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

Optimal. Leaf size=279 \[ \frac{e n \left (\sqrt{-a} e+\sqrt{c} d\right ) (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 \sqrt{-a} \sqrt{c} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}+\frac{e n \left (\sqrt{-a} \sqrt{c} d+a e\right ) (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 a \sqrt{c} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}-\frac{(d-e x) (d+e x)^{n+1}}{2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]

[Out]

-((d - e*x)*(d + e*x)^(1 + n))/(2*(c*d^2 + a*e^2)*(a + c*x^2)) + (e*(Sqrt[c]*d +
Sqrt[-a]*e)*n*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d
+ e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(4*Sqrt[-a]*Sqrt[c]*(Sqrt[c]*d - Sqrt[-a]*e)*
(c*d^2 + a*e^2)*(1 + n)) + (e*(Sqrt[-a]*Sqrt[c]*d + a*e)*n*(d + e*x)^(1 + n)*Hyp
ergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(
4*a*Sqrt[c]*(Sqrt[c]*d + Sqrt[-a]*e)*(c*d^2 + a*e^2)*(1 + n))

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Rubi [A]  time = 0.663582, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{e n \left (\sqrt{-a} e+\sqrt{c} d\right ) (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 \sqrt{-a} \sqrt{c} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}+\frac{e n \left (\sqrt{-a} \sqrt{c} d+a e\right ) (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 a \sqrt{c} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}-\frac{(d-e x) (d+e x)^{n+1}}{2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((d - e*x)*(d + e*x)^(1 + n))/(2*(c*d^2 + a*e^2)*(a + c*x^2)) + (e*(Sqrt[c]*d +
Sqrt[-a]*e)*n*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d
+ e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(4*Sqrt[-a]*Sqrt[c]*(Sqrt[c]*d - Sqrt[-a]*e)*
(c*d^2 + a*e^2)*(1 + n)) + (e*(Sqrt[-a]*Sqrt[c]*d + a*e)*n*(d + e*x)^(1 + n)*Hyp
ergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(
4*a*Sqrt[c]*(Sqrt[c]*d + Sqrt[-a]*e)*(c*d^2 + a*e^2)*(1 + n))

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Rubi in Sympy [A]  time = 104.559, size = 224, normalized size = 0.8 \[ - \frac{\left (d - e x\right ) \left (d + e x\right )^{n + 1}}{2 \left (a + c x^{2}\right ) \left (a e^{2} + c d^{2}\right )} + \frac{e n \left (d + e x\right )^{n + 1} \left (a e - \sqrt{c} d \sqrt{- a}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}}} \right )}}{4 a \sqrt{c} \left (n + 1\right ) \left (a e^{2} + c d^{2}\right ) \left (\sqrt{c} d - e \sqrt{- a}\right )} + \frac{e n \left (d + e x\right )^{n + 1} \left (a e + \sqrt{c} d \sqrt{- a}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}}} \right )}}{4 a \sqrt{c} \left (n + 1\right ) \left (a e^{2} + c d^{2}\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d - e*x)*(d + e*x)**(n + 1)/(2*(a + c*x**2)*(a*e**2 + c*d**2)) + e*n*(d + e*x)
**(n + 1)*(a*e - sqrt(c)*d*sqrt(-a))*hyper((1, n + 1), (n + 2,), sqrt(c)*(d + e*
x)/(sqrt(c)*d - e*sqrt(-a)))/(4*a*sqrt(c)*(n + 1)*(a*e**2 + c*d**2)*(sqrt(c)*d -
e*sqrt(-a))) + e*n*(d + e*x)**(n + 1)*(a*e + sqrt(c)*d*sqrt(-a))*hyper((1, n +
1), (n + 2,), sqrt(c)*(d + e*x)/(sqrt(c)*d + e*sqrt(-a)))/(4*a*sqrt(c)*(n + 1)*(
a*e**2 + c*d**2)*(sqrt(c)*d + e*sqrt(-a)))

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Mathematica [A]  time = 0.097271, size = 0, normalized size = 0. \[ \int \frac{x (d+e x)^n}{\left (a+c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.08, size = 0, normalized size = 0. \[ \int{\frac{x \left ( ex+d \right ) ^{n}}{ \left ( c{x}^{2}+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(e*x+d)^n/(c*x^2+a)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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