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

Optimal. Leaf size=399 \[ \frac{(g x)^{m+1} (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} \left (a^2 e^4 (m+n+2) (m+n+3) (m+n+4) (m+n+5)+c d^2 (m+1) (m+2) \left (2 a e^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+c d^2 \left (m^2+7 m+12\right )\right )\right ) \, _2F_1\left (m+1,-n;m+2;-\frac{e x}{d}\right )}{e^4 g (m+1) (m+n+2) (m+n+3) (m+n+4) (m+n+5)}-\frac{c d (m+2) (g x)^{m+1} (d+e x)^{n+1} \left (2 a e^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+c d^2 \left (m^2+7 m+12\right )\right )}{e^4 g (m+n+2) (m+n+3) (m+n+4) (m+n+5)}+\frac{c (g x)^{m+2} (d+e x)^{n+1} \left (2 a e^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+c d^2 \left (m^2+7 m+12\right )\right )}{e^3 g^2 (m+n+3) (m+n+4) (m+n+5)}-\frac{c^2 d (m+4) (g x)^{m+3} (d+e x)^{n+1}}{e^2 g^3 (m+n+4) (m+n+5)}+\frac{c^2 (g x)^{m+4} (d+e x)^{n+1}}{e g^4 (m+n+5)} \]

[Out]

-((c*d*(2 + m)*(c*d^2*(12 + 7*m + m^2) + 2*a*e^2*(20 + m^2 + 9*n + n^2 + m*(9 +
2*n)))*(g*x)^(1 + m)*(d + e*x)^(1 + n))/(e^4*g*(2 + m + n)*(3 + m + n)*(4 + m +
n)*(5 + m + n))) + (c*(c*d^2*(12 + 7*m + m^2) + 2*a*e^2*(20 + m^2 + 9*n + n^2 +
m*(9 + 2*n)))*(g*x)^(2 + m)*(d + e*x)^(1 + n))/(e^3*g^2*(3 + m + n)*(4 + m + n)*
(5 + m + n)) - (c^2*d*(4 + m)*(g*x)^(3 + m)*(d + e*x)^(1 + n))/(e^2*g^3*(4 + m +
n)*(5 + m + n)) + (c^2*(g*x)^(4 + m)*(d + e*x)^(1 + n))/(e*g^4*(5 + m + n)) + (
(a^2*e^4*(2 + m + n)*(3 + m + n)*(4 + m + n)*(5 + m + n) + c*d^2*(1 + m)*(2 + m)
*(c*d^2*(12 + 7*m + m^2) + 2*a*e^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n))))*(g*x)^
(1 + m)*(d + e*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((e*x)/d)])/(e^4*g*(1 +
m)*(2 + m + n)*(3 + m + n)*(4 + m + n)*(5 + m + n)*(1 + (e*x)/d)^n)

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Rubi [A]  time = 1.45151, antiderivative size = 377, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(g x)^{m+1} (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} \left (\frac{a^2}{m+1}+\frac{c d^2 (m+2) \left (2 a e^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+c d^2 \left (m^2+7 m+12\right )\right )}{e^4 (m+n+2) (m+n+3) (m+n+4) (m+n+5)}\right ) \, _2F_1\left (m+1,-n;m+2;-\frac{e x}{d}\right )}{g}-\frac{c d (m+2) (g x)^{m+1} (d+e x)^{n+1} \left (2 a e^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+c d^2 \left (m^2+7 m+12\right )\right )}{e^4 g (m+n+2) (m+n+3) (m+n+4) (m+n+5)}+\frac{c (g x)^{m+2} (d+e x)^{n+1} \left (2 a e^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+c d^2 \left (m^2+7 m+12\right )\right )}{e^3 g^2 (m+n+3) (m+n+4) (m+n+5)}-\frac{c^2 d (m+4) (g x)^{m+3} (d+e x)^{n+1}}{e^2 g^3 (m+n+4) (m+n+5)}+\frac{c^2 (g x)^{m+4} (d+e x)^{n+1}}{e g^4 (m+n+5)} \]

Antiderivative was successfully verified.

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

[Out]

-((c*d*(2 + m)*(c*d^2*(12 + 7*m + m^2) + 2*a*e^2*(20 + m^2 + 9*n + n^2 + m*(9 +
2*n)))*(g*x)^(1 + m)*(d + e*x)^(1 + n))/(e^4*g*(2 + m + n)*(3 + m + n)*(4 + m +
n)*(5 + m + n))) + (c*(c*d^2*(12 + 7*m + m^2) + 2*a*e^2*(20 + m^2 + 9*n + n^2 +
m*(9 + 2*n)))*(g*x)^(2 + m)*(d + e*x)^(1 + n))/(e^3*g^2*(3 + m + n)*(4 + m + n)*
(5 + m + n)) - (c^2*d*(4 + m)*(g*x)^(3 + m)*(d + e*x)^(1 + n))/(e^2*g^3*(4 + m +
n)*(5 + m + n)) + (c^2*(g*x)^(4 + m)*(d + e*x)^(1 + n))/(e*g^4*(5 + m + n)) + (
(a^2/(1 + m) + (c*d^2*(2 + m)*(c*d^2*(12 + 7*m + m^2) + 2*a*e^2*(20 + m^2 + 9*n
+ n^2 + m*(9 + 2*n))))/(e^4*(2 + m + n)*(3 + m + n)*(4 + m + n)*(5 + m + n)))*(g
*x)^(1 + m)*(d + e*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((e*x)/d)])/(g*(1 +
(e*x)/d)^n)

