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

Optimal. Leaf size=205 \[ -\frac{(g x)^{m+1} \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (a e^2 (m+1)-c d^2 (m+2 p+3)\right ) \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{c x^2}{a}\right )}{c g (m+1) (m+2 p+3)}+\frac{2 d e (g x)^{m+2} \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+2}{2},-p;\frac{m+4}{2};-\frac{c x^2}{a}\right )}{g^2 (m+2)}+\frac{e^2 (g x)^{m+1} \left (a+c x^2\right )^{p+1}}{c g (m+2 p+3)} \]

[Out]

(e^2*(g*x)^(1 + m)*(a + c*x^2)^(1 + p))/(c*g*(3 + m + 2*p)) - ((a*e^2*(1 + m) -
c*d^2*(3 + m + 2*p))*(g*x)^(1 + m)*(a + c*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p
, (3 + m)/2, -((c*x^2)/a)])/(c*g*(1 + m)*(3 + m + 2*p)*(1 + (c*x^2)/a)^p) + (2*d
*e*(g*x)^(2 + m)*(a + c*x^2)^p*Hypergeometric2F1[(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 = 0.401063, antiderivative size = 194, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(g x)^{m+1} \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (\frac{d^2}{m+1}-\frac{a e^2}{c (m+2 p+3)}\right ) \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{c x^2}{a}\right )}{g}+\frac{2 d e (g x)^{m+2} \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+2}{2},-p;\frac{m+4}{2};-\frac{c x^2}{a}\right )}{g^2 (m+2)}+\frac{e^2 (g x)^{m+1} \left (a+c x^2\right )^{p+1}}{c g (m+2 p+3)} \]

Antiderivative was successfully verified.

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

[Out]

(e^2*(g*x)^(1 + m)*(a + c*x^2)^(1 + p))/(c*g*(3 + m + 2*p)) + ((d^2/(1 + m) - (a
*e^2)/(c*(3 + m + 2*p)))*(g*x)^(1 + m)*(a + c*x^2)^p*Hypergeometric2F1[(1 + m)/2
, -p, (3 + m)/2, -((c*x^2)/a)])/(g*(1 + (c*x^2)/a)^p) + (2*d*e*(g*x)^(2 + m)*(a
+ c*x^2)^p*Hypergeometric2F1[(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 = 35.3942, size = 167, normalized size = 0.81 \[ \frac{2 d e \left (g x\right )^{m + 2} \left (1 + \frac{c x^{2}}{a}\right )^{- p} \left (a + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{g^{2} \left (m + 2\right )} + \frac{e^{2} \left (g x\right )^{m + 1} \left (a + c x^{2}\right )^{p + 1}}{c g \left (m + 2 p + 3\right )} - \frac{\left (g x\right )^{m + 1} \left (1 + \frac{c x^{2}}{a}\right )^{- p} \left (a + c x^{2}\right )^{p} \left (a e^{2} \left (m + 1\right ) - c d^{2} \left (m + 2 p + 3\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{c g \left (m + 1\right ) \left (m + 2 p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.168019, size = 162, normalized size = 0.79 \[ \frac{x (g x)^m \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left ((m+2) \left (d^2 (m+3) \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{c x^2}{a}\right )+e^2 (m+1) x^2 \, _2F_1\left (\frac{m+3}{2},-p;\frac{m+5}{2};-\frac{c x^2}{a}\right )\right )+2 d e \left (m^2+4 m+3\right ) x \, _2F_1\left (\frac{m}{2}+1,-p;\frac{m}{2}+2;-\frac{c x^2}{a}\right )\right )}{(m+1) (m+2) (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

(x*(g*x)^m*(a + c*x^2)^p*(2*d*e*(3 + 4*m + m^2)*x*Hypergeometric2F1[1 + m/2, -p,
2 + m/2, -((c*x^2)/a)] + (2 + m)*(d^2*(3 + m)*Hypergeometric2F1[(1 + m)/2, -p,
(3 + m)/2, -((c*x^2)/a)] + e^2*(1 + m)*x^2*Hypergeometric2F1[(3 + m)/2, -p, (5 +
m)/2, -((c*x^2)/a)])))/((1 + m)*(2 + m)*(3 + m)*(1 + (c*x^2)/a)^p)

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int \left ( gx \right ) ^{m} \left ( ex+d \right ) ^{2} \left ( c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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