3.432 \(\int (g x)^m (d+e x) \left (a+c x^2\right )^p \, dx\)

Optimal. Leaf size=135 \[ \frac{d (g x)^{m+1} \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{c x^2}{a}\right )}{g (m+1)}+\frac{e (g x)^{m+2} \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+2}{2},-p;\frac{m+4}{2};-\frac{c x^2}{a}\right )}{g^2 (m+2)} \]

[Out]

(d*(g*x)^(1 + m)*(a + c*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((c*
x^2)/a)])/(g*(1 + m)*(1 + (c*x^2)/a)^p) + (e*(g*x)^(2 + m)*(a + c*x^2)^p*Hyperge
ometric2F1[(2 + m)/2, -p, (4 + m)/2, -((c*x^2)/a)])/(g^2*(2 + m)*(1 + (c*x^2)/a)
^p)

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Rubi [A]  time = 0.150914, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{d (g x)^{m+1} \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{c x^2}{a}\right )}{g (m+1)}+\frac{e (g x)^{m+2} \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+2}{2},-p;\frac{m+4}{2};-\frac{c x^2}{a}\right )}{g^2 (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

(d*(g*x)^(1 + m)*(a + c*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((c*
x^2)/a)])/(g*(1 + m)*(1 + (c*x^2)/a)^p) + (e*(g*x)^(2 + m)*(a + c*x^2)^p*Hyperge
ometric2F1[(2 + m)/2, -p, (4 + m)/2, -((c*x^2)/a)])/(g^2*(2 + m)*(1 + (c*x^2)/a)
^p)

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Rubi in Sympy [A]  time = 20.8313, size = 105, normalized size = 0.78 \[ \frac{d \left (g x\right )^{m + 1} \left (1 + \frac{c x^{2}}{a}\right )^{- p} \left (a + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{g \left (m + 1\right )} + \frac{e \left (g x\right )^{m + 2} \left (1 + \frac{c x^{2}}{a}\right )^{- p} \left (a + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{g^{2} \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*(g*x)**(m + 1)*(1 + c*x**2/a)**(-p)*(a + c*x**2)**p*hyper((-p, m/2 + 1/2), (m/
2 + 3/2,), -c*x**2/a)/(g*(m + 1)) + e*(g*x)**(m + 2)*(1 + c*x**2/a)**(-p)*(a + c
*x**2)**p*hyper((-p, m/2 + 1), (m/2 + 2,), -c*x**2/a)/(g**2*(m + 2))

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Mathematica [A]  time = 0.0996037, size = 106, normalized size = 0.79 \[ \frac{x (g x)^m \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (d (m+2) \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{c x^2}{a}\right )+e (m+1) x \, _2F_1\left (\frac{m}{2}+1,-p;\frac{m}{2}+2;-\frac{c x^2}{a}\right )\right )}{(m+1) (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

(x*(g*x)^m*(a + c*x^2)^p*(e*(1 + m)*x*Hypergeometric2F1[1 + m/2, -p, 2 + m/2, -(
(c*x^2)/a)] + d*(2 + m)*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((c*x^2)/a)
]))/((1 + m)*(2 + m)*(1 + (c*x^2)/a)^p)

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \[ \int \left ( gx \right ) ^{m} \left ( ex+d \right ) \left ( c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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