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

Optimal. Leaf size=148 \[ \frac{(g x)^{m+1} (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{e x}{d},-\frac{\sqrt{c} x}{\sqrt{-a}}\right )}{2 a g (m+1)}+\frac{(g x)^{m+1} (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{e x}{d},\frac{\sqrt{c} x}{\sqrt{-a}}\right )}{2 a g (m+1)} \]

[Out]

((g*x)^(1 + m)*(d + e*x)^n*AppellF1[1 + m, -n, 1, 2 + m, -((e*x)/d), -((Sqrt[c]*
x)/Sqrt[-a])])/(2*a*g*(1 + m)*(1 + (e*x)/d)^n) + ((g*x)^(1 + m)*(d + e*x)^n*Appe
llF1[1 + m, -n, 1, 2 + m, -((e*x)/d), (Sqrt[c]*x)/Sqrt[-a]])/(2*a*g*(1 + m)*(1 +
(e*x)/d)^n)

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Rubi [A]  time = 0.411911, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{(g x)^{m+1} (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{e x}{d},-\frac{\sqrt{c} x}{\sqrt{-a}}\right )}{2 a g (m+1)}+\frac{(g x)^{m+1} (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{e x}{d},\frac{\sqrt{c} x}{\sqrt{-a}}\right )}{2 a g (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[((g*x)^m*(d + e*x)^n)/(a + c*x^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])])/(2*a*g*(1 + m)*(1 + (e*x)/d)^n) + ((g*x)^(1 + m)*(d + e*x)^n*Appe
llF1[1 + m, -n, 1, 2 + m, -((e*x)/d), (Sqrt[c]*x)/Sqrt[-a]])/(2*a*g*(1 + m)*(1 +
(e*x)/d)^n)

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Rubi in Sympy [A]  time = 61.0681, size = 112, normalized size = 0.76 \[ \frac{\left (g x\right )^{m + 1} \left (1 + \frac{e x}{d}\right )^{- n} \left (d + e x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,1,- n,m + 2,- \frac{\sqrt{c} x}{\sqrt{- a}},- \frac{e x}{d} \right )}}{2 a g \left (m + 1\right )} + \frac{\left (g x\right )^{m + 1} \left (1 + \frac{e x}{d}\right )^{- n} \left (d + e x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,1,- n,m + 2,\frac{\sqrt{c} x}{\sqrt{- a}},- \frac{e x}{d} \right )}}{2 a g \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x)**m*(e*x+d)**n/(c*x**2+a),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)/(2*a*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)/(2*a*g*(m + 1
))

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Mathematica [A]  time = 0.0845539, size = 0, normalized size = 0. \[ \int \frac{(g x)^m (d+e x)^n}{a+c x^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \[ \int{\frac{ \left ( gx \right ) ^{m} \left ( ex+d \right ) ^{n}}{c{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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