3.198 \(\int \frac{x^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=244 \[ \frac{15 a^2 (a+b x) \log (a+b x)}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a x (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^2 (a+b x)}{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^6}{4 b^7 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^5}{b^7 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 a^4}{2 b^7 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{20 a^3}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(20*a^3)/(b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - a^6/(4*b^7*(a + b*x)^3*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) + (2*a^5)/(b^7*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(15*a^4)/(2*b^7*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*a*x*(a + b*x))/(b
^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (x^2*(a + b*x))/(2*b^5*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (15*a^2*(a + b*x)*Log[a + b*x])/(b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.292032, antiderivative size = 244, 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 \[ \frac{15 a^2 (a+b x) \log (a+b x)}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a x (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^2 (a+b x)}{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^6}{4 b^7 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^5}{b^7 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 a^4}{2 b^7 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{20 a^3}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(20*a^3)/(b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - a^6/(4*b^7*(a + b*x)^3*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) + (2*a^5)/(b^7*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(15*a^4)/(2*b^7*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*a*x*(a + b*x))/(b
^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (x^2*(a + b*x))/(2*b^5*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (15*a^2*(a + b*x)*Log[a + b*x])/(b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 32.9729, size = 238, normalized size = 0.98 \[ \frac{15 a^{2} \left (a + b x\right ) \log{\left (a + b x \right )}}{b^{7} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{15 a \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{7}} - \frac{x^{6} \left (2 a + 2 b x\right )}{8 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{x^{5}}{2 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{5 x^{4} \left (2 a + 2 b x\right )}{8 b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{5 x^{3}}{b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{15 x^{2} \left (2 a + 2 b x\right )}{4 b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

15*a**2*(a + b*x)*log(a + b*x)/(b**7*sqrt(a**2 + 2*a*b*x + b**2*x**2)) - 15*a*sq
rt(a**2 + 2*a*b*x + b**2*x**2)/b**7 - x**6*(2*a + 2*b*x)/(8*b*(a**2 + 2*a*b*x +
b**2*x**2)**(5/2)) - x**5/(2*b**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)) - 5*x**4*
(2*a + 2*b*x)/(8*b**3*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)) - 5*x**3/(b**4*sqrt(a
**2 + 2*a*b*x + b**2*x**2)) + 15*x**2*(2*a + 2*b*x)/(4*b**5*sqrt(a**2 + 2*a*b*x
+ b**2*x**2))

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Mathematica [A]  time = 0.0627359, size = 106, normalized size = 0.43 \[ \frac{57 a^6+168 a^5 b x+132 a^4 b^2 x^2-32 a^3 b^3 x^3-68 a^2 b^4 x^4+60 a^2 (a+b x)^4 \log (a+b x)-12 a b^5 x^5+2 b^6 x^6}{4 b^7 (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(57*a^6 + 168*a^5*b*x + 132*a^4*b^2*x^2 - 32*a^3*b^3*x^3 - 68*a^2*b^4*x^4 - 12*a
*b^5*x^5 + 2*b^6*x^6 + 60*a^2*(a + b*x)^4*Log[a + b*x])/(4*b^7*(a + b*x)^3*Sqrt[
(a + b*x)^2])

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Maple [A]  time = 0.022, size = 158, normalized size = 0.7 \[{\frac{ \left ( 2\,{b}^{6}{x}^{6}+60\,\ln \left ( bx+a \right ){x}^{4}{a}^{2}{b}^{4}-12\,{x}^{5}a{b}^{5}+240\,\ln \left ( bx+a \right ){x}^{3}{a}^{3}{b}^{3}-68\,{x}^{4}{a}^{2}{b}^{4}+360\,\ln \left ( bx+a \right ){x}^{2}{a}^{4}{b}^{2}-32\,{x}^{3}{a}^{3}{b}^{3}+240\,\ln \left ( bx+a \right ) x{a}^{5}b+132\,{x}^{2}{a}^{4}{b}^{2}+60\,\ln \left ( bx+a \right ){a}^{6}+168\,x{a}^{5}b+57\,{a}^{6} \right ) \left ( bx+a \right ) }{4\,{b}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/4*(2*b^6*x^6+60*ln(b*x+a)*x^4*a^2*b^4-12*x^5*a*b^5+240*ln(b*x+a)*x^3*a^3*b^3-6
8*x^4*a^2*b^4+360*ln(b*x+a)*x^2*a^4*b^2-32*x^3*a^3*b^3+240*ln(b*x+a)*x*a^5*b+132
*x^2*a^4*b^2+60*ln(b*x+a)*a^6+168*x*a^5*b+57*a^6)*(b*x+a)/b^7/((b*x+a)^2)^(5/2)

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Maxima [A]  time = 0.764227, size = 170, normalized size = 0.7 \[ \frac{2 \, b^{6} x^{6} - 12 \, a b^{5} x^{5} - 68 \, a^{2} b^{4} x^{4} - 32 \, a^{3} b^{3} x^{3} + 132 \, a^{4} b^{2} x^{2} + 168 \, a^{5} b x + 57 \, a^{6}}{4 \,{\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}} + \frac{15 \, a^{2} \log \left (b x + a\right )}{b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

1/4*(2*b^6*x^6 - 12*a*b^5*x^5 - 68*a^2*b^4*x^4 - 32*a^3*b^3*x^3 + 132*a^4*b^2*x^
2 + 168*a^5*b*x + 57*a^6)/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9*x^2 + 4*a^3*b^8*x
+ a^4*b^7) + 15*a^2*log(b*x + a)/b^7

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Fricas [A]  time = 0.227531, size = 219, normalized size = 0.9 \[ \frac{2 \, b^{6} x^{6} - 12 \, a b^{5} x^{5} - 68 \, a^{2} b^{4} x^{4} - 32 \, a^{3} b^{3} x^{3} + 132 \, a^{4} b^{2} x^{2} + 168 \, a^{5} b x + 57 \, a^{6} + 60 \,{\left (a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{3} + 6 \, a^{4} b^{2} x^{2} + 4 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right )}{4 \,{\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")

[Out]

1/4*(2*b^6*x^6 - 12*a*b^5*x^5 - 68*a^2*b^4*x^4 - 32*a^3*b^3*x^3 + 132*a^4*b^2*x^
2 + 168*a^5*b*x + 57*a^6 + 60*(a^2*b^4*x^4 + 4*a^3*b^3*x^3 + 6*a^4*b^2*x^2 + 4*a
^5*b*x + a^6)*log(b*x + a))/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9*x^2 + 4*a^3*b^8
*x + a^4*b^7)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral(x**6/((a + b*x)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.561622, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x