\(\int \frac {(d+e x^3) (a+b x^3+c x^6)^p}{x^2} \, dx\) [79]

Optimal result

Integrand size = 25, antiderivative size = 277 \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^2} \, dx=-\frac {d \left (1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (-\frac {1}{3},-p,-p,\frac {2}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{x}+\frac {1}{2} e x^2 \left (1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (\frac {2}{3},-p,-p,\frac {5}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right ) \] Output:

-d*(c*x^6+b*x^3+a)^p*AppellF1(-1/3,-p,-p,2/3,-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/((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)+1/2*e*x^2*(c*x^6+b*x^3+a)^p*Appe 
llF1(2/3,-p,-p,5/3,-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] (warning: unable to verify)

Time = 0.66 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.84 \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^2} \, dx=\frac {\left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \left (-2 d \operatorname {AppellF1}\left (-\frac {1}{3},-p,-p,\frac {2}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+e x^3 \operatorname {AppellF1}\left (\frac {2}{3},-p,-p,\frac {5}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right )}{2 x} \] Input:

Integrate[((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x^2,x]
 

Output:

((a + b*x^3 + c*x^6)^p*(-2*d*AppellF1[-1/3, -p, -p, 2/3, (-2*c*x^3)/(b + S 
qrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + e*x^3*AppellF1[2/ 
3, -p, -p, 5/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b 
^2 - 4*a*c])]))/(2*x*((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 = 0.44 (sec) , antiderivative size = 277, 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.

\(\displaystyle \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^2} \, dx\)

\(\Big \downarrow \) 1864

\(\displaystyle \int \left (\frac {d \left (a+b x^3+c x^6\right )^p}{x^2}+e x \left (a+b x^3+c x^6\right )^p\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} e x^2 \left (\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (\frac {2}{3},-p,-p,\frac {5}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )-\frac {d \left (\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (-\frac {1}{3},-p,-p,\frac {2}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{x}\)

Input:

Int[((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x^2,x]
 

Output:

-((d*(a + b*x^3 + c*x^6)^p*AppellF1[-1/3, -p, -p, 2/3, (-2*c*x^3)/(b - Sqr 
t[b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/(x*(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^2*(a + b*x^3 + c*x^6)^p*AppellF1[2/3, -p, -p, 5/3, (-2*c*x^3)/(b - Sqrt[ 
b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/(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)
 

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]

\[\int \frac {\left (e \,x^{3}+d \right ) \left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{x^{2}}d x\]

Input:

int((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^2,x)
 

Output:

int((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^2,x)
 

Fricas [F]

\[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^2} \, dx=\int { \frac {{\left (e x^{3} + d\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{2}} \,d x } \] Input:

integrate((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^2,x, algorithm="fricas")
 

Output:

integral((e*x^3 + d)*(c*x^6 + b*x^3 + a)^p/x^2, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^2} \, dx=\text {Timed out} \] Input:

integrate((e*x**3+d)*(c*x**6+b*x**3+a)**p/x**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^2} \, dx=\int { \frac {{\left (e x^{3} + d\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{2}} \,d x } \] Input:

integrate((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^2,x, algorithm="maxima")
 

Output:

integrate((e*x^3 + d)*(c*x^6 + b*x^3 + a)^p/x^2, x)
 

Giac [F]

\[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^2} \, dx=\int { \frac {{\left (e x^{3} + d\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{2}} \,d x } \] Input:

integrate((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^2,x, algorithm="giac")
 

Output:

integrate((e*x^3 + d)*(c*x^6 + b*x^3 + a)^p/x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^2} \, dx=\int \frac {\left (e\,x^3+d\right )\,{\left (c\,x^6+b\,x^3+a\right )}^p}{x^2} \,d x \] Input:

int(((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x^2,x)
 

Output:

int(((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x^2, x)
 

Reduce [F]

\[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^2} \, dx =\text {Too large to display} \] Input:

int((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^2,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 + 2*( 
a + b*x**3 + c*x**6)**p*b*d + 3*(a + b*x**3 + c*x**6)**p*b*e*p*x**3 - (a + 
 b*x**3 + c*x**6)**p*b*e*x**3 + 54*int((a + b*x**3 + c*x**6)**p/(9*a*p**2* 
x**2 - a*x**2 + 9*b*p**2*x**5 - b*x**5 + 9*c*p**2*x**8 - c*x**8),x)*a**2*e 
*p**3*x - 6*int((a + b*x**3 + c*x**6)**p/(9*a*p**2*x**2 - a*x**2 + 9*b*p** 
2*x**5 - b*x**5 + 9*c*p**2*x**8 - c*x**8),x)*a**2*e*p*x + 162*int((a + b*x 
**3 + c*x**6)**p/(9*a*p**2*x**2 - a*x**2 + 9*b*p**2*x**5 - b*x**5 + 9*c*p* 
*2*x**8 - c*x**8),x)*a*b*d*p**4*x + 54*int((a + b*x**3 + c*x**6)**p/(9*a*p 
**2*x**2 - a*x**2 + 9*b*p**2*x**5 - b*x**5 + 9*c*p**2*x**8 - c*x**8),x)*a* 
b*d*p**3*x - 18*int((a + b*x**3 + c*x**6)**p/(9*a*p**2*x**2 - a*x**2 + 9*b 
*p**2*x**5 - b*x**5 + 9*c*p**2*x**8 - c*x**8),x)*a*b*d*p**2*x - 6*int((a + 
 b*x**3 + c*x**6)**p/(9*a*p**2*x**2 - a*x**2 + 9*b*p**2*x**5 - b*x**5 + 9* 
c*p**2*x**8 - c*x**8),x)*a*b*d*p*x - 324*int(((a + b*x**3 + c*x**6)**p*x** 
4)/(9*a*p**2 - a + 9*b*p**2*x**3 - b*x**3 + 9*c*p**2*x**6 - c*x**6),x)*a*c 
*e*p**4*x + 54*int(((a + b*x**3 + c*x**6)**p*x**4)/(9*a*p**2 - a + 9*b*p** 
2*x**3 - b*x**3 + 9*c*p**2*x**6 - c*x**6),x)*a*c*e*p**3*x + 36*int(((a + b 
*x**3 + c*x**6)**p*x**4)/(9*a*p**2 - a + 9*b*p**2*x**3 - b*x**3 + 9*c*p**2 
*x**6 - c*x**6),x)*a*c*e*p**2*x - 6*int(((a + b*x**3 + c*x**6)**p*x**4)/(9 
*a*p**2 - a + 9*b*p**2*x**3 - b*x**3 + 9*c*p**2*x**6 - c*x**6),x)*a*c*e*p* 
x + 81*int(((a + b*x**3 + c*x**6)**p*x**4)/(9*a*p**2 - a + 9*b*p**2*x**...