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

Optimal. Leaf size=164 \[ \frac{(g x)^{m+1} (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} \left (a e^2 (m+n+2) (m+n+3)+c d^2 (m+1) (m+2)\right ) \, _2F_1\left (m+1,-n;m+2;-\frac{e x}{d}\right )}{e^2 g (m+1) (m+n+2) (m+n+3)}-\frac{c d (m+2) (g x)^{m+1} (d+e x)^{n+1}}{e^2 g (m+n+2) (m+n+3)}+\frac{c (g x)^{m+2} (d+e x)^{n+1}}{e g^2 (m+n+3)} \]

[Out]

-((c*d*(2 + m)*(g*x)^(1 + m)*(d + e*x)^(1 + n))/(e^2*g*(2 + m + n)*(3 + m + n)))
+ (c*(g*x)^(2 + m)*(d + e*x)^(1 + n))/(e*g^2*(3 + m + n)) + ((c*d^2*(1 + m)*(2
+ m) + a*e^2*(2 + m + n)*(3 + m + n))*(g*x)^(1 + m)*(d + e*x)^n*Hypergeometric2F
1[1 + m, -n, 2 + m, -((e*x)/d)])/(e^2*g*(1 + m)*(2 + m + n)*(3 + m + n)*(1 + (e*
x)/d)^n)

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Rubi [A]  time = 0.302571, antiderivative size = 150, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{(g x)^{m+1} (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} \left (\frac{a}{m+1}+\frac{c d^2 (m+2)}{e^2 (m+n+2) (m+n+3)}\right ) \, _2F_1\left (m+1,-n;m+2;-\frac{e x}{d}\right )}{g}-\frac{c d (m+2) (g x)^{m+1} (d+e x)^{n+1}}{e^2 g (m+n+2) (m+n+3)}+\frac{c (g x)^{m+2} (d+e x)^{n+1}}{e g^2 (m+n+3)} \]

Antiderivative was successfully verified.

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

[Out]

-((c*d*(2 + m)*(g*x)^(1 + m)*(d + e*x)^(1 + n))/(e^2*g*(2 + m + n)*(3 + m + n)))
+ (c*(g*x)^(2 + m)*(d + e*x)^(1 + n))/(e*g^2*(3 + m + n)) + ((a/(1 + m) + (c*d^
2*(2 + m))/(e^2*(2 + m + n)*(3 + m + n)))*(g*x)^(1 + m)*(d + e*x)^n*Hypergeometr
ic2F1[1 + m, -n, 2 + m, -((e*x)/d)])/(g*(1 + (e*x)/d)^n)

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Rubi in Sympy [A]  time = 41.3975, size = 156, normalized size = 0.95 \[ \frac{c d^{2} \left (g x\right )^{m + 1} \left (1 + \frac{e x}{d}\right )^{- n} \left (d + e x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n - 2, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{e x}{d}} \right )}}{e^{2} g \left (m + 1\right )} - \frac{2 c d^{2} \left (g x\right )^{m + 1} \left (1 + \frac{e x}{d}\right )^{- n} \left (d + e x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n - 1, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{e x}{d}} \right )}}{e^{2} g \left (m + 1\right )} + \frac{\left (g x\right )^{m + 1} \left (1 + \frac{e x}{d}\right )^{- n} \left (d + e x\right )^{n} \left (a e^{2} + c d^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{e x}{d}} \right )}}{e^{2} g \left (m + 1\right )} \]

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]

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

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Mathematica [A]  time = 0.0993278, size = 84, normalized size = 0.51 \[ \frac{x (g x)^m (d+e x)^n \left (\frac{e x}{d}+1\right )^{-n} \left (a (m+3) \, _2F_1\left (m+1,-n;m+2;-\frac{e x}{d}\right )+c (m+1) x^2 \, _2F_1\left (m+3,-n;m+4;-\frac{e x}{d}\right )\right )}{(m+1) (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int \left ( gx \right ) ^{m} \left ( ex+d \right ) ^{n} \left ( c{x}^{2}+a \right ) \, dx \]

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. \[ \int{\left (c x^{2} + a\right )}{\left (e x + d\right )}^{n} \left (g x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^2 + a)*(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),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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