3.363 \(\int \frac{x^4 (d+e x)^n}{a+c x^2} \, dx\)

Optimal. Leaf size=250 \[ \frac{\left (c d^2-a e^2\right ) (d+e x)^{n+1}}{c^2 e^3 (n+1)}+\frac{(-a)^{3/2} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 c^2 (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{(-a)^{3/2} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 c^2 (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}-\frac{2 d (d+e x)^{n+2}}{c e^3 (n+2)}+\frac{(d+e x)^{n+3}}{c e^3 (n+3)} \]

[Out]

((c*d^2 - a*e^2)*(d + e*x)^(1 + n))/(c^2*e^3*(1 + n)) - (2*d*(d + e*x)^(2 + n))/
(c*e^3*(2 + n)) + (d + e*x)^(3 + n)/(c*e^3*(3 + n)) + ((-a)^(3/2)*(d + e*x)^(1 +
n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]
*e)])/(2*c^2*(Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) - ((-a)^(3/2)*(d + e*x)^(1 + n)*H
ypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])
/(2*c^2*(Sqrt[c]*d + Sqrt[-a]*e)*(1 + n))

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Rubi [A]  time = 0.696263, antiderivative size = 250, 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 \[ \frac{\left (c d^2-a e^2\right ) (d+e x)^{n+1}}{c^2 e^3 (n+1)}+\frac{(-a)^{3/2} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 c^2 (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{(-a)^{3/2} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 c^2 (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}-\frac{2 d (d+e x)^{n+2}}{c e^3 (n+2)}+\frac{(d+e x)^{n+3}}{c e^3 (n+3)} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(d + e*x)^n)/(a + c*x^2),x]

[Out]

((c*d^2 - a*e^2)*(d + e*x)^(1 + n))/(c^2*e^3*(1 + n)) - (2*d*(d + e*x)^(2 + n))/
(c*e^3*(2 + n)) + (d + e*x)^(3 + n)/(c*e^3*(3 + n)) + ((-a)^(3/2)*(d + e*x)^(1 +
n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]
*e)])/(2*c^2*(Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) - ((-a)^(3/2)*(d + e*x)^(1 + n)*H
ypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])
/(2*c^2*(Sqrt[c]*d + Sqrt[-a]*e)*(1 + n))

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Rubi in Sympy [A]  time = 89.9628, size = 204, normalized size = 0.82 \[ - \frac{2 d \left (d + e x\right )^{n + 2}}{c e^{3} \left (n + 2\right )} + \frac{\left (d + e x\right )^{n + 3}}{c e^{3} \left (n + 3\right )} - \frac{\left (- a\right )^{\frac{3}{2}} \left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}}} \right )}}{2 c^{2} \left (n + 1\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} + \frac{\left (- a\right )^{\frac{3}{2}} \left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}}} \right )}}{2 c^{2} \left (n + 1\right ) \left (\sqrt{c} d - e \sqrt{- a}\right )} - \frac{\left (d + e x\right )^{n + 1} \left (a e^{2} - c d^{2}\right )}{c^{2} e^{3} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(e*x+d)**n/(c*x**2+a),x)

[Out]

-2*d*(d + e*x)**(n + 2)/(c*e**3*(n + 2)) + (d + e*x)**(n + 3)/(c*e**3*(n + 3)) -
(-a)**(3/2)*(d + e*x)**(n + 1)*hyper((1, n + 1), (n + 2,), sqrt(c)*(d + e*x)/(s
qrt(c)*d + e*sqrt(-a)))/(2*c**2*(n + 1)*(sqrt(c)*d + e*sqrt(-a))) + (-a)**(3/2)*
(d + e*x)**(n + 1)*hyper((1, n + 1), (n + 2,), sqrt(c)*(d + e*x)/(sqrt(c)*d - e*
sqrt(-a)))/(2*c**2*(n + 1)*(sqrt(c)*d - e*sqrt(-a))) - (d + e*x)**(n + 1)*(a*e**
2 - c*d**2)/(c**2*e**3*(n + 1))

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Mathematica [C]  time = 0.915572, size = 354, normalized size = 1.42 \[ \frac{(d+e x)^n \left (\frac{i a^{3/2} \sqrt{c} e^3 \left (\left (\frac{\sqrt{c} (d+e x)}{e \left (\sqrt{c} x+i \sqrt{a}\right )}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} x e+i \sqrt{a} e}\right )-\left (\frac{\sqrt{c} (d+e x)}{e \left (\sqrt{c} x-i \sqrt{a}\right )}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac{\sqrt{c} d+i \sqrt{a} e}{i \sqrt{a} e-\sqrt{c} e x}\right )\right )}{n}-\frac{2 a c e^2 (d+e x)}{n+1}+\frac{2 c^2 \left (2 d^3 \left (\left (\frac{e x}{d}+1\right )^n-1\right )-2 d^2 e n x \left (\frac{e x}{d}+1\right )^n+e^3 \left (n^2+3 n+2\right ) x^3 \left (\frac{e x}{d}+1\right )^n+d e^2 n (n+1) x^2 \left (\frac{e x}{d}+1\right )^n\right ) \left (\frac{e x}{d}+1\right )^{-n}}{(n+1) (n+2) (n+3)}\right )}{2 c^3 e^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(d + e*x)^n)/(a + c*x^2),x]

[Out]

((d + e*x)^n*((-2*a*c*e^2*(d + e*x))/(1 + n) + (2*c^2*(-2*d^2*e*n*x*(1 + (e*x)/d
)^n + d*e^2*n*(1 + n)*x^2*(1 + (e*x)/d)^n + e^3*(2 + 3*n + n^2)*x^3*(1 + (e*x)/d
)^n + 2*d^3*(-1 + (1 + (e*x)/d)^n)))/((1 + n)*(2 + n)*(3 + n)*(1 + (e*x)/d)^n) +
(I*a^(3/2)*Sqrt[c]*e^3*(-(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) + Hypergeometric2F1[-n, -n, 1 - n, -((Sqrt[c]*d - I*Sqrt[a]*e)/(I*Sqr
t[a]*e + Sqrt[c]*e*x))]/((Sqrt[c]*(d + e*x))/(e*(I*Sqrt[a] + Sqrt[c]*x)))^n))/n)
)/(2*c^3*e^3)

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Maple [F]  time = 0.091, size = 0, normalized size = 0. \[ \int{\frac{{x}^{4} \left ( ex+d \right ) ^{n}}{c{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(e*x+d)^n/(c*x^2+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{n} x^{4}}{c x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^n*x^4/(c*x^2 + a),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^n*x^4/(c*x^2 + a),x, algorithm="fricas")

[Out]

integral((e*x + d)^n*x^4/(c*x^2 + a), 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(x**4*(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. \[ \int \frac{{\left (e x + d\right )}^{n} x^{4}}{c x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^n*x^4/(c*x^2 + a),x, algorithm="giac")

[Out]

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