Optimal. Leaf size=177 \[ \frac{a^2 d e \left (a+b x^2\right )^{p+1}}{b^3 (p+1)}-\frac{2 a d e \left (a+b x^2\right )^{p+2}}{b^3 (p+2)}+\frac{d e \left (a+b x^2\right )^{p+3}}{b^3 (p+3)}-\frac{x^5 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (5 a e^2-b d^2 (2 p+7)\right ) \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right )}{5 b (2 p+7)}+\frac{e^2 x^5 \left (a+b x^2\right )^{p+1}}{b (2 p+7)} \]
[Out]
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Rubi [A] time = 0.324414, antiderivative size = 169, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{a^2 d e \left (a+b x^2\right )^{p+1}}{b^3 (p+1)}-\frac{2 a d e \left (a+b x^2\right )^{p+2}}{b^3 (p+2)}+\frac{d e \left (a+b x^2\right )^{p+3}}{b^3 (p+3)}+\frac{1}{5} x^5 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (d^2-\frac{5 a e^2}{2 b p+7 b}\right ) \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right )+\frac{e^2 x^5 \left (a+b x^2\right )^{p+1}}{b (2 p+7)} \]
Antiderivative was successfully verified.
[In] Int[x^4*(d + e*x)^2*(a + b*x^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 45.1335, size = 151, normalized size = 0.85 \[ \frac{a^{2} d e \left (a + b x^{2}\right )^{p + 1}}{b^{3} \left (p + 1\right )} - \frac{2 a d e \left (a + b x^{2}\right )^{p + 2}}{b^{3} \left (p + 2\right )} + \frac{d^{2} x^{5} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{5} + \frac{e^{2} x^{7} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{7} + \frac{d e \left (a + b x^{2}\right )^{p + 3}}{b^{3} \left (p + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(e*x+d)**2*(b*x**2+a)**p,x)
[Out]
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Mathematica [A] time = 0.291189, size = 229, normalized size = 1.29 \[ \frac{\left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (5 e \left (7 d \left (2 a^3 \left (\left (\frac{b x^2}{a}+1\right )^p-1\right )-2 a^2 b p x^2 \left (\frac{b x^2}{a}+1\right )^p+b^3 \left (p^2+3 p+2\right ) x^6 \left (\frac{b x^2}{a}+1\right )^p+a b^2 p (p+1) x^4 \left (\frac{b x^2}{a}+1\right )^p\right )+b^3 e \left (p^3+6 p^2+11 p+6\right ) x^7 \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right )\right )+7 b^3 d^2 \left (p^3+6 p^2+11 p+6\right ) x^5 \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right )\right )}{35 b^3 (p+1) (p+2) (p+3)} \]
Antiderivative was successfully verified.
[In] Integrate[x^4*(d + e*x)^2*(a + b*x^2)^p,x]
[Out]
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Maple [F] time = 0.085, size = 0, normalized size = 0. \[ \int{x}^{4} \left ( ex+d \right ) ^{2} \left ( b{x}^{2}+a \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(e*x+d)^2*(b*x^2+a)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}^{2}{\left (b x^{2} + a\right )}^{p} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2*(b*x^2 + a)^p*x^4,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{2} x^{6} + 2 \, d e x^{5} + d^{2} x^{4}\right )}{\left (b x^{2} + a\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2*(b*x^2 + a)^p*x^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 151.911, size = 1047, normalized size = 5.92 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(e*x+d)**2*(b*x**2+a)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}^{2}{\left (b x^{2} + a\right )}^{p} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2*(b*x^2 + a)^p*x^4,x, algorithm="giac")
[Out]