\(\int \frac {(g x)^m (a+c x^2)^p}{(d+e x)^3} \, dx\) [436]

Optimal result

Integrand size = 22, antiderivative size = 321 \[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^3} \, dx=\frac {x (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m}{2},-p,3,\frac {3+m}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^3 (1+m)}-\frac {3 e x^2 (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {2+m}{2},-p,3,\frac {4+m}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^4 (2+m)}+\frac {3 e^2 x^3 (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3+m}{2},-p,3,\frac {5+m}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^5 (3+m)}-\frac {e^3 x^4 (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {4+m}{2},-p,3,\frac {6+m}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^6 (4+m)} \]

[Out]

x*(g*x)^m*(c*x^2+a)^p*AppellF1(1/2+1/2*m,3,-p,3/2+1/2*m,e^2*x^2/d^2,-c*x^2/a)/d^3/(1+m)/((1+c*x^2/a)^p)-3*e*x^
2*(g*x)^m*(c*x^2+a)^p*AppellF1(1+1/2*m,3,-p,2+1/2*m,e^2*x^2/d^2,-c*x^2/a)/d^4/(2+m)/((1+c*x^2/a)^p)+3*e^2*x^3*
(g*x)^m*(c*x^2+a)^p*AppellF1(3/2+1/2*m,3,-p,5/2+1/2*m,e^2*x^2/d^2,-c*x^2/a)/d^5/(3+m)/((1+c*x^2/a)^p)-e^3*x^4*
(g*x)^m*(c*x^2+a)^p*AppellF1(2+1/2*m,3,-p,3+1/2*m,e^2*x^2/d^2,-c*x^2/a)/d^6/(4+m)/((1+c*x^2/a)^p)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {976, 525, 524} \[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^3} \, dx=-\frac {e^3 x^4 (g x)^m \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+4}{2},-p,3,\frac {m+6}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^6 (m+4)}+\frac {3 e^2 x^3 (g x)^m \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+3}{2},-p,3,\frac {m+5}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^5 (m+3)}-\frac {3 e x^2 (g x)^m \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+2}{2},-p,3,\frac {m+4}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^4 (m+2)}+\frac {x (g x)^m \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+1}{2},-p,3,\frac {m+3}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^3 (m+1)} \]

[In]

Int[((g*x)^m*(a + c*x^2)^p)/(d + e*x)^3,x]

[Out]

(x*(g*x)^m*(a + c*x^2)^p*AppellF1[(1 + m)/2, -p, 3, (3 + m)/2, -((c*x^2)/a), (e^2*x^2)/d^2])/(d^3*(1 + m)*(1 +
(c*x^2)/a)^p) - (3*e*x^2*(g*x)^m*(a + c*x^2)^p*AppellF1[(2 + m)/2, -p, 3, (4 + m)/2, -((c*x^2)/a), (e^2*x^2)/
d^2])/(d^4*(2 + m)*(1 + (c*x^2)/a)^p) + (3*e^2*x^3*(g*x)^m*(a + c*x^2)^p*AppellF1[(3 + m)/2, -p, 3, (5 + m)/2,
-((c*x^2)/a), (e^2*x^2)/d^2])/(d^5*(3 + m)*(1 + (c*x^2)/a)^p) - (e^3*x^4*(g*x)^m*(a + c*x^2)^p*AppellF1[(4 +
m)/2, -p, 3, (6 + m)/2, -((c*x^2)/a), (e^2*x^2)/d^2])/(d^6*(4 + 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 976

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

Rubi steps \begin{align*} \text {integral}& = \left (x^{-m} (g x)^m\right ) \int \left (\frac {d^3 x^m \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^3}-\frac {3 d^2 e x^{1+m} \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^3}+\frac {3 d e^2 x^{2+m} \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^3}+\frac {e^3 x^{3+m} \left (a+c x^2\right )^p}{\left (-d^2+e^2 x^2\right )^3}\right ) \, dx \\ & = \left (d^3 x^{-m} (g x)^m\right ) \int \frac {x^m \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^3} \, dx-\left (3 d^2 e x^{-m} (g x)^m\right ) \int \frac {x^{1+m} \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^3} \, dx+\left (3 d e^2 x^{-m} (g x)^m\right ) \int \frac {x^{2+m} \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^3} \, dx+\left (e^3 x^{-m} (g x)^m\right ) \int \frac {x^{3+m} \left (a+c x^2\right )^p}{\left (-d^2+e^2 x^2\right )^3} \, dx \\ & = \left (d^3 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}{\left (d^2-e^2 x^2\right )^3} \, dx-\left (3 d^2 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}{\left (d^2-e^2 x^2\right )^3} \, dx+\left (3 d e^2 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^{2+m} \left (1+\frac {c x^2}{a}\right )^p}{\left (d^2-e^2 x^2\right )^3} \, dx+\left (e^3 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^{3+m} \left (1+\frac {c x^2}{a}\right )^p}{\left (-d^2+e^2 x^2\right )^3} \, 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,3;\frac {3+m}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^3 (1+m)}-\frac {3 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,3;\frac {4+m}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^4 (2+m)}+\frac {3 e^2 x^3 (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {3+m}{2};-p,3;\frac {5+m}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^5 (3+m)}-\frac {e^3 x^4 (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {4+m}{2};-p,3;\frac {6+m}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^6 (4+m)} \\ \end{align*}

Mathematica [F]

\[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^3} \, dx=\int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^3} \, dx \]

[In]

Integrate[((g*x)^m*(a + c*x^2)^p)/(d + e*x)^3,x]

[Out]

Integrate[((g*x)^m*(a + c*x^2)^p)/(d + e*x)^3, x]

Maple [F]

\[\int \frac {\left (g x \right )^{m} \left (c \,x^{2}+a \right )^{p}}{\left (e x +d \right )^{3}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}}{{\left (e x + d\right )}^{3}} \,d x } \]

[In]

integrate((g*x)^m*(c*x^2+a)^p/(e*x+d)^3,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^3} \, dx=\text {Timed out} \]

[In]

integrate((g*x)**m*(c*x**2+a)**p/(e*x+d)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}}{{\left (e x + d\right )}^{3}} \,d x } \]

[In]

integrate((g*x)^m*(c*x^2+a)^p/(e*x+d)^3,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}}{{\left (e x + d\right )}^{3}} \,d x } \]

[In]

integrate((g*x)^m*(c*x^2+a)^p/(e*x+d)^3,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^3} \, dx=\int \frac {{\left (g\,x\right )}^m\,{\left (c\,x^2+a\right )}^p}{{\left (d+e\,x\right )}^3} \,d x \]

[In]

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

[Out]

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