\(\int \frac {1}{(d+e x) (a+b x+c x^2)^{7/3}} \, dx\) [2493]

Optimal result

Integrand size = 22, antiderivative size = 182 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{7/3}} \, dx=-\frac {3 \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{7/3} \operatorname {AppellF1}\left (\frac {14}{3},\frac {7}{3},\frac {7}{3},\frac {17}{3},\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{224\ 2^{2/3} e \left (a+b x+c x^2\right )^{7/3}} \]

[Out]

-3/448*AppellF1(14/3,7/3,7/3,17/3,1/2*(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/(c*x^2+b*x+a)^(7/3)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {772, 138} \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{7/3}} \, dx=-\frac {3 \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \operatorname {AppellF1}\left (\frac {14}{3},\frac {7}{3},\frac {7}{3},\frac {17}{3},\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{224\ 2^{2/3} e \left (a+b x+c x^2\right )^{7/3}} \]

[In]

Int[1/((d + e*x)*(a + b*x + c*x^2)^(7/3)),x]

[Out]

(-3*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(7/3)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x
)))^(7/3)*AppellF1[14/3, 7/3, 7/3, 17/3, (2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*d - ((b + Sqr
t[b^2 - 4*a*c])*e)/c)/(2*(d + e*x))])/(224*2^(2/3)*e*(a + b*x + c*x^2)^(7/3))

Rule 138

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> 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] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c,
2]}, Dist[(-(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] && NeQ[b^2 - 4*a*c,
0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] &&  !IntegerQ[p] && ILtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (\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}\right ) \text {Subst}\left (\int \frac {x^{11/3}}{\left (1-\frac {1}{2} \left (2 d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{c}\right ) x\right )^{7/3} \left (1-\frac {1}{2} \left (2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}\right ) x\right )^{7/3}} \, dx,x,\frac {1}{d+e x}\right )}{16\ 2^{2/3} e \left (\frac {1}{d+e x}\right )^{14/3} \left (a+b x+c x^2\right )^{7/3}} \\ & = -\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} F_1\left (\frac {14}{3};\frac {7}{3},\frac {7}{3};\frac {17}{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 )}{224\ 2^{2/3} e \left (a+b x+c x^2\right )^{7/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.45 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{7/3}} \, dx=-\frac {3 \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{7/3} \operatorname {AppellF1}\left (\frac {14}{3},\frac {7}{3},\frac {7}{3},\frac {17}{3},\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 c d-b e+\sqrt {b^2-4 a c} e}{2 c d+2 c e x}\right )}{224\ 2^{2/3} e (a+x (b+c x))^{7/3}} \]

[In]

Integrate[1/((d + e*x)*(a + b*x + c*x^2)^(7/3)),x]

[Out]

(-3*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(7/3)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x
)))^(7/3)*AppellF1[14/3, 7/3, 7/3, 17/3, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + S
qrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(224*2^(2/3)*e*(a + x*(b + c*x))^(7/3))

Maple [F]

\[\int \frac {1}{\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{3}}}d x\]

[In]

int(1/(e*x+d)/(c*x^2+b*x+a)^(7/3),x)

[Out]

int(1/(e*x+d)/(c*x^2+b*x+a)^(7/3),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{7/3}} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)/(c*x^2+b*x+a)^(7/3),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{7/3}} \, dx=\int \frac {1}{\left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac {7}{3}}}\, dx \]

[In]

integrate(1/(e*x+d)/(c*x**2+b*x+a)**(7/3),x)

[Out]

Integral(1/((d + e*x)*(a + b*x + c*x**2)**(7/3)), x)

Maxima [F]

\[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}} {\left (e x + d\right )}} \,d x } \]

[In]

integrate(1/(e*x+d)/(c*x^2+b*x+a)^(7/3),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + b*x + a)^(7/3)*(e*x + d)), x)

Giac [F]

\[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}} {\left (e x + d\right )}} \,d x } \]

[In]

integrate(1/(e*x+d)/(c*x^2+b*x+a)^(7/3),x, algorithm="giac")

[Out]

integrate(1/((c*x^2 + b*x + a)^(7/3)*(e*x + d)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{7/3}} \, dx=\int \frac {1}{\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{7/3}} \,d x \]

[In]

int(1/((d + e*x)*(a + b*x + c*x^2)^(7/3)),x)

[Out]

int(1/((d + e*x)*(a + b*x + c*x^2)^(7/3)), x)