\(\int \frac {(g x)^m (a+c x^2)^p}{d+e x} \, dx\) [434]

Optimal result

Integrand size = 22, antiderivative size = 157 \[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{d+e x} \, dx=\frac {x (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m}{2},-p,1,\frac {3+m}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d (1+m)}-\frac {e x^2 (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {2+m}{2},-p,1,\frac {4+m}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2 (2+m)} \]

[Out]

x*(g*x)^m*(c*x^2+a)^p*AppellF1(1/2+1/2*m,1,-p,3/2+1/2*m,e^2*x^2/d^2,-c*x^2/a)/d/(1+m)/((1+c*x^2/a)^p)-e*x^2*(g
*x)^m*(c*x^2+a)^p*AppellF1(1+1/2*m,1,-p,2+1/2*m,e^2*x^2/d^2,-c*x^2/a)/d^2/(2+m)/((1+c*x^2/a)^p)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {973, 525, 524} \[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{d+e x} \, dx=\frac {x (g x)^m \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+1}{2},-p,1,\frac {m+3}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d (m+1)}-\frac {e x^2 (g x)^m \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+2}{2},-p,1,\frac {m+4}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2 (m+2)} \]

[In]

Int[((g*x)^m*(a + c*x^2)^p)/(d + e*x),x]

[Out]

(x*(g*x)^m*(a + c*x^2)^p*AppellF1[(1 + m)/2, -p, 1, (3 + m)/2, -((c*x^2)/a), (e^2*x^2)/d^2])/(d*(1 + m)*(1 + (
c*x^2)/a)^p) - (e*x^2*(g*x)^m*(a + c*x^2)^p*AppellF1[(2 + m)/2, -p, 1, (4 + m)/2, -((c*x^2)/a), (e^2*x^2)/d^2]
)/(d^2*(2 + m)*(1 + (c*x^2)/a)^p)

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 973

Int[(((g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[d*((g*x)^n/x^n), In
t[(x^n*(a + c*x^2)^p)/(d^2 - e^2*x^2), x], x] - Dist[e*((g*x)^n/x^n), Int[(x^(n + 1)*(a + c*x^2)^p)/(d^2 - e^2
*x^2), x], x] /; FreeQ[{a, c, d, e, g, n, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] &&  !IntegersQ[n, 2
*p]

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

Mathematica [F]

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

[In]

Integrate[((g*x)^m*(a + c*x^2)^p)/(d + e*x),x]

[Out]

Integrate[((g*x)^m*(a + c*x^2)^p)/(d + e*x), x]

Maple [F]

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

[In]

int((g*x)^m*(c*x^2+a)^p/(e*x+d),x)

[Out]

int((g*x)^m*(c*x^2+a)^p/(e*x+d),x)

Fricas [F]

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

[In]

integrate((g*x)^m*(c*x^2+a)^p/(e*x+d),x, algorithm="fricas")

[Out]

integral((c*x^2 + a)^p*(g*x)^m/(e*x + d), x)

Sympy [F(-1)]

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

[In]

integrate((g*x)**m*(c*x**2+a)**p/(e*x+d),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate((g*x)^m*(c*x^2+a)^p/(e*x+d),x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^p*(g*x)^m/(e*x + d), x)

Giac [F]

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

[In]

integrate((g*x)^m*(c*x^2+a)^p/(e*x+d),x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^p*(g*x)^m/(e*x + d), x)

Mupad [F(-1)]

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

[In]

int(((g*x)^m*(a + c*x^2)^p)/(d + e*x),x)

[Out]

int(((g*x)^m*(a + c*x^2)^p)/(d + e*x), x)