Optimal. Leaf size=297 \[ -\frac{(d+e x)^{n+1} \left (\sqrt{-a} \sqrt{c} d e n+a e^2 (n+2)+2 c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 c^{3/2} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}+\frac{(d+e x)^{n+1} \left (\sqrt{-a} d e n-\frac{a e^2 (n+2)+2 c d^2}{\sqrt{c}}\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 c (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}+\frac{a (d-e x) (d+e x)^{n+1}}{2 c \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.769574, antiderivative size = 297, 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 \[ -\frac{(d+e x)^{n+1} \left (\sqrt{-a} \sqrt{c} d e n+a e^2 (n+2)+2 c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 c^{3/2} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}+\frac{(d+e x)^{n+1} \left (\sqrt{-a} d e n-\frac{a e^2 (n+2)+2 c d^2}{\sqrt{c}}\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 c (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}+\frac{a (d-e x) (d+e x)^{n+1}}{2 c \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(d + e*x)^n)/(a + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 165.105, size = 354, normalized size = 1.19 \[ \frac{a \left (d - e x\right ) \left (d + e x\right )^{n + 1}}{2 c \left (a + c x^{2}\right ) \left (a e^{2} + c d^{2}\right )} - \frac{e n \left (d + e x\right )^{n + 1} \left (a e - \sqrt{c} d \sqrt{- a}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}}} \right )}}{4 c^{\frac{3}{2}} \left (n + 1\right ) \left (a e^{2} + c d^{2}\right ) \left (\sqrt{c} d - e \sqrt{- a}\right )} - \frac{e n \left (d + e x\right )^{n + 1} \left (a e + \sqrt{c} d \sqrt{- a}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}}} \right )}}{4 c^{\frac{3}{2}} \left (n + 1\right ) \left (a e^{2} + c d^{2}\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} - \frac{\left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}}} \right )}}{2 c^{\frac{3}{2}} \left (n + 1\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} - \frac{\left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}}} \right )}}{2 c^{\frac{3}{2}} \left (n + 1\right ) \left (\sqrt{c} d - e \sqrt{- a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(e*x+d)**n/(c*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.140881, size = 0, normalized size = 0. \[ \int \frac{x^3 (d+e x)^n}{\left (a+c x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(x^3*(d + e*x)^n)/(a + c*x^2)^2,x]
[Out]
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Maple [F] time = 0.087, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3} \left ( ex+d \right ) ^{n}}{ \left ( c{x}^{2}+a \right ) ^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(e*x+d)^n/(c*x^2+a)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{n} x^{3}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^n*x^3/(c*x^2 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{n} x^{3}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^n*x^3/(c*x^2 + a)^2,x, algorithm="fricas")
[Out]
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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**3*(e*x+d)**n/(c*x**2+a)**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{n} x^{3}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^n*x^3/(c*x^2 + a)^2,x, algorithm="giac")
[Out]