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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]