3.181 \(\int \frac{x^4}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=182 \[ \frac{x^4 (a+b x)}{4 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a x^3 (a+b x)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^2 x^2 (a+b x)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^4 (a+b x) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^3 x (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

-((a^3*x*(a + b*x))/(b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) + (a^2*x^2*(a + b*x))/(
2*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (a*x^3*(a + b*x))/(3*b^2*Sqrt[a^2 + 2*a*b
*x + b^2*x^2]) + (x^4*(a + b*x))/(4*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (a^4*(a +
b*x)*Log[a + b*x])/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.180366, antiderivative size = 182, 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 \[ \frac{x^4 (a+b x)}{4 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a x^3 (a+b x)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^2 x^2 (a+b x)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^4 (a+b x) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^3 x (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

-((a^3*x*(a + b*x))/(b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) + (a^2*x^2*(a + b*x))/(
2*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (a*x^3*(a + b*x))/(3*b^2*Sqrt[a^2 + 2*a*b
*x + b^2*x^2]) + (x^4*(a + b*x))/(4*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (a^4*(a +
b*x)*Log[a + b*x])/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 30.2263, size = 180, normalized size = 0.99 \[ \frac{a^{4} \left (a + b x\right ) \log{\left (a + b x \right )}}{b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{a^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{5}} + \frac{a^{2} x^{2} \left (2 a + 2 b x\right )}{4 b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{a x^{3} \left (2 a + 2 b x\right )}{6 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{x^{4} \left (2 a + 2 b x\right )}{8 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/((b*x+a)**2)**(1/2),x)

[Out]

a**4*(a + b*x)*log(a + b*x)/(b**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)) - a**3*sqrt(
a**2 + 2*a*b*x + b**2*x**2)/b**5 + a**2*x**2*(2*a + 2*b*x)/(4*b**3*sqrt(a**2 + 2
*a*b*x + b**2*x**2)) - a*x**3*(2*a + 2*b*x)/(6*b**2*sqrt(a**2 + 2*a*b*x + b**2*x
**2)) + x**4*(2*a + 2*b*x)/(8*b*sqrt(a**2 + 2*a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.0439541, size = 68, normalized size = 0.37 \[ \frac{(a+b x) \left (12 a^4 \log (a+b x)+b x \left (-12 a^3+6 a^2 b x-4 a b^2 x^2+3 b^3 x^3\right )\right )}{12 b^5 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

((a + b*x)*(b*x*(-12*a^3 + 6*a^2*b*x - 4*a*b^2*x^2 + 3*b^3*x^3) + 12*a^4*Log[a +
b*x]))/(12*b^5*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.012, size = 67, normalized size = 0.4 \[{\frac{ \left ( bx+a \right ) \left ( 3\,{b}^{4}{x}^{4}-4\,{x}^{3}a{b}^{3}+6\,{x}^{2}{a}^{2}{b}^{2}+12\,{a}^{4}\ln \left ( bx+a \right ) -12\,x{a}^{3}b \right ) }{12\,{b}^{5}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/((b*x+a)^2)^(1/2),x)

[Out]

1/12*(b*x+a)*(3*b^4*x^4-4*x^3*a*b^3+6*x^2*a^2*b^2+12*a^4*ln(b*x+a)-12*x*a^3*b)/(
(b*x+a)^2)^(1/2)/b^5

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Maxima [A]  time = 0.740903, size = 201, normalized size = 1.1 \[ \frac{13 \, a^{4} \log \left (x + \frac{a}{b}\right )}{6 \,{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{13 \, a^{3} x}{6 \,{\left (b^{2}\right )}^{\frac{3}{2}} b} + \frac{13 \, a^{2} x^{2}}{12 \, \sqrt{b^{2}} b^{2}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} x^{3}}{4 \, b^{2}} - \frac{7 \, a^{4} \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{6 \, b^{4}} - \frac{7 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a x^{2}}{12 \, b^{3}} + \frac{7 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3}}{6 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt((b*x + a)^2),x, algorithm="maxima")

[Out]

13/6*a^4*log(x + a/b)/(b^2)^(5/2) - 13/6*a^3*x/((b^2)^(3/2)*b) + 13/12*a^2*x^2/(
sqrt(b^2)*b^2) + 1/4*sqrt(b^2*x^2 + 2*a*b*x + a^2)*x^3/b^2 - 7/6*a^4*sqrt(b^(-2)
)*log(x + a/b)/b^4 - 7/12*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*x^2/b^3 + 7/6*sqrt(b^2
*x^2 + 2*a*b*x + a^2)*a^3/b^5

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Fricas [A]  time = 0.225762, size = 70, normalized size = 0.38 \[ \frac{3 \, b^{4} x^{4} - 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a^{3} b x + 12 \, a^{4} \log \left (b x + a\right )}{12 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt((b*x + a)^2),x, algorithm="fricas")

[Out]

1/12*(3*b^4*x^4 - 4*a*b^3*x^3 + 6*a^2*b^2*x^2 - 12*a^3*b*x + 12*a^4*log(b*x + a)
)/b^5

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Sympy [A]  time = 1.19583, size = 49, normalized size = 0.27 \[ \frac{a^{4} \log{\left (a + b x \right )}}{b^{5}} - \frac{a^{3} x}{b^{4}} + \frac{a^{2} x^{2}}{2 b^{3}} - \frac{a x^{3}}{3 b^{2}} + \frac{x^{4}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/((b*x+a)**2)**(1/2),x)

[Out]

a**4*log(a + b*x)/b**5 - a**3*x/b**4 + a**2*x**2/(2*b**3) - a*x**3/(3*b**2) + x*
*4/(4*b)

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GIAC/XCAS [A]  time = 0.207681, size = 112, normalized size = 0.62 \[ \frac{a^{4}{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (b x + a\right )}{b^{5}} + \frac{3 \, b^{3} x^{4}{\rm sign}\left (b x + a\right ) - 4 \, a b^{2} x^{3}{\rm sign}\left (b x + a\right ) + 6 \, a^{2} b x^{2}{\rm sign}\left (b x + a\right ) - 12 \, a^{3} x{\rm sign}\left (b x + a\right )}{12 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt((b*x + a)^2),x, algorithm="giac")

[Out]

a^4*ln(abs(b*x + a))*sign(b*x + a)/b^5 + 1/12*(3*b^3*x^4*sign(b*x + a) - 4*a*b^2
*x^3*sign(b*x + a) + 6*a^2*b*x^2*sign(b*x + a) - 12*a^3*x*sign(b*x + a))/b^4