3.213 \(\int \frac{x^4}{(a+b x)^7} \, dx\)

Optimal. Leaf size=35 \[ \frac{x^5}{30 a^2 (a+b x)^5}+\frac{x^5}{6 a (a+b x)^6} \]

[Out]

x^5/(6*a*(a + b*x)^6) + x^5/(30*a^2*(a + b*x)^5)

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Rubi [A]  time = 0.0235098, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x^5}{30 a^2 (a+b x)^5}+\frac{x^5}{6 a (a+b x)^6} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x)^7,x]

[Out]

x^5/(6*a*(a + b*x)^6) + x^5/(30*a^2*(a + b*x)^5)

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Rubi in Sympy [A]  time = 4.14424, size = 27, normalized size = 0.77 \[ \frac{x^{5}}{6 a \left (a + b x\right )^{6}} + \frac{x^{5}}{30 a^{2} \left (a + b x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x+a)**7,x)

[Out]

x**5/(6*a*(a + b*x)**6) + x**5/(30*a**2*(a + b*x)**5)

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Mathematica [A]  time = 0.0159163, size = 53, normalized size = 1.51 \[ -\frac{a^4+6 a^3 b x+15 a^2 b^2 x^2+20 a b^3 x^3+15 b^4 x^4}{30 b^5 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x)^7,x]

[Out]

-(a^4 + 6*a^3*b*x + 15*a^2*b^2*x^2 + 20*a*b^3*x^3 + 15*b^4*x^4)/(30*b^5*(a + b*x
)^6)

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Maple [B]  time = 0.008, size = 72, normalized size = 2.1 \[ -{\frac{{a}^{4}}{6\,{b}^{5} \left ( bx+a \right ) ^{6}}}+{\frac{4\,a}{3\,{b}^{5} \left ( bx+a \right ) ^{3}}}-{\frac{1}{2\, \left ( bx+a \right ) ^{2}{b}^{5}}}+{\frac{4\,{a}^{3}}{5\,{b}^{5} \left ( bx+a \right ) ^{5}}}-{\frac{3\,{a}^{2}}{2\,{b}^{5} \left ( bx+a \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x+a)^7,x)

[Out]

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

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Maxima [A]  time = 1.32237, size = 147, normalized size = 4.2 \[ -\frac{15 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x^{2} + 6 \, a^{3} b x + a^{4}}{30 \,{\left (b^{11} x^{6} + 6 \, a b^{10} x^{5} + 15 \, a^{2} b^{9} x^{4} + 20 \, a^{3} b^{8} x^{3} + 15 \, a^{4} b^{7} x^{2} + 6 \, a^{5} b^{6} x + a^{6} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x + a)^7,x, algorithm="maxima")

[Out]

-1/30*(15*b^4*x^4 + 20*a*b^3*x^3 + 15*a^2*b^2*x^2 + 6*a^3*b*x + a^4)/(b^11*x^6 +
6*a*b^10*x^5 + 15*a^2*b^9*x^4 + 20*a^3*b^8*x^3 + 15*a^4*b^7*x^2 + 6*a^5*b^6*x +
a^6*b^5)

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Fricas [A]  time = 0.203021, size = 147, normalized size = 4.2 \[ -\frac{15 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x^{2} + 6 \, a^{3} b x + a^{4}}{30 \,{\left (b^{11} x^{6} + 6 \, a b^{10} x^{5} + 15 \, a^{2} b^{9} x^{4} + 20 \, a^{3} b^{8} x^{3} + 15 \, a^{4} b^{7} x^{2} + 6 \, a^{5} b^{6} x + a^{6} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x + a)^7,x, algorithm="fricas")

[Out]

-1/30*(15*b^4*x^4 + 20*a*b^3*x^3 + 15*a^2*b^2*x^2 + 6*a^3*b*x + a^4)/(b^11*x^6 +
6*a*b^10*x^5 + 15*a^2*b^9*x^4 + 20*a^3*b^8*x^3 + 15*a^4*b^7*x^2 + 6*a^5*b^6*x +
a^6*b^5)

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Sympy [A]  time = 2.56095, size = 116, normalized size = 3.31 \[ - \frac{a^{4} + 6 a^{3} b x + 15 a^{2} b^{2} x^{2} + 20 a b^{3} x^{3} + 15 b^{4} x^{4}}{30 a^{6} b^{5} + 180 a^{5} b^{6} x + 450 a^{4} b^{7} x^{2} + 600 a^{3} b^{8} x^{3} + 450 a^{2} b^{9} x^{4} + 180 a b^{10} x^{5} + 30 b^{11} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x+a)**7,x)

[Out]

-(a**4 + 6*a**3*b*x + 15*a**2*b**2*x**2 + 20*a*b**3*x**3 + 15*b**4*x**4)/(30*a**
6*b**5 + 180*a**5*b**6*x + 450*a**4*b**7*x**2 + 600*a**3*b**8*x**3 + 450*a**2*b*
*9*x**4 + 180*a*b**10*x**5 + 30*b**11*x**6)

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GIAC/XCAS [A]  time = 0.209, size = 69, normalized size = 1.97 \[ -\frac{15 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x^{2} + 6 \, a^{3} b x + a^{4}}{30 \,{\left (b x + a\right )}^{6} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x + a)^7,x, algorithm="giac")

[Out]

-1/30*(15*b^4*x^4 + 20*a*b^3*x^3 + 15*a^2*b^2*x^2 + 6*a^3*b*x + a^4)/((b*x + a)^
6*b^5)