\(\int \frac {(a+b x^4)^p}{c+e x^2} \, dx\) [180]

Optimal result

Integrand size = 19, antiderivative size = 123 \[ \int \frac {\left (a+b x^4\right )^p}{c+e x^2} \, dx=\frac {x \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{4},-p,1,\frac {5}{4},-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{c}-\frac {e x^3 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{4},-p,1,\frac {7}{4},-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{3 c^2} \]

[Out]

x*(b*x^4+a)^p*AppellF1(1/4,1,-p,5/4,e^2*x^4/c^2,-b*x^4/a)/c/((1+b*x^4/a)^p)-1/3*e*x^3*(b*x^4+a)^p*AppellF1(3/4
,1,-p,7/4,e^2*x^4/c^2,-b*x^4/a)/c^2/((1+b*x^4/a)^p)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1254, 441, 440, 525, 524} \[ \int \frac {\left (a+b x^4\right )^p}{c+e x^2} \, dx=\frac {x \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{4},-p,1,\frac {5}{4},-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{c}-\frac {e x^3 \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{4},-p,1,\frac {7}{4},-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{3 c^2} \]

[In]

Int[(a + b*x^4)^p/(c + e*x^2),x]

[Out]

(x*(a + b*x^4)^p*AppellF1[1/4, -p, 1, 5/4, -((b*x^4)/a), (e^2*x^4)/c^2])/(c*(1 + (b*x^4)/a)^p) - (e*x^3*(a + b
*x^4)^p*AppellF1[3/4, -p, 1, 7/4, -((b*x^4)/a), (e^2*x^4)/c^2])/(3*c^2*(1 + (b*x^4)/a)^p)

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 524

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

Rule 525

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

Rule 1254

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^4)^p, (d/
(d^2 - e^2*x^4) - e*(x^2/(d^2 - e^2*x^4)))^(-q), x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&&  !IntegerQ[p] && ILtQ[q, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c \left (a+b x^4\right )^p}{c^2-e^2 x^4}+\frac {e x^2 \left (a+b x^4\right )^p}{-c^2+e^2 x^4}\right ) \, dx \\ & = c \int \frac {\left (a+b x^4\right )^p}{c^2-e^2 x^4} \, dx+e \int \frac {x^2 \left (a+b x^4\right )^p}{-c^2+e^2 x^4} \, dx \\ & = \left (c \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {b x^4}{a}\right )^p}{c^2-e^2 x^4} \, dx+\left (e \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int \frac {x^2 \left (1+\frac {b x^4}{a}\right )^p}{-c^2+e^2 x^4} \, dx \\ & = \frac {x \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} F_1\left (\frac {1}{4};-p,1;\frac {5}{4};-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{c}-\frac {e x^3 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} F_1\left (\frac {3}{4};-p,1;\frac {7}{4};-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{3 c^2} \\ \end{align*}

Mathematica [F]

\[ \int \frac {\left (a+b x^4\right )^p}{c+e x^2} \, dx=\int \frac {\left (a+b x^4\right )^p}{c+e x^2} \, dx \]

[In]

Integrate[(a + b*x^4)^p/(c + e*x^2),x]

[Out]

Integrate[(a + b*x^4)^p/(c + e*x^2), x]

Maple [F]

\[\int \frac {\left (b \,x^{4}+a \right )^{p}}{e \,x^{2}+c}d x\]

[In]

int((b*x^4+a)^p/(e*x^2+c),x)

[Out]

int((b*x^4+a)^p/(e*x^2+c),x)

Fricas [F]

\[ \int \frac {\left (a+b x^4\right )^p}{c+e x^2} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{p}}{e x^{2} + c} \,d x } \]

[In]

integrate((b*x^4+a)^p/(e*x^2+c),x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^p/(e*x^2 + c), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^4\right )^p}{c+e x^2} \, dx=\text {Timed out} \]

[In]

integrate((b*x**4+a)**p/(e*x**2+c),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a+b x^4\right )^p}{c+e x^2} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{p}}{e x^{2} + c} \,d x } \]

[In]

integrate((b*x^4+a)^p/(e*x^2+c),x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^p/(e*x^2 + c), x)

Giac [F]

\[ \int \frac {\left (a+b x^4\right )^p}{c+e x^2} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{p}}{e x^{2} + c} \,d x } \]

[In]

integrate((b*x^4+a)^p/(e*x^2+c),x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^p/(e*x^2 + c), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^4\right )^p}{c+e x^2} \, dx=\int \frac {{\left (b\,x^4+a\right )}^p}{e\,x^2+c} \,d x \]

[In]

int((a + b*x^4)^p/(c + e*x^2),x)

[Out]

int((a + b*x^4)^p/(c + e*x^2), x)