Optimal. Leaf size=23 \[ \frac{\left (a+b x^2\right )^{p+1}}{2 b (p+1)} \]
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Rubi [A] time = 0.0156318, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\left (a+b x^2\right )^{p+1}}{2 b (p+1)} \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*x^2)^p,x]
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Rubi in Sympy [A] time = 2.6632, size = 15, normalized size = 0.65 \[ \frac{\left (a + b x^{2}\right )^{p + 1}}{2 b \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x**2+a)**p,x)
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Mathematica [A] time = 0.00581633, size = 22, normalized size = 0.96 \[ \frac{\left (a+b x^2\right )^{p+1}}{2 b p+2 b} \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*x^2)^p,x]
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Maple [A] time = 0.003, size = 22, normalized size = 1. \[{\frac{ \left ( b{x}^{2}+a \right ) ^{1+p}}{2\,b \left ( 1+p \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x^2+a)^p,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^p*x,x, algorithm="maxima")
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Fricas [A] time = 0.216218, size = 34, normalized size = 1.48 \[ \frac{{\left (b x^{2} + a\right )}{\left (b x^{2} + a\right )}^{p}}{2 \,{\left (b p + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^p*x,x, algorithm="fricas")
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Sympy [A] time = 2.13288, size = 97, normalized size = 4.22 \[ \begin{cases} \frac{x^{2}}{2 a} & \text{for}\: b = 0 \wedge p = -1 \\\frac{a^{p} x^{2}}{2} & \text{for}\: b = 0 \\\frac{\log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 b} + \frac{\log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 b} & \text{for}\: p = -1 \\\frac{a \left (a + b x^{2}\right )^{p}}{2 b p + 2 b} + \frac{b x^{2} \left (a + b x^{2}\right )^{p}}{2 b p + 2 b} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x**2+a)**p,x)
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GIAC/XCAS [A] time = 0.227714, size = 28, normalized size = 1.22 \[ \frac{{\left (b x^{2} + a\right )}^{p + 1}}{2 \, b{\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^p*x,x, algorithm="giac")
[Out]