3.200 \(\int \frac{x^4}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=171 \[ -\frac{3 a^2}{b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{4 a}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^4}{4 b^5 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{4 a^3}{3 b^5 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.209664, antiderivative size = 171, 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 \[ -\frac{3 a^2}{b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{4 a}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^4}{4 b^5 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{4 a^3}{3 b^5 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.0363, size = 162, normalized size = 0.95 \[ - \frac{x^{4} \left (2 a + 2 b x\right )}{8 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{x^{3}}{3 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{x^{2} \left (2 a + 2 b x\right )}{4 b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{x}{b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{\left (a + b x\right ) \log{\left (a + b x \right )}}{b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0503004, size = 73, normalized size = 0.43 \[ \frac{a \left (25 a^3+88 a^2 b x+108 a b^2 x^2+48 b^3 x^3\right )+12 (a+b x)^4 \log (a+b x)}{12 b^5 (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(a*(25*a^3 + 88*a^2*b*x + 108*a*b^2*x^2 + 48*b^3*x^3) + 12*(a + b*x)^4*Log[a + b
*x])/(12*b^5*(a + b*x)^3*Sqrt[(a + b*x)^2])

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 123, normalized size = 0.7 \[{\frac{ \left ( 12\,\ln \left ( bx+a \right ){x}^{4}{b}^{4}+48\,\ln \left ( bx+a \right ){x}^{3}a{b}^{3}+72\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}+48\,{x}^{3}a{b}^{3}+48\,\ln \left ( bx+a \right ) x{a}^{3}b+108\,{x}^{2}{a}^{2}{b}^{2}+12\,{a}^{4}\ln \left ( bx+a \right ) +88\,x{a}^{3}b+25\,{a}^{4} \right ) \left ( bx+a \right ) }{12\,{b}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/12*(12*ln(b*x+a)*x^4*b^4+48*ln(b*x+a)*x^3*a*b^3+72*ln(b*x+a)*x^2*a^2*b^2+48*x^
3*a*b^3+48*ln(b*x+a)*x*a^3*b+108*x^2*a^2*b^2+12*a^4*ln(b*x+a)+88*x*a^3*b+25*a^4)
*(b*x+a)/b^5/((b*x+a)^2)^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.830752, size = 124, normalized size = 0.73 \[ \frac{48 \, a b^{3} x^{3} + 108 \, a^{2} b^{2} x^{2} + 88 \, a^{3} b x + 25 \, a^{4}}{12 \,{\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}} + \frac{\log \left (b x + a\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

1/12*(48*a*b^3*x^3 + 108*a^2*b^2*x^2 + 88*a^3*b*x + 25*a^4)/(b^9*x^4 + 4*a*b^8*x
^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5) + log(b*x + a)/b^5

_______________________________________________________________________________________

Fricas [A]  time = 0.23905, size = 171, normalized size = 1. \[ \frac{48 \, a b^{3} x^{3} + 108 \, a^{2} b^{2} x^{2} + 88 \, a^{3} b x + 25 \, a^{4} + 12 \,{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{12 \,{\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")

[Out]

1/12*(48*a*b^3*x^3 + 108*a^2*b^2*x^2 + 88*a^3*b*x + 25*a^4 + 12*(b^4*x^4 + 4*a*b
^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4)*log(b*x + a))/(b^9*x^4 + 4*a*b^8*x^3 +
6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral(x**4/((a + b*x)**2)**(5/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.551313, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x