3.364 \(\int \frac{x^3 (d+e x)^n}{a+c x^2} \, dx\)

Optimal. Leaf size=209 \[ \frac{a (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 c^{3/2} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}+\frac{a (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 c^{3/2} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}-\frac{d (d+e x)^{n+1}}{c e^2 (n+1)}+\frac{(d+e x)^{n+2}}{c e^2 (n+2)} \]

[Out]

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

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Rubi [A]  time = 0.40901, antiderivative size = 209, 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 \[ \frac{a (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 c^{3/2} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}+\frac{a (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 c^{3/2} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}-\frac{d (d+e x)^{n+1}}{c e^2 (n+1)}+\frac{(d+e x)^{n+2}}{c e^2 (n+2)} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(d + e*x)^n)/(a + c*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 59.8406, size = 167, normalized size = 0.8 \[ \frac{a \left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}}} \right )}}{2 c^{\frac{3}{2}} \left (n + 1\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} + \frac{a \left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}}} \right )}}{2 c^{\frac{3}{2}} \left (n + 1\right ) \left (\sqrt{c} d - e \sqrt{- a}\right )} - \frac{d \left (d + e x\right )^{n + 1}}{c e^{2} \left (n + 1\right )} + \frac{\left (d + e x\right )^{n + 2}}{c e^{2} \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(e*x+d)**n/(c*x**2+a),x)

[Out]

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

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Mathematica [C]  time = 1.1137, size = 275, normalized size = 1.32 \[ \frac{(d+e x)^n \left (-a \left (n^2+3 n+2\right ) \left (\frac{\sqrt{c} (d+e x)}{e \left (\sqrt{c} x-i \sqrt{a}\right )}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac{\sqrt{c} d+i \sqrt{a} e}{i \sqrt{a} e-\sqrt{c} e x}\right )-a \left (n^2+3 n+2\right ) \left (\frac{\sqrt{c} (d+e x)}{e \left (\sqrt{c} x+i \sqrt{a}\right )}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} x e+i \sqrt{a} e}\right )+\frac{2 c d^2 n \left (\left (\frac{e x}{d}+1\right )^{-n}-1\right )}{e^2}+\frac{2 c d n^2 x}{e}+2 c n (n+1) x^2\right )}{2 c^2 n (n+1) (n+2)} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(d + e*x)^n)/(a + c*x^2),x]

[Out]

((d + e*x)^n*((2*c*d*n^2*x)/e + 2*c*n*(1 + n)*x^2 + (2*c*d^2*n*(-1 + (1 + (e*x)/
d)^(-n)))/e^2 - (a*(2 + 3*n + n^2)*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 - (a*(2 + 3*n + n^2)*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*S
qrt[a] + Sqrt[c]*x)))^n))/(2*c^2*n*(1 + n)*(2 + n))

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3} \left ( ex+d \right ) ^{n}}{c{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(e*x+d)^n/(c*x^2+a),x)

[Out]

int(x^3*(e*x+d)^n/(c*x^2+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{n} x^{3}}{c x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^n*x^3/(c*x^2 + a),x, algorithm="maxima")

[Out]

integrate((e*x + d)^n*x^3/(c*x^2 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{n} x^{3}}{c x^{2} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^n*x^3/(c*x^2 + a),x, algorithm="fricas")

[Out]

integral((e*x + d)^n*x^3/(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**3*(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. \[ \int \frac{{\left (e x + d\right )}^{n} x^{3}}{c x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^n*x^3/(c*x^2 + a),x, algorithm="giac")

[Out]

integrate((e*x + d)^n*x^3/(c*x^2 + a), x)