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

Optimal. Leaf size=513 \[ -\frac{c (d+e x)^{n+1} (a e+c d x)}{2 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac{e (d+e x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{e x}{d}+1\right )}{a^2 d^2 (n+1)}-\frac{c (d+e x)^{n+1} \left (\sqrt{-a} \sqrt{c} d e n+a e^2 (1-n)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 (-a)^{5/2} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}+\frac{c (d+e x)^{n+1} \left (-\sqrt{-a} \sqrt{c} d e n+a e^2 (1-n)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 (-a)^{5/2} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}-\frac{c (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 (-a)^{5/2} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}+\frac{c (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 (-a)^{5/2} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )} \]

[Out]

-(c*(a*e + c*d*x)*(d + e*x)^(1 + n))/(2*a^2*(c*d^2 + a*e^2)*(a + c*x^2)) - (c*(d
+ e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*
d - Sqrt[-a]*e)])/(2*(-a)^(5/2)*(Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) - (c*(c*d^2 +
a*e^2*(1 - n) + Sqrt[-a]*Sqrt[c]*d*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*(-a)^(5/2)*(Sqrt[
c]*d - Sqrt[-a]*e)*(c*d^2 + a*e^2)*(1 + n)) + (c*(d + e*x)^(1 + n)*Hypergeometri
c2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(2*(-a)^(5/
2)*(Sqrt[c]*d + Sqrt[-a]*e)*(1 + n)) + (c*(c*d^2 + a*e^2*(1 - n) - Sqrt[-a]*Sqrt
[c]*d*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*(-a)^(5/2)*(Sqrt[c]*d + Sqrt[-a]*e)*(c*d^2 + a
*e^2)*(1 + n)) + (e*(d + e*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (e*
x)/d])/(a^2*d^2*(1 + n))

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Rubi [A]  time = 1.5029, antiderivative size = 513, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{c (d+e x)^{n+1} (a e+c d x)}{2 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac{e (d+e x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{e x}{d}+1\right )}{a^2 d^2 (n+1)}-\frac{c (d+e x)^{n+1} \left (\sqrt{-a} \sqrt{c} d e n+a e^2 (1-n)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 (-a)^{5/2} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}+\frac{c (d+e x)^{n+1} \left (-\sqrt{-a} \sqrt{c} d e n+a e^2 (1-n)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 (-a)^{5/2} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}-\frac{c (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 (-a)^{5/2} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}+\frac{c (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 (-a)^{5/2} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(c*(a*e + c*d*x)*(d + e*x)^(1 + n))/(2*a^2*(c*d^2 + a*e^2)*(a + c*x^2)) - (c*(d
+ e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*
d - Sqrt[-a]*e)])/(2*(-a)^(5/2)*(Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) - (c*(c*d^2 +
a*e^2*(1 - n) + Sqrt[-a]*Sqrt[c]*d*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*(-a)^(5/2)*(Sqrt[
c]*d - Sqrt[-a]*e)*(c*d^2 + a*e^2)*(1 + n)) + (c*(d + e*x)^(1 + n)*Hypergeometri
c2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(2*(-a)^(5/
2)*(Sqrt[c]*d + Sqrt[-a]*e)*(1 + n)) + (c*(c*d^2 + a*e^2*(1 - n) - Sqrt[-a]*Sqrt
[c]*d*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*(-a)^(5/2)*(Sqrt[c]*d + Sqrt[-a]*e)*(c*d^2 + a
*e^2)*(1 + n)) + (e*(d + e*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (e*
x)/d])/(a^2*d^2*(1 + n))

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Rubi in Sympy [A]  time = 175.627, size = 422, normalized size = 0.82 \[ \frac{c \left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}}} \right )}}{2 \left (- a\right )^{\frac{5}{2}} \left (n + 1\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} - \frac{c \left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}}} \right )}}{2 \left (- a\right )^{\frac{5}{2}} \left (n + 1\right ) \left (\sqrt{c} d - e \sqrt{- a}\right )} - \frac{c \left (d + e x\right )^{n + 1} \left (a e + c d x\right )}{2 a^{2} \left (a + c x^{2}\right ) \left (a e^{2} + c d^{2}\right )} + \frac{e \left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{e x}{d}} \right )}}{a^{2} d^{2} \left (n + 1\right )} - \frac{c \left (d + e x\right )^{n + 1} \left (a \sqrt{c} d e n + \sqrt{- a} \left (a e^{2} \left (- n + 1\right ) + c d^{2}\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}}} \right )}}{4 a^{3} \left (n + 1\right ) \left (a e^{2} + c d^{2}\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} + \frac{c \left (d + e x\right )^{n + 1} \left (- a \sqrt{c} d e n + \sqrt{- a} \left (a e^{2} \left (- n + 1\right ) + c d^{2}\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}}} \right )}}{4 a^{3} \left (n + 1\right ) \left (a e^{2} + c d^{2}\right ) \left (\sqrt{c} d - e \sqrt{- a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*(d + e*x)**(n + 1)*hyper((1, n + 1), (n + 2,), sqrt(c)*(d + e*x)/(sqrt(c)*d +
e*sqrt(-a)))/(2*(-a)**(5/2)*(n + 1)*(sqrt(c)*d + e*sqrt(-a))) - c*(d + e*x)**(n
+ 1)*hyper((1, n + 1), (n + 2,), sqrt(c)*(d + e*x)/(sqrt(c)*d - e*sqrt(-a)))/(2*
(-a)**(5/2)*(n + 1)*(sqrt(c)*d - e*sqrt(-a))) - c*(d + e*x)**(n + 1)*(a*e + c*d*
x)/(2*a**2*(a + c*x**2)*(a*e**2 + c*d**2)) + e*(d + e*x)**(n + 1)*hyper((2, n +
1), (n + 2,), 1 + e*x/d)/(a**2*d**2*(n + 1)) - c*(d + e*x)**(n + 1)*(a*sqrt(c)*d
*e*n + sqrt(-a)*(a*e**2*(-n + 1) + c*d**2))*hyper((1, n + 1), (n + 2,), sqrt(c)*
(d + e*x)/(sqrt(c)*d + e*sqrt(-a)))/(4*a**3*(n + 1)*(a*e**2 + c*d**2)*(sqrt(c)*d
+ e*sqrt(-a))) + c*(d + e*x)**(n + 1)*(-a*sqrt(c)*d*e*n + sqrt(-a)*(a*e**2*(-n
+ 1) + c*d**2))*hyper((1, n + 1), (n + 2,), sqrt(c)*(d + e*x)/(sqrt(c)*d - e*sqr
t(-a)))/(4*a**3*(n + 1)*(a*e**2 + c*d**2)*(sqrt(c)*d - e*sqrt(-a)))

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Mathematica [A]  time = 0.121062, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^n}{x^2 \left (a+c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.106, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{n}}{{x}^{2} \left ( c{x}^{2}+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{n}}{c^{2} x^{6} + 2 \, a c x^{4} + a^{2} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x + d)^n/(c^2*x^6 + 2*a*c*x^4 + a^2*x^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((e*x+d)**n/x**2/(c*x**2+a)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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