\(\int \frac {(a+b x^3+c x^6)^p}{x^4} \, dx\) [252]

Optimal result

Integrand size = 18, antiderivative size = 164 \[ \int \frac {\left (a+b x^3+c x^6\right )^p}{x^4} \, dx=-\frac {4^p \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (1-2 p,-p,-p,2 (1-p),-\frac {b-\sqrt {b^2-4 a c}}{2 c x^3},-\frac {b+\sqrt {b^2-4 a c}}{2 c x^3}\right )}{3 (1-2 p) x^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^3+c x^6\right )^p}{x^4} \, dx=\frac {4^p \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (1-2 p,-p,-p,2-2 p,-\frac {b+\sqrt {b^2-4 a c}}{2 c x^3},\frac {-b+\sqrt {b^2-4 a c}}{2 c x^3}\right )}{3 (-1+2 p) x^3} \] Input:

Integrate[(a + b*x^3 + c*x^6)^p/x^4,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1693, 1178, 150}

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.

\(\displaystyle \int \frac {\left (a+b x^3+c x^6\right )^p}{x^4} \, dx\)

\(\Big \downarrow \) 1693

\(\displaystyle \frac {1}{3} \int \frac {\left (c x^6+b x^3+a\right )^p}{x^6}dx^3\)

\(\Big \downarrow \) 1178

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

\(\Big \downarrow \) 150

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

Input:

Int[(a + b*x^3 + c*x^6)^p/x^4,x]
 

Output:

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

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 1178

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(-(1/(d + e*x))^(2*p))*((a + 
b*x + c*x^2)^p/(e*(e*((b - q + 2*c*x)/(2*c*(d + e*x))))^p*(e*((b + q + 2*c* 
x)/(2*c*(d + e*x))))^p))   Subst[Int[x^(-m - 2*(p + 1))*Simp[1 - (d - e*((b 
 - q)/(2*c)))*x, x]^p*Simp[1 - (d - e*((b + q)/(2*c)))*x, x]^p, x], x, 1/(d 
 + e*x)], x]] /; FreeQ[{a, b, c, d, e, p}, x] && ILtQ[m, 0]
 

rule 1693

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, 
x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && IntegerQ 
[Simplify[(m + 1)/n]]
 

Maple [F]

\[\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{x^{4}}d x\]

Input:

int((c*x^6+b*x^3+a)^p/x^4,x)
 

Output:

int((c*x^6+b*x^3+a)^p/x^4,x)
 

Fricas [F]

\[ \int \frac {\left (a+b x^3+c x^6\right )^p}{x^4} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{4}} \,d x } \] Input:

integrate((c*x^6+b*x^3+a)^p/x^4,x, algorithm="fricas")
 

Output:

integral((c*x^6 + b*x^3 + a)^p/x^4, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3+c x^6\right )^p}{x^4} \, dx=\text {Timed out} \] Input:

integrate((c*x**6+b*x**3+a)**p/x**4,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^3+c x^6\right )^p}{x^4} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{4}} \,d x } \] Input:

integrate((c*x^6+b*x^3+a)^p/x^4,x, algorithm="maxima")
 

Output:

integrate((c*x^6 + b*x^3 + a)^p/x^4, x)
 

Giac [F]

\[ \int \frac {\left (a+b x^3+c x^6\right )^p}{x^4} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{4}} \,d x } \] Input:

integrate((c*x^6+b*x^3+a)^p/x^4,x, algorithm="giac")
 

Output:

integrate((c*x^6 + b*x^3 + a)^p/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3+c x^6\right )^p}{x^4} \, dx=\int \frac {{\left (c\,x^6+b\,x^3+a\right )}^p}{x^4} \,d x \] Input:

int((a + b*x^3 + c*x^6)^p/x^4,x)
 

Output:

int((a + b*x^3 + c*x^6)^p/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^3+c x^6\right )^p}{x^4} \, dx=\frac {-\left (c \,x^{6}+b \,x^{3}+a \right )^{p} b p +\left (c \,x^{6}+b \,x^{3}+a \right )^{p} b -\left (c \,x^{6}+b \,x^{3}+a \right )^{p} c \,x^{3}+3 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{c p \,x^{7}-c \,x^{7}+b p \,x^{4}-b \,x^{4}+a p x -a x}d x \right ) b^{2} p^{3} x^{3}-6 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{c p \,x^{7}-c \,x^{7}+b p \,x^{4}-b \,x^{4}+a p x -a x}d x \right ) b^{2} p^{2} x^{3}+3 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{c p \,x^{7}-c \,x^{7}+b p \,x^{4}-b \,x^{4}+a p x -a x}d x \right ) b^{2} p \,x^{3}+6 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p} x^{5}}{c p \,x^{6}-c \,x^{6}+b p \,x^{3}-b \,x^{3}+a p -a}d x \right ) c^{2} p^{2} x^{3}-6 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p} x^{5}}{c p \,x^{6}-c \,x^{6}+b p \,x^{3}-b \,x^{3}+a p -a}d x \right ) c^{2} p \,x^{3}+6 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p} x^{2}}{c p \,x^{6}-c \,x^{6}+b p \,x^{3}-b \,x^{3}+a p -a}d x \right ) b c \,p^{3} x^{3}-9 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p} x^{2}}{c p \,x^{6}-c \,x^{6}+b p \,x^{3}-b \,x^{3}+a p -a}d x \right ) b c \,p^{2} x^{3}+3 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p} x^{2}}{c p \,x^{6}-c \,x^{6}+b p \,x^{3}-b \,x^{3}+a p -a}d x \right ) b c p \,x^{3}}{3 b \,x^{3} \left (p -1\right )} \] Input:

int((c*x^6+b*x^3+a)^p/x^4,x)
 

Output:

( - (a + b*x**3 + c*x**6)**p*b*p + (a + b*x**3 + c*x**6)**p*b - (a + b*x** 
3 + c*x**6)**p*c*x**3 + 3*int((a + b*x**3 + c*x**6)**p/(a*p*x - a*x + b*p* 
x**4 - b*x**4 + c*p*x**7 - c*x**7),x)*b**2*p**3*x**3 - 6*int((a + b*x**3 + 
 c*x**6)**p/(a*p*x - a*x + b*p*x**4 - b*x**4 + c*p*x**7 - c*x**7),x)*b**2* 
p**2*x**3 + 3*int((a + b*x**3 + c*x**6)**p/(a*p*x - a*x + b*p*x**4 - b*x** 
4 + c*p*x**7 - c*x**7),x)*b**2*p*x**3 + 6*int(((a + b*x**3 + c*x**6)**p*x* 
*5)/(a*p - a + b*p*x**3 - b*x**3 + c*p*x**6 - c*x**6),x)*c**2*p**2*x**3 - 
6*int(((a + b*x**3 + c*x**6)**p*x**5)/(a*p - a + b*p*x**3 - b*x**3 + c*p*x 
**6 - c*x**6),x)*c**2*p*x**3 + 6*int(((a + b*x**3 + c*x**6)**p*x**2)/(a*p 
- a + b*p*x**3 - b*x**3 + c*p*x**6 - c*x**6),x)*b*c*p**3*x**3 - 9*int(((a 
+ b*x**3 + c*x**6)**p*x**2)/(a*p - a + b*p*x**3 - b*x**3 + c*p*x**6 - c*x* 
*6),x)*b*c*p**2*x**3 + 3*int(((a + b*x**3 + c*x**6)**p*x**2)/(a*p - a + b* 
p*x**3 - b*x**3 + c*p*x**6 - c*x**6),x)*b*c*p*x**3)/(3*b*x**3*(p - 1))