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

Optimal. Leaf size=167 \[ \frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 \sqrt{-a} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 \sqrt{-a} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )} \]

[Out]

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

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Rubi [A]  time = 0.237423, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 \sqrt{-a} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 \sqrt{-a} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^n/(a + c*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 42.5933, size = 131, normalized size = 0.78 \[ - \frac{\left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}}} \right )}}{2 \sqrt{- a} \left (n + 1\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} + \frac{\left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}}} \right )}}{2 \sqrt{- a} \left (n + 1\right ) \left (\sqrt{c} d - e \sqrt{- a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.122853, size = 210, normalized size = 1.26 \[ -\frac{i (d+e x)^n \left (\left (\frac{\sqrt{c} (d+e x)}{e \left (\sqrt{c} x-i \sqrt{a}\right )}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac{\sqrt{c} d+i \sqrt{a} e}{i \sqrt{a} e-\sqrt{c} e x}\right )-\left (\frac{\sqrt{c} (d+e x)}{e \left (\sqrt{c} x+i \sqrt{a}\right )}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} x e+i \sqrt{a} e}\right )\right )}{2 \sqrt{a} \sqrt{c} n} \]

Antiderivative was successfully verified.

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

[Out]

((-I/2)*(d + e*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 - 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))/(Sqrt[a]*
Sqrt[c]*n)

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{n}}{c{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^n/(c*x^2+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^n/(c*x^2 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x + d)^n/(c*x^2 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^n/(c*x^2 + a), x)