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

Optimal. Leaf size=238 \[ \frac{x (g x)^m \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{m+1}{2};-p,2;\frac{m+3}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^2 (m+1)}+\frac{e^2 x^3 (g x)^m \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{m+3}{2};-p,2;\frac{m+5}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^4 (m+3)}-\frac{2 e x^2 (g x)^m \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{m+2}{2};-p,2;\frac{m+4}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^3 (m+2)} \]

[Out]

(x*(g*x)^m*(a + c*x^2)^p*AppellF1[(1 + m)/2, -p, 2, (3 + m)/2, -((c*x^2)/a), (e^
2*x^2)/d^2])/(d^2*(1 + m)*(1 + (c*x^2)/a)^p) - (2*e*x^2*(g*x)^m*(a + c*x^2)^p*Ap
pellF1[(2 + m)/2, -p, 2, (4 + m)/2, -((c*x^2)/a), (e^2*x^2)/d^2])/(d^3*(2 + m)*(
1 + (c*x^2)/a)^p) + (e^2*x^3*(g*x)^m*(a + c*x^2)^p*AppellF1[(3 + m)/2, -p, 2, (5
+ m)/2, -((c*x^2)/a), (e^2*x^2)/d^2])/(d^4*(3 + m)*(1 + (c*x^2)/a)^p)

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Rubi [A]  time = 0.65451, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{x (g x)^m \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{m+1}{2};-p,2;\frac{m+3}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^2 (m+1)}+\frac{e^2 x^3 (g x)^m \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{m+3}{2};-p,2;\frac{m+5}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^4 (m+3)}-\frac{2 e x^2 (g x)^m \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{m+2}{2};-p,2;\frac{m+4}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^3 (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

(x*(g*x)^m*(a + c*x^2)^p*AppellF1[(1 + m)/2, -p, 2, (3 + m)/2, -((c*x^2)/a), (e^
2*x^2)/d^2])/(d^2*(1 + m)*(1 + (c*x^2)/a)^p) - (2*e*x^2*(g*x)^m*(a + c*x^2)^p*Ap
pellF1[(2 + m)/2, -p, 2, (4 + m)/2, -((c*x^2)/a), (e^2*x^2)/d^2])/(d^3*(2 + m)*(
1 + (c*x^2)/a)^p) + (e^2*x^3*(g*x)^m*(a + c*x^2)^p*AppellF1[(3 + m)/2, -p, 2, (5
+ m)/2, -((c*x^2)/a), (e^2*x^2)/d^2])/(d^4*(3 + m)*(1 + (c*x^2)/a)^p)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.109583, size = 0, normalized size = 0. \[ \int \frac{(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.081, size = 0, normalized size = 0. \[ \int{\frac{ \left ( gx \right ) ^{m} \left ( c{x}^{2}+a \right ) ^{p}}{ \left ( ex+d \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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