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

Optimal. Leaf size=306 \[ \frac{(d+e x)^{n+1} \left (-\sqrt{-a} \sqrt{c} d e n+a e^2 (n+1)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 \sqrt{-a} c (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}-\frac{(d+e x)^{n+1} \left (\sqrt{-a} \sqrt{c} d e n+a e^2 (n+1)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 \sqrt{-a} c (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}-\frac{(d+e x)^{n+1} (a e+c d x)}{2 c \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 1.05898, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{(d+e x)^{n+1} \left (-\sqrt{-a} \sqrt{c} d e n+a e^2 (n+1)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 \sqrt{-a} c (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}-\frac{(d+e x)^{n+1} \left (\sqrt{-a} \sqrt{c} d e n+a e^2 (n+1)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 \sqrt{-a} c (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}-\frac{(d+e x)^{n+1} (a e+c d x)}{2 c \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 161.51, size = 382, normalized size = 1.25 \[ - \frac{\left (d + e x\right )^{n + 1} \left (a e + c d x\right )}{2 c \left (a + c x^{2}\right ) \left (a e^{2} + c d^{2}\right )} - \frac{\left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}}} \right )}}{2 c \sqrt{- a} \left (n + 1\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} + \frac{\left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}}} \right )}}{2 c \sqrt{- a} \left (n + 1\right ) \left (\sqrt{c} d - e \sqrt{- a}\right )} - \frac{\left (d + e x\right )^{n + 1} \left (a \sqrt{c} d e n + \sqrt{- a} \left (a e^{2} \left (- n + 1\right ) + c d^{2}\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}}} \right )}}{4 a c \left (n + 1\right ) \left (a e^{2} + c d^{2}\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} + \frac{\left (d + e x\right )^{n + 1} \left (- a \sqrt{c} d e n + \sqrt{- a} \left (a e^{2} \left (- n + 1\right ) + c d^{2}\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}}} \right )}}{4 a c \left (n + 1\right ) \left (a e^{2} + c d^{2}\right ) \left (\sqrt{c} d - e \sqrt{- a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.128038, size = 0, normalized size = 0. \[ \int \frac{x^2 (d+e x)^n}{\left (a+c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2} \left ( ex+d \right ) ^{n}}{ \left ( c{x}^{2}+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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