Integrand size = 25, antiderivative size = 279 \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^6} \, dx=-\frac {d \left (1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (-\frac {5}{3},-p,-p,-\frac {2}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{5 x^5}-\frac {e \left (1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (-\frac {2}{3},-p,-p,\frac {1}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{2 x^2} \] Output:
-1/5*d*(c*x^6+b*x^3+a)^p*AppellF1(-5/3,-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^5/((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*(c*x^6+b*x^3+a)^p*A ppellF1(-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)
Time = 0.69 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.84 \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^6} \, dx=-\frac {\left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \left (2 d \operatorname {AppellF1}\left (-\frac {5}{3},-p,-p,-\frac {2}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+5 e x^3 \operatorname {AppellF1}\left (-\frac {2}{3},-p,-p,\frac {1}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right )}{10 x^5} \] Input:
Integrate[((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x^6,x]
Output:
-1/10*((a + b*x^3 + c*x^6)^p*(2*d*AppellF1[-5/3, -p, -p, -2/3, (-2*c*x^3)/ (b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + 5*e*x^3*App ellF1[-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^5*((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)
Time = 0.46 (sec) , antiderivative size = 279, 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.
\(\displaystyle \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^6} \, dx\) |
\(\Big \downarrow \) 1864 |
\(\displaystyle \int \left (\frac {d \left (a+b x^3+c x^6\right )^p}{x^6}+\frac {e \left (a+b x^3+c x^6\right )^p}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d \left (\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (a+b x^3+c x^6\right )^p \left (\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \operatorname {AppellF1}\left (-\frac {5}{3},-p,-p,-\frac {2}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{5 x^5}-\frac {e \left (\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (a+b x^3+c x^6\right )^p \left (\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \operatorname {AppellF1}\left (-\frac {2}{3},-p,-p,\frac {1}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{2 x^2}\) |
Input:
Int[((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x^6,x]
Output:
-1/5*(d*(a + b*x^3 + c*x^6)^p*AppellF1[-5/3, -p, -p, -2/3, (-2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/(x^5*(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*(a + b*x^3 + c*x^6)^p*AppellF1[-2/3, -p, -p, 1/3, (-2*c*x^3)/(b - Sqr t[b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/(2*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)
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]
\[\int \frac {\left (e \,x^{3}+d \right ) \left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{x^{6}}d x\]
Input:
int((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^6,x)
Output:
int((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^6,x)
\[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^6} \, dx=\int { \frac {{\left (e x^{3} + d\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{6}} \,d x } \] Input:
integrate((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^6,x, algorithm="fricas")
Output:
integral((e*x^3 + d)*(c*x^6 + b*x^3 + a)^p/x^6, x)
Timed out. \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^6} \, dx=\text {Timed out} \] Input:
integrate((e*x**3+d)*(c*x**6+b*x**3+a)**p/x**6,x)
Output:
Timed out
\[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^6} \, dx=\int { \frac {{\left (e x^{3} + d\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{6}} \,d x } \] Input:
integrate((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^6,x, algorithm="maxima")
Output:
integrate((e*x^3 + d)*(c*x^6 + b*x^3 + a)^p/x^6, x)
\[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^6} \, dx=\int { \frac {{\left (e x^{3} + d\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{6}} \,d x } \] Input:
integrate((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^6,x, algorithm="giac")
Output:
integrate((e*x^3 + d)*(c*x^6 + b*x^3 + a)^p/x^6, x)
Timed out. \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^6} \, dx=\int \frac {\left (e\,x^3+d\right )\,{\left (c\,x^6+b\,x^3+a\right )}^p}{x^6} \,d x \] Input:
int(((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x^6,x)
Output:
int(((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x^6, x)
\[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^6} \, dx =\text {Too large to display} \] Input:
int((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^6,x)
Output:
( - 6*(a + b*x**3 + c*x**6)**p*d*p + 2*(a + b*x**3 + c*x**6)**p*d + 5*(a + b*x**3 + c*x**6)**p*e*x**3 + 270*int((a + b*x**3 + c*x**6)**p/(9*a*p**2*x **3 - 18*a*p*x**3 + 5*a*x**3 + 9*b*p**2*x**6 - 18*b*p*x**6 + 5*b*x**6 + 9* c*p**2*x**9 - 18*c*p*x**9 + 5*c*x**9),x)*a*e*p**3*x**5 - 540*int((a + b*x* *3 + c*x**6)**p/(9*a*p**2*x**3 - 18*a*p*x**3 + 5*a*x**3 + 9*b*p**2*x**6 - 18*b*p*x**6 + 5*b*x**6 + 9*c*p**2*x**9 - 18*c*p*x**9 + 5*c*x**9),x)*a*e*p* *2*x**5 + 150*int((a + b*x**3 + c*x**6)**p/(9*a*p**2*x**3 - 18*a*p*x**3 + 5*a*x**3 + 9*b*p**2*x**6 - 18*b*p*x**6 + 5*b*x**6 + 9*c*p**2*x**9 - 18*c*p *x**9 + 5*c*x**9),x)*a*e*p*x**5 + 162*int((a + b*x**3 + c*x**6)**p/(9*a*p* *2*x**3 - 18*a*p*x**3 + 5*a*x**3 + 9*b*p**2*x**6 - 18*b*p*x**6 + 5*b*x**6 + 9*c*p**2*x**9 - 18*c*p*x**9 + 5*c*x**9),x)*b*d*p**4*x**5 - 378*int((a + b*x**3 + c*x**6)**p/(9*a*p**2*x**3 - 18*a*p*x**3 + 5*a*x**3 + 9*b*p**2*x** 6 - 18*b*p*x**6 + 5*b*x**6 + 9*c*p**2*x**9 - 18*c*p*x**9 + 5*c*x**9),x)*b* d*p**3*x**5 + 198*int((a + b*x**3 + c*x**6)**p/(9*a*p**2*x**3 - 18*a*p*x** 3 + 5*a*x**3 + 9*b*p**2*x**6 - 18*b*p*x**6 + 5*b*x**6 + 9*c*p**2*x**9 - 18 *c*p*x**9 + 5*c*x**9),x)*b*d*p**2*x**5 - 30*int((a + b*x**3 + c*x**6)**p/( 9*a*p**2*x**3 - 18*a*p*x**3 + 5*a*x**3 + 9*b*p**2*x**6 - 18*b*p*x**6 + 5*b *x**6 + 9*c*p**2*x**9 - 18*c*p*x**9 + 5*c*x**9),x)*b*d*p*x**5 + 135*int((a + b*x**3 + c*x**6)**p/(9*a*p**2 - 18*a*p + 5*a + 9*b*p**2*x**3 - 18*b*p*x **3 + 5*b*x**3 + 9*c*p**2*x**6 - 18*c*p*x**6 + 5*c*x**6),x)*b*e*p**3*x*...