3.199 \(\int \frac{x^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=205 \[ -\frac{10 a^2}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a (a+b x) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^5}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a^4}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 a^3}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(-10*a^2)/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + a^5/(4*b^6*(a + b*x)^3*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) - (5*a^4)/(3*b^6*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
) + (5*a^3)/(b^6*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (x*(a + b*x))/(b^5*S
qrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*a*(a + b*x)*Log[a + b*x])/(b^6*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])

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Rubi [A]  time = 0.241021, antiderivative size = 205, 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 \[ -\frac{10 a^2}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a (a+b x) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^5}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a^4}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 a^3}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(-10*a^2)/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + a^5/(4*b^6*(a + b*x)^3*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) - (5*a^4)/(3*b^6*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
) + (5*a^3)/(b^6*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (x*(a + b*x))/(b^5*S
qrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*a*(a + b*x)*Log[a + b*x])/(b^6*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 24.5336, size = 199, normalized size = 0.97 \[ - \frac{5 a \left (a + b x\right ) \log{\left (a + b x \right )}}{b^{6} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{x^{5} \left (2 a + 2 b x\right )}{8 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{5 x^{4}}{12 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{5 x^{3} \left (2 a + 2 b x\right )}{12 b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{5 x^{2}}{2 b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{5 \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

-5*a*(a + b*x)*log(a + b*x)/(b**6*sqrt(a**2 + 2*a*b*x + b**2*x**2)) - x**5*(2*a
+ 2*b*x)/(8*b*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)) - 5*x**4/(12*b**2*(a**2 + 2*a
*b*x + b**2*x**2)**(3/2)) - 5*x**3*(2*a + 2*b*x)/(12*b**3*(a**2 + 2*a*b*x + b**2
*x**2)**(3/2)) - 5*x**2/(2*b**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)) + 5*sqrt(a**2
+ 2*a*b*x + b**2*x**2)/b**6

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Mathematica [A]  time = 0.0595892, size = 93, normalized size = 0.45 \[ \frac{-77 a^5-248 a^4 b x-252 a^3 b^2 x^2-48 a^2 b^3 x^3+48 a b^4 x^4-60 a (a+b x)^4 \log (a+b x)+12 b^5 x^5}{12 b^6 (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(-77*a^5 - 248*a^4*b*x - 252*a^3*b^2*x^2 - 48*a^2*b^3*x^3 + 48*a*b^4*x^4 + 12*b^
5*x^5 - 60*a*(a + b*x)^4*Log[a + b*x])/(12*b^6*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.01, size = 145, normalized size = 0.7 \[ -{\frac{ \left ( 60\,\ln \left ( bx+a \right ){x}^{4}a{b}^{4}-12\,{b}^{5}{x}^{5}+240\,\ln \left ( bx+a \right ){x}^{3}{a}^{2}{b}^{3}-48\,a{b}^{4}{x}^{4}+360\,\ln \left ( bx+a \right ){x}^{2}{a}^{3}{b}^{2}+48\,{a}^{2}{b}^{3}{x}^{3}+240\,\ln \left ( bx+a \right ) x{a}^{4}b+252\,{a}^{3}{b}^{2}{x}^{2}+60\,\ln \left ( bx+a \right ){a}^{5}+248\,{a}^{4}bx+77\,{a}^{5} \right ) \left ( bx+a \right ) }{12\,{b}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/12*(60*ln(b*x+a)*x^4*a*b^4-12*b^5*x^5+240*ln(b*x+a)*x^3*a^2*b^3-48*a*b^4*x^4+
360*ln(b*x+a)*x^2*a^3*b^2+48*a^2*b^3*x^3+240*ln(b*x+a)*x*a^4*b+252*a^3*b^2*x^2+6
0*ln(b*x+a)*a^5+248*a^4*b*x+77*a^5)*(b*x+a)/b^6/((b*x+a)^2)^(5/2)

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Maxima [A]  time = 0.75791, size = 153, normalized size = 0.75 \[ \frac{12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5}}{12 \,{\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}} - \frac{5 \, a \log \left (b x + a\right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

1/12*(12*b^5*x^5 + 48*a*b^4*x^4 - 48*a^2*b^3*x^3 - 252*a^3*b^2*x^2 - 248*a^4*b*x
- 77*a^5)/(b^10*x^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6) - 5*
a*log(b*x + a)/b^6

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Fricas [A]  time = 0.22903, size = 201, normalized size = 0.98 \[ \frac{12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5} - 60 \,{\left (a b^{4} x^{4} + 4 \, a^{2} b^{3} x^{3} + 6 \, a^{3} b^{2} x^{2} + 4 \, a^{4} b x + a^{5}\right )} \log \left (b x + a\right )}{12 \,{\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")

[Out]

1/12*(12*b^5*x^5 + 48*a*b^4*x^4 - 48*a^2*b^3*x^3 - 252*a^3*b^2*x^2 - 248*a^4*b*x
- 77*a^5 - 60*(a*b^4*x^4 + 4*a^2*b^3*x^3 + 6*a^3*b^2*x^2 + 4*a^4*b*x + a^5)*log
(b*x + a))/(b^10*x^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral(x**5/((a + b*x)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.558379, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x