Integrand size = 22, antiderivative size = 189 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{7/3}} \, dx=-\frac {3 \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{7/3} \operatorname {AppellF1}\left (\frac {20}{3},\frac {7}{3},\frac {7}{3},\frac {23}{3},\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{320\ 2^{2/3} e (d+e x)^2 \left (a+b x+c x^2\right )^{7/3}} \]
-3/640*AppellF1(20/3,7/3,7/3,23/3,1/2*(2*d-e*(b+(-4*a*c+b^2)^(1/2))/c)/(e* x+d),1/2*(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))/c/(e*x+d))*(e*(b+2*c*x-(-4*a*c+b ^2)^(1/2))/c/(e*x+d))^(7/3)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/c/(e*x+d))^(7/ 3)*2^(1/3)/e/(e*x+d)^2/(c*x^2+b*x+a)^(7/3)
Time = 12.19 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{7/3}} \, dx=-\frac {3 e^3 \sqrt [3]{\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}} \sqrt [3]{\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}} \operatorname {AppellF1}\left (\frac {20}{3},\frac {7}{3},\frac {7}{3},\frac {23}{3},\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 c d-b e+\sqrt {b^2-4 a c} e}{2 c d+2 c e x}\right )}{20\ 2^{2/3} c^2 (d+e x)^6 \sqrt [3]{a+x (b+c x)}} \]
(-3*e^3*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/3)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/3)*AppellF1[20/3, 7/3, 7/3, 23/3, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(20*2^(2/3)*c^2*(d + e*x)^6*(a + x*(b + c*x))^(1/3))
Time = 0.30 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1178, 27, 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.
\(\displaystyle \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{7/3}} \, dx\) |
\(\Big \downarrow \) 1178 |
\(\displaystyle -\frac {\left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \int \frac {16\ 2^{2/3} \left (\frac {1}{d+e x}\right )^{17/3}}{\left (2-\frac {2 d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{c}}{d+e x}\right )^{7/3} \left (2-\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{d+e x}\right )^{7/3}}d\frac {1}{d+e x}}{16\ 2^{2/3} e \left (\frac {1}{d+e x}\right )^{14/3} \left (a+b x+c x^2\right )^{7/3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \int \frac {\left (\frac {1}{d+e x}\right )^{17/3}}{\left (2-\frac {2 d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{c}}{d+e x}\right )^{7/3} \left (2-\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{d+e x}\right )^{7/3}}d\frac {1}{d+e x}}{e \left (\frac {1}{d+e x}\right )^{14/3} \left (a+b x+c x^2\right )^{7/3}}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle -\frac {3 \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \operatorname {AppellF1}\left (\frac {20}{3},\frac {7}{3},\frac {7}{3},\frac {23}{3},\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{320\ 2^{2/3} e (d+e x)^2 \left (a+b x+c x^2\right )^{7/3}}\) |
(-3*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(7/3)*((e*(b + Sqr t[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(7/3)*AppellF1[20/3, 7/3, 7/3, 23/ 3, (2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*d - ((b + Sqrt[ b^2 - 4*a*c])*e)/c)/(2*(d + e*x))])/(320*2^(2/3)*e*(d + e*x)^2*(a + b*x + c*x^2)^(7/3))
3.25.95.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
\[\int \frac {1}{\left (e x +d \right )^{3} \left (c \,x^{2}+b x +a \right )^{\frac {7}{3}}}d x\]
Timed out. \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{7/3}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{7/3}} \, dx=\int \frac {1}{\left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac {7}{3}}}\, dx \]
\[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}} {\left (e x + d\right )}^{3}} \,d x } \]
\[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}} {\left (e x + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{7/3}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{7/3}} \,d x \]