\(\int \frac {(d+e x^n)^q}{a+b x^n+c x^{2 n}} \, dx\) [148]

Optimal result

Integrand size = 26, antiderivative size = 194 \[ \int \frac {\left (d+e x^n\right )^q}{a+b x^n+c x^{2 n}} \, dx=-\frac {2 c x \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{n},1,-q,1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {2 c x \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{n},1,-q,1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}} \]

[Out]

-2*c*x*(d+e*x^n)^q*AppellF1(1/n,1,-q,1+1/n,-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)),-e*x^n/d)/((1+e*x^n/d)^q)/(b^2-4*a*
c-b*(-4*a*c+b^2)^(1/2))-2*c*x*(d+e*x^n)^q*AppellF1(1/n,1,-q,1+1/n,-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)),-e*x^n/d)/((
1+e*x^n/d)^q)/(b^2-4*a*c+b*(-4*a*c+b^2)^(1/2))

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1442, 441, 440} \[ \int \frac {\left (d+e x^n\right )^q}{a+b x^n+c x^{2 n}} \, dx=-\frac {2 c x \left (d+e x^n\right )^q \left (\frac {e x^n}{d}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{n},1,-q,1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{-b \sqrt {b^2-4 a c}-4 a c+b^2}-\frac {2 c x \left (d+e x^n\right )^q \left (\frac {e x^n}{d}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{n},1,-q,1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{b \sqrt {b^2-4 a c}-4 a c+b^2} \]

[In]

Int[(d + e*x^n)^q/(a + b*x^n + c*x^(2*n)),x]

[Out]

(-2*c*x*(d + e*x^n)^q*AppellF1[n^(-1), 1, -q, 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), -((e*x^n)/d)])/(
(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(1 + (e*x^n)/d)^q) - (2*c*x*(d + e*x^n)^q*AppellF1[n^(-1), 1, -q, 1 + n^(-
1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), -((e*x^n)/d)])/((b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(1 + (e*x^n)/d)^q)

Rule 440

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 441

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^F
racPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 1442

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{r = Rt[b^2 -
4*a*c, 2]}, Dist[2*(c/r), Int[(d + e*x^n)^q/(b - r + 2*c*x^n), x], x] - Dist[2*(c/r), Int[(d + e*x^n)^q/(b +
r + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] &&  !IntegerQ[q]

Rubi steps \begin{align*} \text {integral}& = \frac {(2 c) \int \frac {\left (d+e x^n\right )^q}{b-\sqrt {b^2-4 a c}+2 c x^n} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {\left (d+e x^n\right )^q}{b+\sqrt {b^2-4 a c}+2 c x^n} \, dx}{\sqrt {b^2-4 a c}} \\ & = \frac {\left (2 c \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q}\right ) \int \frac {\left (1+\frac {e x^n}{d}\right )^q}{b-\sqrt {b^2-4 a c}+2 c x^n} \, dx}{\sqrt {b^2-4 a c}}-\frac {\left (2 c \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q}\right ) \int \frac {\left (1+\frac {e x^n}{d}\right )^q}{b+\sqrt {b^2-4 a c}+2 c x^n} \, dx}{\sqrt {b^2-4 a c}} \\ & = -\frac {2 c x \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q} F_1\left (\frac {1}{n};1,-q;1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {2 c x \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q} F_1\left (\frac {1}{n};1,-q;1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {\left (d+e x^n\right )^q}{a+b x^n+c x^{2 n}} \, dx=\int \frac {\left (d+e x^n\right )^q}{a+b x^n+c x^{2 n}} \, dx \]

[In]

Integrate[(d + e*x^n)^q/(a + b*x^n + c*x^(2*n)),x]

[Out]

Integrate[(d + e*x^n)^q/(a + b*x^n + c*x^(2*n)), x]

Maple [F]

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

[In]

int((d+e*x^n)^q/(a+b*x^n+c*x^(2*n)),x)

[Out]

int((d+e*x^n)^q/(a+b*x^n+c*x^(2*n)),x)

Fricas [F]

\[ \int \frac {\left (d+e x^n\right )^q}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{q}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]

[In]

integrate((d+e*x^n)^q/(a+b*x^n+c*x^(2*n)),x, algorithm="fricas")

[Out]

integral((e*x^n + d)^q/(c*x^(2*n) + b*x^n + a), x)

Sympy [F(-1)]

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

[In]

integrate((d+e*x**n)**q/(a+b*x**n+c*x**(2*n)),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (d+e x^n\right )^q}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{q}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]

[In]

integrate((d+e*x^n)^q/(a+b*x^n+c*x^(2*n)),x, algorithm="maxima")

[Out]

integrate((e*x^n + d)^q/(c*x^(2*n) + b*x^n + a), x)

Giac [F]

\[ \int \frac {\left (d+e x^n\right )^q}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{q}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]

[In]

integrate((d+e*x^n)^q/(a+b*x^n+c*x^(2*n)),x, algorithm="giac")

[Out]

integrate((e*x^n + d)^q/(c*x^(2*n) + b*x^n + a), x)

Mupad [F(-1)]

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

[In]

int((d + e*x^n)^q/(a + b*x^n + c*x^(2*n)),x)

[Out]

int((d + e*x^n)^q/(a + b*x^n + c*x^(2*n)), x)