3.365 \(\int \frac{x^2 (d+e x)^n}{a+c x^2} \, dx\)

Optimal. Leaf size=194 \[ \frac{\sqrt{-a} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 c (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{\sqrt{-a} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 c (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}+\frac{(d+e x)^{n+1}}{c e (n+1)} \]

[Out]

(d + e*x)^(1 + n)/(c*e*(1 + n)) + (Sqrt[-a]*(d + e*x)^(1 + n)*Hypergeometric2F1[
1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(2*c*(Sqrt[c]*d
- Sqrt[-a]*e)*(1 + n)) - (Sqrt[-a]*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n,
2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(2*c*(Sqrt[c]*d + Sqrt[-a
]*e)*(1 + n))

_______________________________________________________________________________________

Rubi [A]  time = 0.470028, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\sqrt{-a} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 c (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{\sqrt{-a} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 c (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}+\frac{(d+e x)^{n+1}}{c e (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

(d + e*x)^(1 + n)/(c*e*(1 + n)) + (Sqrt[-a]*(d + e*x)^(1 + n)*Hypergeometric2F1[
1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(2*c*(Sqrt[c]*d
- Sqrt[-a]*e)*(1 + n)) - (Sqrt[-a]*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n,
2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(2*c*(Sqrt[c]*d + Sqrt[-a
]*e)*(1 + n))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 77.3805, size = 150, normalized size = 0.77 \[ - \frac{\sqrt{- a} \left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}}} \right )}}{2 c \left (n + 1\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} + \frac{\sqrt{- a} \left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}}} \right )}}{2 c \left (n + 1\right ) \left (\sqrt{c} d - e \sqrt{- a}\right )} + \frac{\left (d + e x\right )^{n + 1}}{c e \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.442627, size = 233, normalized size = 1.2 \[ \frac{(d+e x)^n \left (\frac{2 c (d+e x)}{n+1}-\frac{i \sqrt{a} \sqrt{c} e \left (\left (\frac{\sqrt{c} (d+e x)}{e \left (\sqrt{c} x+i \sqrt{a}\right )}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} x e+i \sqrt{a} e}\right )-\left (\frac{\sqrt{c} (d+e x)}{e \left (\sqrt{c} x-i \sqrt{a}\right )}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac{\sqrt{c} d+i \sqrt{a} e}{i \sqrt{a} e-\sqrt{c} e x}\right )\right )}{n}\right )}{2 c^2 e} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^n*((2*c*(d + e*x))/(1 + n) - (I*Sqrt[a]*Sqrt[c]*e*(-(Hypergeometric2F
1[-n, -n, 1 - n, (Sqrt[c]*d + I*Sqrt[a]*e)/(I*Sqrt[a]*e - Sqrt[c]*e*x)]/((Sqrt[c
]*(d + e*x))/(e*((-I)*Sqrt[a] + Sqrt[c]*x)))^n) + Hypergeometric2F1[-n, -n, 1 -
n, -((Sqrt[c]*d - I*Sqrt[a]*e)/(I*Sqrt[a]*e + Sqrt[c]*e*x))]/((Sqrt[c]*(d + e*x)
)/(e*(I*Sqrt[a] + Sqrt[c]*x)))^n))/n))/(2*c^2*e)

_______________________________________________________________________________________

Maple [F]  time = 0.073, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2} \left ( ex+d \right ) ^{n}}{c{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(e*x+d)^n/(c*x^2+a),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(e*x+d)**n/(c*x**2+a),x)

[Out]

Integral(x**2*(d + e*x)**n/(a + c*x**2), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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