3.188 \(\int x (a+b x)^{p/2} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 (a+b x)^{\frac{p+4}{2}}}{b^2 (p+4)}-\frac{2 a (a+b x)^{\frac{p+2}{2}}}{b^2 (p+2)} \]

[Out]

(-2*a*(a + b*x)^((2 + p)/2))/(b^2*(2 + p)) + (2*(a + b*x)^((4 + p)/2))/(b^2*(4 +
p))

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Rubi [A]  time = 0.0360198, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2 (a+b x)^{\frac{p+4}{2}}}{b^2 (p+4)}-\frac{2 a (a+b x)^{\frac{p+2}{2}}}{b^2 (p+2)} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x)^(p/2),x]

[Out]

(-2*a*(a + b*x)^((2 + p)/2))/(b^2*(2 + p)) + (2*(a + b*x)^((4 + p)/2))/(b^2*(4 +
p))

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Rubi in Sympy [A]  time = 3.7542, size = 37, normalized size = 0.77 \[ - \frac{2 a \left (a + b x\right )^{\frac{p}{2} + 1}}{b^{2} \left (p + 2\right )} + \frac{2 \left (a + b x\right )^{\frac{p}{2} + 2}}{b^{2} \left (p + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*((b*x+a)**(1/2))**p,x)

[Out]

-2*a*(a + b*x)**(p/2 + 1)/(b**2*(p + 2)) + 2*(a + b*x)**(p/2 + 2)/(b**2*(p + 4))

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Mathematica [A]  time = 0.0207298, size = 38, normalized size = 0.79 \[ \frac{2 (a+b x)^{\frac{p}{2}+1} (b (p+2) x-2 a)}{b^2 (p+2) (p+4)} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x)^(p/2),x]

[Out]

(2*(a + b*x)^(1 + p/2)*(-2*a + b*(2 + p)*x))/(b^2*(2 + p)*(4 + p))

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Maple [A]  time = 0.003, size = 43, normalized size = 0.9 \[ -2\,{\frac{ \left ( \sqrt{bx+a} \right ) ^{p} \left ( -xpb-2\,bx+2\,a \right ) \left ( bx+a \right ) }{{b}^{2} \left ({p}^{2}+6\,p+8 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*((b*x+a)^(1/2))^p,x)

[Out]

-2*((b*x+a)^(1/2))^p*(-b*p*x-2*b*x+2*a)*(b*x+a)/b^2/(p^2+6*p+8)

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Maxima [A]  time = 1.36924, size = 61, normalized size = 1.27 \[ \frac{2 \,{\left (b^{2}{\left (p + 2\right )} x^{2} + a b p x - 2 \, a^{2}\right )}{\left (b x + a\right )}^{\frac{1}{2} \, p}}{{\left (p^{2} + 6 \, p + 8\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b^2*(p + 2)*x^2 + a*b*p*x - 2*a^2)*(b*x + a)^(1/2*p)/((p^2 + 6*p + 8)*b^2)

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Fricas [A]  time = 0.234875, size = 78, normalized size = 1.62 \[ \frac{2 \,{\left (a b p x +{\left (b^{2} p + 2 \, b^{2}\right )} x^{2} - 2 \, a^{2}\right )} \sqrt{b x + a}^{p}}{b^{2} p^{2} + 6 \, b^{2} p + 8 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a*b*p*x + (b^2*p + 2*b^2)*x^2 - 2*a^2)*sqrt(b*x + a)^p/(b^2*p^2 + 6*b^2*p + 8
*b^2)

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Sympy [A]  time = 0.919282, size = 216, normalized size = 4.5 \[ \begin{cases} \frac{a^{\frac{p}{2}} x^{2}}{2} & \text{for}\: b = 0 \\\frac{a \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac{a}{a b^{2} + b^{3} x} + \frac{b x \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} & \text{for}\: p = -4 \\- \frac{a \log{\left (\frac{a}{b} + x \right )}}{b^{2}} + \frac{x}{b} & \text{for}\: p = -2 \\- \frac{4 a^{2} \left (a + b x\right )^{\frac{p}{2}}}{b^{2} p^{2} + 6 b^{2} p + 8 b^{2}} + \frac{2 a b p x \left (a + b x\right )^{\frac{p}{2}}}{b^{2} p^{2} + 6 b^{2} p + 8 b^{2}} + \frac{2 b^{2} p x^{2} \left (a + b x\right )^{\frac{p}{2}}}{b^{2} p^{2} + 6 b^{2} p + 8 b^{2}} + \frac{4 b^{2} x^{2} \left (a + b x\right )^{\frac{p}{2}}}{b^{2} p^{2} + 6 b^{2} p + 8 b^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*((b*x+a)**(1/2))**p,x)

[Out]

Piecewise((a**(p/2)*x**2/2, Eq(b, 0)), (a*log(a/b + x)/(a*b**2 + b**3*x) + a/(a*
b**2 + b**3*x) + b*x*log(a/b + x)/(a*b**2 + b**3*x), Eq(p, -4)), (-a*log(a/b + x
)/b**2 + x/b, Eq(p, -2)), (-4*a**2*(a + b*x)**(p/2)/(b**2*p**2 + 6*b**2*p + 8*b*
*2) + 2*a*b*p*x*(a + b*x)**(p/2)/(b**2*p**2 + 6*b**2*p + 8*b**2) + 2*b**2*p*x**2
*(a + b*x)**(p/2)/(b**2*p**2 + 6*b**2*p + 8*b**2) + 4*b**2*x**2*(a + b*x)**(p/2)
/(b**2*p**2 + 6*b**2*p + 8*b**2), True))

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GIAC/XCAS [A]  time = 0.203847, size = 122, normalized size = 2.54 \[ \frac{2 \,{\left (b^{2} p x^{2} e^{\left (\frac{1}{2} \, p{\rm ln}\left (b x + a\right )\right )} + a b p x e^{\left (\frac{1}{2} \, p{\rm ln}\left (b x + a\right )\right )} + 2 \, b^{2} x^{2} e^{\left (\frac{1}{2} \, p{\rm ln}\left (b x + a\right )\right )} - 2 \, a^{2} e^{\left (\frac{1}{2} \, p{\rm ln}\left (b x + a\right )\right )}\right )}}{b^{2} p^{2} + 6 \, b^{2} p + 8 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b^2*p*x^2*e^(1/2*p*ln(b*x + a)) + a*b*p*x*e^(1/2*p*ln(b*x + a)) + 2*b^2*x^2*e
^(1/2*p*ln(b*x + a)) - 2*a^2*e^(1/2*p*ln(b*x + a)))/(b^2*p^2 + 6*b^2*p + 8*b^2)