3.380 \(\int \frac{(g x)^m (d+e x)^n}{\left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=295 \[ \frac{(g x)^{m+1} (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{e x}{d},-\frac{\sqrt{c} x}{\sqrt{-a}}\right )}{4 a^2 g (m+1)}+\frac{(g x)^{m+1} (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{e x}{d},\frac{\sqrt{c} x}{\sqrt{-a}}\right )}{4 a^2 g (m+1)}+\frac{(g x)^{m+1} (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac{e x}{d},-\frac{\sqrt{c} x}{\sqrt{-a}}\right )}{4 a^2 g (m+1)}+\frac{(g x)^{m+1} (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac{e x}{d},\frac{\sqrt{c} x}{\sqrt{-a}}\right )}{4 a^2 g (m+1)} \]

[Out]

((g*x)^(1 + m)*(d + e*x)^n*AppellF1[1 + m, -n, 1, 2 + m, -((e*x)/d), -((Sqrt[c]*
x)/Sqrt[-a])])/(4*a^2*g*(1 + m)*(1 + (e*x)/d)^n) + ((g*x)^(1 + m)*(d + e*x)^n*Ap
pellF1[1 + m, -n, 1, 2 + m, -((e*x)/d), (Sqrt[c]*x)/Sqrt[-a]])/(4*a^2*g*(1 + m)*
(1 + (e*x)/d)^n) + ((g*x)^(1 + m)*(d + e*x)^n*AppellF1[1 + m, -n, 2, 2 + m, -((e
*x)/d), -((Sqrt[c]*x)/Sqrt[-a])])/(4*a^2*g*(1 + m)*(1 + (e*x)/d)^n) + ((g*x)^(1
+ m)*(d + e*x)^n*AppellF1[1 + m, -n, 2, 2 + m, -((e*x)/d), (Sqrt[c]*x)/Sqrt[-a]]
)/(4*a^2*g*(1 + m)*(1 + (e*x)/d)^n)

_______________________________________________________________________________________

Rubi [A]  time = 0.962102, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(g x)^{m+1} (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{e x}{d},-\frac{\sqrt{c} x}{\sqrt{-a}}\right )}{4 a^2 g (m+1)}+\frac{(g x)^{m+1} (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{e x}{d},\frac{\sqrt{c} x}{\sqrt{-a}}\right )}{4 a^2 g (m+1)}+\frac{(g x)^{m+1} (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac{e x}{d},-\frac{\sqrt{c} x}{\sqrt{-a}}\right )}{4 a^2 g (m+1)}+\frac{(g x)^{m+1} (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac{e x}{d},\frac{\sqrt{c} x}{\sqrt{-a}}\right )}{4 a^2 g (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

((g*x)^(1 + m)*(d + e*x)^n*AppellF1[1 + m, -n, 1, 2 + m, -((e*x)/d), -((Sqrt[c]*
x)/Sqrt[-a])])/(4*a^2*g*(1 + m)*(1 + (e*x)/d)^n) + ((g*x)^(1 + m)*(d + e*x)^n*Ap
pellF1[1 + m, -n, 1, 2 + m, -((e*x)/d), (Sqrt[c]*x)/Sqrt[-a]])/(4*a^2*g*(1 + m)*
(1 + (e*x)/d)^n) + ((g*x)^(1 + m)*(d + e*x)^n*AppellF1[1 + m, -n, 2, 2 + m, -((e
*x)/d), -((Sqrt[c]*x)/Sqrt[-a])])/(4*a^2*g*(1 + m)*(1 + (e*x)/d)^n) + ((g*x)^(1
+ m)*(d + e*x)^n*AppellF1[1 + m, -n, 2, 2 + m, -((e*x)/d), (Sqrt[c]*x)/Sqrt[-a]]
)/(4*a^2*g*(1 + m)*(1 + (e*x)/d)^n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 152.239, size = 233, normalized size = 0.79 \[ \frac{\left (g x\right )^{m + 1} \left (1 + \frac{e x}{d}\right )^{- n} \left (d + e x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,1,- n,m + 2,- \frac{\sqrt{c} x}{\sqrt{- a}},- \frac{e x}{d} \right )}}{4 a^{2} g \left (m + 1\right )} + \frac{\left (g x\right )^{m + 1} \left (1 + \frac{e x}{d}\right )^{- n} \left (d + e x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,1,- n,m + 2,\frac{\sqrt{c} x}{\sqrt{- a}},- \frac{e x}{d} \right )}}{4 a^{2} g \left (m + 1\right )} + \frac{\left (g x\right )^{m + 1} \left (1 + \frac{e x}{d}\right )^{- n} \left (d + e x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,2,- n,m + 2,- \frac{\sqrt{c} x}{\sqrt{- a}},- \frac{e x}{d} \right )}}{4 a^{2} g \left (m + 1\right )} + \frac{\left (g x\right )^{m + 1} \left (1 + \frac{e x}{d}\right )^{- n} \left (d + e x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,2,- n,m + 2,\frac{\sqrt{c} x}{\sqrt{- a}},- \frac{e x}{d} \right )}}{4 a^{2} g \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x)**m*(e*x+d)**n/(c*x**2+a)**2,x)

[Out]

(g*x)**(m + 1)*(1 + e*x/d)**(-n)*(d + e*x)**n*appellf1(m + 1, 1, -n, m + 2, -sqr
t(c)*x/sqrt(-a), -e*x/d)/(4*a**2*g*(m + 1)) + (g*x)**(m + 1)*(1 + e*x/d)**(-n)*(
d + e*x)**n*appellf1(m + 1, 1, -n, m + 2, sqrt(c)*x/sqrt(-a), -e*x/d)/(4*a**2*g*
(m + 1)) + (g*x)**(m + 1)*(1 + e*x/d)**(-n)*(d + e*x)**n*appellf1(m + 1, 2, -n,
m + 2, -sqrt(c)*x/sqrt(-a), -e*x/d)/(4*a**2*g*(m + 1)) + (g*x)**(m + 1)*(1 + e*x
/d)**(-n)*(d + e*x)**n*appellf1(m + 1, 2, -n, m + 2, sqrt(c)*x/sqrt(-a), -e*x/d)
/(4*a**2*g*(m + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.109072, size = 0, normalized size = 0. \[ \int \frac{(g x)^m (d+e x)^n}{\left (a+c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.084, size = 0, normalized size = 0. \[ \int{\frac{ \left ( gx \right ) ^{m} \left ( ex+d \right ) ^{n}}{ \left ( c{x}^{2}+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x)^m*(e*x+d)^n/(c*x^2+a)^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{n} \left (g x\right )^{m}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{n} \left (g x\right )^{m}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x + d)^n*(g*x)^m/(c^2*x^4 + 2*a*c*x^2 + a^2), x)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x)**m*(e*x+d)**n/(c*x**2+a)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{n} \left (g x\right )^{m}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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