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Rubi in Sympy [A]  time = 88.0008, size = 289, normalized size = 0.72 \[ \frac{c^{2} d^{4} \left (g x\right )^{m + 1} \left (1 + \frac{e x}{d}\right )^{- n} \left (d + e x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n - 4, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{e x}{d}} \right )}}{e^{4} g \left (m + 1\right )} - \frac{4 c^{2} d^{4} \left (g x\right )^{m + 1} \left (1 + \frac{e x}{d}\right )^{- n} \left (d + e x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n - 3, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{e x}{d}} \right )}}{e^{4} g \left (m + 1\right )} - \frac{4 c d^{2} \left (g x\right )^{m + 1} \left (1 + \frac{e x}{d}\right )^{- n} \left (d + e x\right )^{n} \left (a e^{2} + c d^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - n - 1, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{e x}{d}} \right )}}{e^{4} g \left (m + 1\right )} + \frac{2 c d^{2} \left (g x\right )^{m + 1} \left (1 + \frac{e x}{d}\right )^{- n} \left (d + e x\right )^{n} \left (a e^{2} + 3 c d^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - n - 2, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{e x}{d}} \right )}}{e^{4} g \left (m + 1\right )} + \frac{\left (g x\right )^{m + 1} \left (1 + \frac{e x}{d}\right )^{- n} \left (d + e x\right )^{n} \left (a e^{2} + c d^{2}\right )^{2}{{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{e x}{d}} \right )}}{e^{4} g \left (m + 1\right )} \]

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]

c**2*d**4*(g*x)**(m + 1)*(1 + e*x/d)**(-n)*(d + e*x)**n*hyper((-n - 4, m + 1), (
m + 2,), -e*x/d)/(e**4*g*(m + 1)) - 4*c**2*d**4*(g*x)**(m + 1)*(1 + e*x/d)**(-n)
*(d + e*x)**n*hyper((-n - 3, m + 1), (m + 2,), -e*x/d)/(e**4*g*(m + 1)) - 4*c*d*
*2*(g*x)**(m + 1)*(1 + e*x/d)**(-n)*(d + e*x)**n*(a*e**2 + c*d**2)*hyper((-n - 1
, m + 1), (m + 2,), -e*x/d)/(e**4*g*(m + 1)) + 2*c*d**2*(g*x)**(m + 1)*(1 + e*x/
d)**(-n)*(d + e*x)**n*(a*e**2 + 3*c*d**2)*hyper((-n - 2, m + 1), (m + 2,), -e*x/
d)/(e**4*g*(m + 1)) + (g*x)**(m + 1)*(1 + e*x/d)**(-n)*(d + e*x)**n*(a*e**2 + c*
d**2)**2*hyper((-n, m + 1), (m + 2,), -e*x/d)/(e**4*g*(m + 1))

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Mathematica [A]  time = 0.231068, size = 128, normalized size = 0.32 \[ \frac{x (g x)^m (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} \left (a^2 \left (m^2+8 m+15\right ) \, _2F_1\left (m+1,-n;m+2;-\frac{e x}{d}\right )+c (m+1) x^2 \left (2 a (m+5) \, _2F_1\left (m+3,-n;m+4;-\frac{e x}{d}\right )+c (m+3) x^2 \, _2F_1\left (m+5,-n;m+6;-\frac{e x}{d}\right )\right )\right )}{(m+1) (m+3) (m+5)} \]

Antiderivative was successfully verified.

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

[Out]

(x*(g*x)^m*(d + e*x)^n*(a^2*(15 + 8*m + m^2)*Hypergeometric2F1[1 + m, -n, 2 + m,
-((e*x)/d)] + c*(1 + m)*x^2*(2*a*(5 + m)*Hypergeometric2F1[3 + m, -n, 4 + m, -(
(e*x)/d)] + c*(3 + m)*x^2*Hypergeometric2F1[5 + m, -n, 6 + m, -((e*x)/d)])))/((1
+ m)*(3 + m)*(5 + m)*(1 + (e*x)/d)^n)

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Maple [F]  time = 0.111, size = 0, normalized size = 0. \[ \int \left ( gx \right ) ^{m} \left ( ex+d \right ) ^{n} \left ( c{x}^{2}+a \right ) ^{2}\, dx \]

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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )}{\left (e x + d\right )}^{n} \left (g x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c^2*x^4 + 2*a*c*x^2 + a^2)*(e*x + d)^n*(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)**n*(c*x**2+a)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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