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

Optimal. Leaf size=207 \[ \frac{c (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 (-a)^{3/2} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{c (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 (-a)^{3/2} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}+\frac{e (d+e x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{e x}{d}+1\right )}{a d^2 (n+1)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.468437, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{c (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 (-a)^{3/2} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{c (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 (-a)^{3/2} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}+\frac{e (d+e x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{e x}{d}+1\right )}{a d^2 (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 72.9256, size = 165, normalized size = 0.8 \[ - \frac{c \left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}}} \right )}}{2 \left (- a\right )^{\frac{3}{2}} \left (n + 1\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} + \frac{c \left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}}} \right )}}{2 \left (- a\right )^{\frac{3}{2}} \left (n + 1\right ) \left (\sqrt{c} d - e \sqrt{- a}\right )} + \frac{e \left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{e x}{d}} \right )}}{a d^{2} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**n/x**2/(c*x**2+a),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)**(3/2)*(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)**(3/2)*(n + 1)*(sqrt(c)*d - e*sqrt(-a))) + e*(d + e*x)**(n + 1)*hyper((2,
n + 1), (n + 2,), 1 + e*x/d)/(a*d**2*(n + 1))

_______________________________________________________________________________________

Mathematica [C]  time = 0.584545, size = 263, normalized size = 1.27 \[ \frac{(d+e x)^n \left (\frac{2 \left (\frac{d}{e x}+1\right )^{-n} \, _2F_1\left (1-n,-n;2-n;-\frac{d}{e x}\right )}{(n-1) x}+\frac{i \sqrt{c} \left (\left (\frac{\sqrt{c} (d+e x)}{e \left (\sqrt{c} x-i \sqrt{a}\right )}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac{\sqrt{c} d+i \sqrt{a} e}{i \sqrt{a} e-\sqrt{c} e x}\right )-\left (\frac{\sqrt{c} (d+e x)}{e \left (\sqrt{c} x+i \sqrt{a}\right )}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} x e+i \sqrt{a} e}\right )\right )}{\sqrt{a} n}\right )}{2 a} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.081, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{n}}{{x}^{2} \left ( c{x}^{2}+a \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^n/x^2/(c*x^2+a),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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