Integrand size = 22, antiderivative size = 189 \[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(d+e x)^2} \, dx=\frac {12\ 2^{2/3} \left (a+b x+c x^2\right )^{4/3} \operatorname {AppellF1}\left (-\frac {5}{3},-\frac {4}{3},-\frac {4}{3},-\frac {2}{3},\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{5 e \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{4/3} (d+e x)} \]
12/5*2^(2/3)*(c*x^2+b*x+a)^(4/3)*AppellF1(-5/3,-4/3,-4/3,-2/3,1/2*(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/(e*x+d)/(e*(b+2*c*x-(-4*a*c+b^2)^(1/2))/c/(e*x+d))^(4/3)/(e*(b+2* c*x+(-4*a*c+b^2)^(1/2))/c/(e*x+d))^(4/3)
\[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(d+e x)^2} \, dx=\int \frac {\left (a+b x+c x^2\right )^{4/3}}{(d+e x)^2} \, dx \]
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.
\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(d+e x)^2} \, dx\) |
\(\Big \downarrow \) 1178 |
\(\displaystyle -\frac {4\ 2^{2/3} \left (\frac {1}{d+e x}\right )^{8/3} \left (a+b x+c x^2\right )^{4/3} \int \frac {\left (2-\frac {2 d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{c}}{d+e x}\right )^{4/3} \left (2-\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{d+e x}\right )^{4/3}}{4\ 2^{2/3} \left (\frac {1}{d+e x}\right )^{8/3}}d\frac {1}{d+e x}}{e \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{d+e x}\right )^{8/3} \left (a+b x+c x^2\right )^{4/3} \int \frac {\left (2-\frac {2 d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{c}}{d+e x}\right )^{4/3} \left (2-\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{d+e x}\right )^{4/3}}{\left (\frac {1}{d+e x}\right )^{8/3}}d\frac {1}{d+e x}}{e \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3}}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {12\ 2^{2/3} \left (a+b x+c x^2\right )^{4/3} \operatorname {AppellF1}\left (-\frac {5}{3},-\frac {4}{3},-\frac {4}{3},-\frac {2}{3},\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{5 e (d+e x) \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3}}\) |
(12*2^(2/3)*(a + b*x + c*x^2)^(4/3)*AppellF1[-5/3, -4/3, -4/3, -2/3, (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))])/(5*e*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*( d + e*x)))^(4/3)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(4/3) *(d + e*x))
3.25.87.3.1 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]
\[\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {4}{3}}}{\left (e x +d \right )^{2}}d x\]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(d+e x)^2} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(d+e x)^2} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {4}{3}}}{\left (d + e x\right )^{2}}\, dx \]
\[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(d+e x)^2} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {4}{3}}}{{\left (e x + d\right )}^{2}} \,d x } \]
\[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(d+e x)^2} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {4}{3}}}{{\left (e x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(d+e x)^2} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{4/3}}{{\left (d+e\,x\right )}^2} \,d x \]