3.215 \(\int \frac{x^2}{(a+b x)^7} \, dx\)

Optimal. Leaf size=47 \[ -\frac{a^2}{6 b^3 (a+b x)^6}+\frac{2 a}{5 b^3 (a+b x)^5}-\frac{1}{4 b^3 (a+b x)^4} \]

[Out]

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

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Rubi [A]  time = 0.0503404, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a^2}{6 b^3 (a+b x)^6}+\frac{2 a}{5 b^3 (a+b x)^5}-\frac{1}{4 b^3 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x)^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.58587, size = 42, normalized size = 0.89 \[ - \frac{a^{2}}{6 b^{3} \left (a + b x\right )^{6}} + \frac{2 a}{5 b^{3} \left (a + b x\right )^{5}} - \frac{1}{4 b^{3} \left (a + b x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x+a)**7,x)

[Out]

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

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Mathematica [A]  time = 0.0126672, size = 31, normalized size = 0.66 \[ -\frac{a^2+6 a b x+15 b^2 x^2}{60 b^3 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b*x)^7,x]

[Out]

-(a^2 + 6*a*b*x + 15*b^2*x^2)/(60*b^3*(a + b*x)^6)

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Maple [A]  time = 0.009, size = 42, normalized size = 0.9 \[ -{\frac{{a}^{2}}{6\,{b}^{3} \left ( bx+a \right ) ^{6}}}+{\frac{2\,a}{5\,{b}^{3} \left ( bx+a \right ) ^{5}}}-{\frac{1}{4\,{b}^{3} \left ( bx+a \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x+a)^7,x)

[Out]

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

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Maxima [A]  time = 1.34074, size = 117, normalized size = 2.49 \[ -\frac{15 \, b^{2} x^{2} + 6 \, a b x + a^{2}}{60 \,{\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x + a)^7,x, algorithm="maxima")

[Out]

-1/60*(15*b^2*x^2 + 6*a*b*x + a^2)/(b^9*x^6 + 6*a*b^8*x^5 + 15*a^2*b^7*x^4 + 20*
a^3*b^6*x^3 + 15*a^4*b^5*x^2 + 6*a^5*b^4*x + a^6*b^3)

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Fricas [A]  time = 0.203165, size = 117, normalized size = 2.49 \[ -\frac{15 \, b^{2} x^{2} + 6 \, a b x + a^{2}}{60 \,{\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x + a)^7,x, algorithm="fricas")

[Out]

-1/60*(15*b^2*x^2 + 6*a*b*x + a^2)/(b^9*x^6 + 6*a*b^8*x^5 + 15*a^2*b^7*x^4 + 20*
a^3*b^6*x^3 + 15*a^4*b^5*x^2 + 6*a^5*b^4*x + a^6*b^3)

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Sympy [A]  time = 2.3188, size = 92, normalized size = 1.96 \[ - \frac{a^{2} + 6 a b x + 15 b^{2} x^{2}}{60 a^{6} b^{3} + 360 a^{5} b^{4} x + 900 a^{4} b^{5} x^{2} + 1200 a^{3} b^{6} x^{3} + 900 a^{2} b^{7} x^{4} + 360 a b^{8} x^{5} + 60 b^{9} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x+a)**7,x)

[Out]

-(a**2 + 6*a*b*x + 15*b**2*x**2)/(60*a**6*b**3 + 360*a**5*b**4*x + 900*a**4*b**5
*x**2 + 1200*a**3*b**6*x**3 + 900*a**2*b**7*x**4 + 360*a*b**8*x**5 + 60*b**9*x**
6)

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GIAC/XCAS [A]  time = 0.22031, size = 39, normalized size = 0.83 \[ -\frac{15 \, b^{2} x^{2} + 6 \, a b x + a^{2}}{60 \,{\left (b x + a\right )}^{6} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x + a)^7,x, algorithm="giac")

[Out]

-1/60*(15*b^2*x^2 + 6*a*b*x + a^2)/((b*x + a)^6*b^3)