3.266 \(\int \frac{\left (a+b x^3\right )^5}{x^7} \, dx\)

Optimal. Leaf size=66 \[ -\frac{a^5}{6 x^6}-\frac{5 a^4 b}{3 x^3}+10 a^3 b^2 \log (x)+\frac{10}{3} a^2 b^3 x^3+\frac{5}{6} a b^4 x^6+\frac{b^5 x^9}{9} \]

[Out]

-a^5/(6*x^6) - (5*a^4*b)/(3*x^3) + (10*a^2*b^3*x^3)/3 + (5*a*b^4*x^6)/6 + (b^5*x
^9)/9 + 10*a^3*b^2*Log[x]

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Rubi [A]  time = 0.0857663, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{a^5}{6 x^6}-\frac{5 a^4 b}{3 x^3}+10 a^3 b^2 \log (x)+\frac{10}{3} a^2 b^3 x^3+\frac{5}{6} a b^4 x^6+\frac{b^5 x^9}{9} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^5/x^7,x]

[Out]

-a^5/(6*x^6) - (5*a^4*b)/(3*x^3) + (10*a^2*b^3*x^3)/3 + (5*a*b^4*x^6)/6 + (b^5*x
^9)/9 + 10*a^3*b^2*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{5}}{6 x^{6}} - \frac{5 a^{4} b}{3 x^{3}} + \frac{10 a^{3} b^{2} \log{\left (x^{3} \right )}}{3} + \frac{10 a^{2} b^{3} x^{3}}{3} + \frac{5 a b^{4} \int ^{x^{3}} x\, dx}{3} + \frac{b^{5} x^{9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5/x**7,x)

[Out]

-a**5/(6*x**6) - 5*a**4*b/(3*x**3) + 10*a**3*b**2*log(x**3)/3 + 10*a**2*b**3*x**
3/3 + 5*a*b**4*Integral(x, (x, x**3))/3 + b**5*x**9/9

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Mathematica [A]  time = 0.0083586, size = 66, normalized size = 1. \[ -\frac{a^5}{6 x^6}-\frac{5 a^4 b}{3 x^3}+10 a^3 b^2 \log (x)+\frac{10}{3} a^2 b^3 x^3+\frac{5}{6} a b^4 x^6+\frac{b^5 x^9}{9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^5/x^7,x]

[Out]

-a^5/(6*x^6) - (5*a^4*b)/(3*x^3) + (10*a^2*b^3*x^3)/3 + (5*a*b^4*x^6)/6 + (b^5*x
^9)/9 + 10*a^3*b^2*Log[x]

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Maple [A]  time = 0.009, size = 57, normalized size = 0.9 \[ -{\frac{{a}^{5}}{6\,{x}^{6}}}-{\frac{5\,{a}^{4}b}{3\,{x}^{3}}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{3}}{3}}+{\frac{5\,a{b}^{4}{x}^{6}}{6}}+{\frac{{b}^{5}{x}^{9}}{9}}+10\,{a}^{3}{b}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5/x^7,x)

[Out]

-1/6*a^5/x^6-5/3*a^4*b/x^3+10/3*a^2*b^3*x^3+5/6*a*b^4*x^6+1/9*b^5*x^9+10*a^3*b^2
*ln(x)

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Maxima [A]  time = 1.44025, size = 80, normalized size = 1.21 \[ \frac{1}{9} \, b^{5} x^{9} + \frac{5}{6} \, a b^{4} x^{6} + \frac{10}{3} \, a^{2} b^{3} x^{3} + \frac{10}{3} \, a^{3} b^{2} \log \left (x^{3}\right ) - \frac{10 \, a^{4} b x^{3} + a^{5}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5/x^7,x, algorithm="maxima")

[Out]

1/9*b^5*x^9 + 5/6*a*b^4*x^6 + 10/3*a^2*b^3*x^3 + 10/3*a^3*b^2*log(x^3) - 1/6*(10
*a^4*b*x^3 + a^5)/x^6

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Fricas [A]  time = 0.212785, size = 82, normalized size = 1.24 \[ \frac{2 \, b^{5} x^{15} + 15 \, a b^{4} x^{12} + 60 \, a^{2} b^{3} x^{9} + 180 \, a^{3} b^{2} x^{6} \log \left (x\right ) - 30 \, a^{4} b x^{3} - 3 \, a^{5}}{18 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5/x^7,x, algorithm="fricas")

[Out]

1/18*(2*b^5*x^15 + 15*a*b^4*x^12 + 60*a^2*b^3*x^9 + 180*a^3*b^2*x^6*log(x) - 30*
a^4*b*x^3 - 3*a^5)/x^6

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Sympy [A]  time = 1.56623, size = 63, normalized size = 0.95 \[ 10 a^{3} b^{2} \log{\left (x \right )} + \frac{10 a^{2} b^{3} x^{3}}{3} + \frac{5 a b^{4} x^{6}}{6} + \frac{b^{5} x^{9}}{9} - \frac{a^{5} + 10 a^{4} b x^{3}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5/x**7,x)

[Out]

10*a**3*b**2*log(x) + 10*a**2*b**3*x**3/3 + 5*a*b**4*x**6/6 + b**5*x**9/9 - (a**
5 + 10*a**4*b*x**3)/(6*x**6)

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GIAC/XCAS [A]  time = 0.225594, size = 93, normalized size = 1.41 \[ \frac{1}{9} \, b^{5} x^{9} + \frac{5}{6} \, a b^{4} x^{6} + \frac{10}{3} \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2}{\rm ln}\left ({\left | x \right |}\right ) - \frac{30 \, a^{3} b^{2} x^{6} + 10 \, a^{4} b x^{3} + a^{5}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5/x^7,x, algorithm="giac")

[Out]

1/9*b^5*x^9 + 5/6*a*b^4*x^6 + 10/3*a^2*b^3*x^3 + 10*a^3*b^2*ln(abs(x)) - 1/6*(30
*a^3*b^2*x^6 + 10*a^4*b*x^3 + a^5)/x^6