Integrand size = 18, antiderivative size = 168 \[ \int \frac {\left (a+b x^3+c x^6\right )^p}{x^7} \, dx=-\frac {2^{-1+2 p} \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (2 (1-p),-p,-p,3-2 p,-\frac {b-\sqrt {b^2-4 a c}}{2 c x^3},-\frac {b+\sqrt {b^2-4 a c}}{2 c x^3}\right )}{3 (1-p) x^6} \]
-1/3*2^(-1+2*p)*(c*x^6+b*x^3+a)^p*AppellF1(2-2*p,-p,-p,3-2*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-p)/x^6/(((b+2*c *x^3-(-4*a*c+b^2)^(1/2))/c/x^3)^p)/(((b+2*c*x^3+(-4*a*c+b^2)^(1/2))/c/x^3) ^p)
Time = 0.31 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^3+c x^6\right )^p}{x^7} \, dx=\frac {2^{-1+2 p} \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (2-2 p,-p,-p,3-2 p,-\frac {b+\sqrt {b^2-4 a c}}{2 c x^3},\frac {-b+\sqrt {b^2-4 a c}}{2 c x^3}\right )}{3 (-1+p) x^6} \]
(2^(-1 + 2*p)*(a + b*x^3 + c*x^6)^p*AppellF1[2 - 2*p, -p, -p, 3 - 2*p, -1/ 2*(b + Sqrt[b^2 - 4*a*c])/(c*x^3), (-b + Sqrt[b^2 - 4*a*c])/(2*c*x^3)])/(3 *(-1 + p)*x^6*((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)
Time = 0.29 (sec) , antiderivative size = 166, 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.
\(\displaystyle \int \frac {\left (a+b x^3+c x^6\right )^p}{x^7} \, dx\) |
\(\Big \downarrow \) 1693 |
\(\displaystyle \frac {1}{3} \int \frac {\left (c x^6+b x^3+a\right )^p}{x^9}dx^3\) |
\(\Big \downarrow \) 1178 |
\(\displaystyle -\frac {1}{3} 4^p \left (\frac {1}{x^3}\right )^{2 p} \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (\frac {\sqrt {b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p \int \left (\frac {b-\sqrt {b^2-4 a c}}{2 c x^3}+1\right )^p \left (\frac {b+\sqrt {b^2-4 a c}}{2 c x^3}+1\right )^p \left (\frac {1}{x^3}\right )^{1-2 p}d\frac {1}{x^3}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle -\frac {2^{2 p-1} \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (\frac {\sqrt {b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (2-2 p,-p,-p,3-2 p,-\frac {b-\sqrt {b^2-4 a c}}{2 c x^3},-\frac {b+\sqrt {b^2-4 a c}}{2 c x^3}\right )}{3 (1-p) x^6}\) |
-1/3*(2^(-1 + 2*p)*(a + b*x^3 + c*x^6)^p*AppellF1[2 - 2*p, -p, -p, 3 - 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 - p)*x^6*((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)
3.3.70.3.1 Defintions of rubi rules used
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])
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]
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]]
\[\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{x^{7}}d x\]
\[ \int \frac {\left (a+b x^3+c x^6\right )^p}{x^7} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{7}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^3+c x^6\right )^p}{x^7} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b x^3+c x^6\right )^p}{x^7} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{7}} \,d x } \]
\[ \int \frac {\left (a+b x^3+c x^6\right )^p}{x^7} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{7}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^3+c x^6\right )^p}{x^7} \, dx=\int \frac {{\left (c\,x^6+b\,x^3+a\right )}^p}{x^7} \,d x \]