Optimal. Leaf size=93 \[ \frac{d x \left (a+b x^2\right )^{p+1}}{b (2 p+3)}-\frac{x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} (a d-b c (2 p+3)) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )}{b (2 p+3)} \]
[Out]
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Rubi [A] time = 0.0961799, antiderivative size = 85, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c-\frac{a d}{2 b p+3 b}\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )+\frac{d x \left (a+b x^2\right )^{p+1}}{b (2 p+3)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^p*(c + d*x^2),x]
[Out]
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Rubi in Sympy [A] time = 12.7575, size = 73, normalized size = 0.78 \[ \frac{d x \left (a + b x^{2}\right )^{p + 1}}{b \left (2 p + 3\right )} - \frac{x \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p} \left (a d - b c \left (2 p + 3\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{b \left (2 p + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**p*(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.030472, size = 75, normalized size = 0.81 \[ \frac{1}{3} x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (3 c \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )+d x^2 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^p*(c + d*x^2),x]
[Out]
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Maple [F] time = 0.044, size = 0, normalized size = 0. \[ \int \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) \, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^p*(d*x^2+c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (d x^{2} + c\right )}{\left (b x^{2} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(b*x^2 + a)^p,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (d x^{2} + c\right )}{\left (b x^{2} + a\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(b*x^2 + a)^p,x, algorithm="fricas")
[Out]
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Sympy [A] time = 39.9354, size = 53, normalized size = 0.57 \[ a^{p} c x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )} + \frac{a^{p} d x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**p*(d*x**2+c),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (d x^{2} + c\right )}{\left (b x^{2} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(b*x^2 + a)^p,x, algorithm="giac")
[Out]