3.1736 \(\int \left (a^2+2 a b x+b^2 x^2\right )^p \, dx\)

Optimal. Leaf size=34 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b (2 p+1)} \]

[Out]

((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^p)/(b*(1 + 2*p))

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Rubi [A]  time = 0.0229978, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b (2 p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^p,x]

[Out]

((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^p)/(b*(1 + 2*p))

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Rubi in Sympy [A]  time = 2.57058, size = 34, normalized size = 1. \[ \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{2 b \left (2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**p,x)

[Out]

(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**p/(2*b*(2*p + 1))

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Mathematica [A]  time = 0.0155313, size = 23, normalized size = 0.68 \[ \frac{(a+b x) \left ((a+b x)^2\right )^p}{2 b p+b} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^p,x]

[Out]

((a + b*x)*((a + b*x)^2)^p)/(b + 2*b*p)

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Maple [A]  time = 0.002, size = 35, normalized size = 1. \[{\frac{ \left ( bx+a \right ) \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{b \left ( 1+2\,p \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^p,x)

[Out]

(b*x+a)*(b^2*x^2+2*a*b*x+a^2)^p/b/(1+2*p)

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Maxima [A]  time = 0.758148, size = 34, normalized size = 1. \[ \frac{{\left (b x + a\right )}{\left (b x + a\right )}^{2 \, p}}{b{\left (2 \, p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^p,x, algorithm="maxima")

[Out]

(b*x + a)*(b*x + a)^(2*p)/(b*(2*p + 1))

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Fricas [A]  time = 0.219757, size = 43, normalized size = 1.26 \[ \frac{{\left (b x + a\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \, b p + b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^p,x, algorithm="fricas")

[Out]

(b*x + a)*(b^2*x^2 + 2*a*b*x + a^2)^p/(2*b*p + b)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**p,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.211098, size = 74, normalized size = 2.18 \[ \frac{b x e^{\left (p{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )\right )} + a e^{\left (p{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )\right )}}{2 \, b p + b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^p,x, algorithm="giac")

[Out]

(b*x*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2)) + a*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2)))/(2
*b*p + b)