3.182 \(\int \frac{x^3}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=144 \[ -\frac{a x^2 (a+b x)}{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^3 (a+b x)}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^2 x (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^3 (a+b x) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(a^2*x*(a + b*x))/(b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (a*x^2*(a + b*x))/(2*b^2
*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (x^3*(a + b*x))/(3*b*Sqrt[a^2 + 2*a*b*x + b^2*
x^2]) - (a^3*(a + b*x)*Log[a + b*x])/(b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.142503, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{a x^2 (a+b x)}{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^3 (a+b x)}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^2 x (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^3 (a+b x) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(a^2*x*(a + b*x))/(b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (a*x^2*(a + b*x))/(2*b^2
*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (x^3*(a + b*x))/(3*b*Sqrt[a^2 + 2*a*b*x + b^2*
x^2]) - (a^3*(a + b*x)*Log[a + b*x])/(b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 21.993, size = 139, normalized size = 0.97 \[ - \frac{a^{3} \left (a + b x\right ) \log{\left (a + b x \right )}}{b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{a^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{4}} - \frac{a x^{2} \left (2 a + 2 b x\right )}{4 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{x^{3} \left (2 a + 2 b x\right )}{6 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/((b*x+a)**2)**(1/2),x)

[Out]

-a**3*(a + b*x)*log(a + b*x)/(b**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)) + a**2*sqrt
(a**2 + 2*a*b*x + b**2*x**2)/b**4 - a*x**2*(2*a + 2*b*x)/(4*b**2*sqrt(a**2 + 2*a
*b*x + b**2*x**2)) + x**3*(2*a + 2*b*x)/(6*b*sqrt(a**2 + 2*a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.0341207, size = 57, normalized size = 0.4 \[ \frac{(a+b x) \left (b x \left (6 a^2-3 a b x+2 b^2 x^2\right )-6 a^3 \log (a+b x)\right )}{6 b^4 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

((a + b*x)*(b*x*(6*a^2 - 3*a*b*x + 2*b^2*x^2) - 6*a^3*Log[a + b*x]))/(6*b^4*Sqrt
[(a + b*x)^2])

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Maple [A]  time = 0.01, size = 56, normalized size = 0.4 \[ -{\frac{ \left ( bx+a \right ) \left ( -2\,{b}^{3}{x}^{3}+3\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( bx+a \right ) -6\,{a}^{2}bx \right ) }{6\,{b}^{4}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/((b*x+a)^2)^(1/2),x)

[Out]

-1/6*(b*x+a)*(-2*b^3*x^3+3*a*b^2*x^2+6*a^3*ln(b*x+a)-6*a^2*b*x)/((b*x+a)^2)^(1/2
)/b^4

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Maxima [A]  time = 0.714077, size = 159, normalized size = 1.1 \[ -\frac{5 \, a^{3} b \log \left (x + \frac{a}{b}\right )}{3 \,{\left (b^{2}\right )}^{\frac{5}{2}}} + \frac{5 \, a^{2} x}{3 \,{\left (b^{2}\right )}^{\frac{3}{2}}} - \frac{5 \, a x^{2}}{6 \, \sqrt{b^{2}} b} + \frac{2 \, a^{3} \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{3 \, b^{3}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} x^{2}}{3 \, b^{2}} - \frac{2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}}{3 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt((b*x + a)^2),x, algorithm="maxima")

[Out]

-5/3*a^3*b*log(x + a/b)/(b^2)^(5/2) + 5/3*a^2*x/(b^2)^(3/2) - 5/6*a*x^2/(sqrt(b^
2)*b) + 2/3*a^3*sqrt(b^(-2))*log(x + a/b)/b^3 + 1/3*sqrt(b^2*x^2 + 2*a*b*x + a^2
)*x^2/b^2 - 2/3*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2/b^4

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Fricas [A]  time = 0.22162, size = 55, normalized size = 0.38 \[ \frac{2 \, b^{3} x^{3} - 3 \, a b^{2} x^{2} + 6 \, a^{2} b x - 6 \, a^{3} \log \left (b x + a\right )}{6 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt((b*x + a)^2),x, algorithm="fricas")

[Out]

1/6*(2*b^3*x^3 - 3*a*b^2*x^2 + 6*a^2*b*x - 6*a^3*log(b*x + a))/b^4

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Sympy [A]  time = 1.16244, size = 37, normalized size = 0.26 \[ - \frac{a^{3} \log{\left (a + b x \right )}}{b^{4}} + \frac{a^{2} x}{b^{3}} - \frac{a x^{2}}{2 b^{2}} + \frac{x^{3}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/((b*x+a)**2)**(1/2),x)

[Out]

-a**3*log(a + b*x)/b**4 + a**2*x/b**3 - a*x**2/(2*b**2) + x**3/(3*b)

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GIAC/XCAS [A]  time = 0.209696, size = 90, normalized size = 0.62 \[ -\frac{a^{3}{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (b x + a\right )}{b^{4}} + \frac{2 \, b^{2} x^{3}{\rm sign}\left (b x + a\right ) - 3 \, a b x^{2}{\rm sign}\left (b x + a\right ) + 6 \, a^{2} x{\rm sign}\left (b x + a\right )}{6 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt((b*x + a)^2),x, algorithm="giac")

[Out]

-a^3*ln(abs(b*x + a))*sign(b*x + a)/b^4 + 1/6*(2*b^2*x^3*sign(b*x + a) - 3*a*b*x
^2*sign(b*x + a) + 6*a^2*x*sign(b*x + a))/b^3