d*p + (a 
+ b*x**3 + c*x**6)**p*b*d + 3*(a + b*x**3 + c*x**6)**p*b*e*p*x**3 - 2*(a + 
 b*x**3 + c*x**6)**p*b*e*x**3 + 216*int((a + b*x**3 + c*x**6)**p/(18*a*p** 
2*x**3 - 9*a*p*x**3 - 2*a*x**3 + 18*b*p**2*x**6 - 9*b*p*x**6 - 2*b*x**6 + 
18*c*p**2*x**9 - 9*c*p*x**9 - 2*c*x**9),x)*a**2*e*p**3*x**2 - 108*int((a + 
 b*x**3 + c*x**6)**p/(18*a*p**2*x**3 - 9*a*p*x**3 - 2*a*x**3 + 18*b*p**2*x 
**6 - 9*b*p*x**6 - 2*b*x**6 + 18*c*p**2*x**9 - 9*c*p*x**9 - 2*c*x**9),x)*a 
**2*e*p**2*x**2 - 24*int((a + b*x**3 + c*x**6)**p/(18*a*p**2*x**3 - 9*a*p* 
x**3 - 2*a*x**3 + 18*b*p**2*x**6 - 9*b*p*x**6 - 2*b*x**6 + 18*c*p**2*x**9 
- 9*c*p*x**9 - 2*c*x**9),x)*a**2*e*p*x**2 + 324*int((a + b*x**3 + c*x**6)* 
*p/(18*a*p**2*x**3 - 9*a*p*x**3 - 2*a*x**3 + 18*b*p**2*x**6 - 9*b*p*x**6 - 
 2*b*x**6 + 18*c*p**2*x**9 - 9*c*p*x**9 - 2*c*x**9),x)*a*b*d*p**4*x**2 - 1 
08*int((a + b*x**3 + c*x**6)**p/(18*a*p**2*x**3 - 9*a*p*x**3 - 2*a*x**3 + 
18*b*p**2*x**6 - 9*b*p*x**6 - 2*b*x**6 + 18*c*p**2*x**9 - 9*c*p*x**9 - 2*c 
*x**9),x)*a*b*d*p**3*x**2 - 63*int((a + b*x**3 + c*x**6)**p/(18*a*p**2*x** 
3 - 9*a*p*x**3 - 2*a*x**3 + 18*b*p**2*x**6 - 9*b*p*x**6 - 2*b*x**6 + 18*c* 
p**2*x**9 - 9*c*p*x**9 - 2*c*x**9),x)*a*b*d*p**2*x**2 - 6*int((a + b*x**3 
+ c*x**6)**p/(18*a*p**2*x**3 - 9*a*p*x**3 - 2*a*x**3 + 18*b*p**2*x**6 - 9* 
b*p*x**6 - 2*b*x**6 + 18*c*p**2*x**9 - 9*c*p*x**9 - 2*c*x**9),x)*a*b*d*p*x 
**2 - 648*int(((a + b*x**3 + c*x**6)**p*x**3)/(18*a*p**2 - 9*a*p - 2*a ...