3.2325 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^{10}}{x^4} \, dx\)

Optimal. Leaf size=131 \[ -\frac{a^{10}}{3 x^3}-\frac{15 a^9 b}{4 x^{8/3}}-\frac{135 a^8 b^2}{7 x^{7/3}}-\frac{60 a^7 b^3}{x^2}-\frac{126 a^6 b^4}{x^{5/3}}-\frac{189 a^5 b^5}{x^{4/3}}-\frac{210 a^4 b^6}{x}-\frac{180 a^3 b^7}{x^{2/3}}-\frac{135 a^2 b^8}{\sqrt [3]{x}}+10 a b^9 \log (x)+3 b^{10} \sqrt [3]{x} \]

[Out]

-a^10/(3*x^3) - (15*a^9*b)/(4*x^(8/3)) - (135*a^8*b^2)/(7*x^(7/3)) - (60*a^7*b^3
)/x^2 - (126*a^6*b^4)/x^(5/3) - (189*a^5*b^5)/x^(4/3) - (210*a^4*b^6)/x - (180*a
^3*b^7)/x^(2/3) - (135*a^2*b^8)/x^(1/3) + 3*b^10*x^(1/3) + 10*a*b^9*Log[x]

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Rubi [A]  time = 0.175049, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{a^{10}}{3 x^3}-\frac{15 a^9 b}{4 x^{8/3}}-\frac{135 a^8 b^2}{7 x^{7/3}}-\frac{60 a^7 b^3}{x^2}-\frac{126 a^6 b^4}{x^{5/3}}-\frac{189 a^5 b^5}{x^{4/3}}-\frac{210 a^4 b^6}{x}-\frac{180 a^3 b^7}{x^{2/3}}-\frac{135 a^2 b^8}{\sqrt [3]{x}}+10 a b^9 \log (x)+3 b^{10} \sqrt [3]{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^10/x^4,x]

[Out]

-a^10/(3*x^3) - (15*a^9*b)/(4*x^(8/3)) - (135*a^8*b^2)/(7*x^(7/3)) - (60*a^7*b^3
)/x^2 - (126*a^6*b^4)/x^(5/3) - (189*a^5*b^5)/x^(4/3) - (210*a^4*b^6)/x - (180*a
^3*b^7)/x^(2/3) - (135*a^2*b^8)/x^(1/3) + 3*b^10*x^(1/3) + 10*a*b^9*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{10}}{3 x^{3}} - \frac{15 a^{9} b}{4 x^{\frac{8}{3}}} - \frac{135 a^{8} b^{2}}{7 x^{\frac{7}{3}}} - \frac{60 a^{7} b^{3}}{x^{2}} - \frac{126 a^{6} b^{4}}{x^{\frac{5}{3}}} - \frac{189 a^{5} b^{5}}{x^{\frac{4}{3}}} - \frac{210 a^{4} b^{6}}{x} - \frac{180 a^{3} b^{7}}{x^{\frac{2}{3}}} - \frac{135 a^{2} b^{8}}{\sqrt [3]{x}} + 30 a b^{9} \log{\left (\sqrt [3]{x} \right )} + 3 \int ^{\sqrt [3]{x}} b^{10}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/3))**10/x**4,x)

[Out]

-a**10/(3*x**3) - 15*a**9*b/(4*x**(8/3)) - 135*a**8*b**2/(7*x**(7/3)) - 60*a**7*
b**3/x**2 - 126*a**6*b**4/x**(5/3) - 189*a**5*b**5/x**(4/3) - 210*a**4*b**6/x -
180*a**3*b**7/x**(2/3) - 135*a**2*b**8/x**(1/3) + 30*a*b**9*log(x**(1/3)) + 3*In
tegral(b**10, (x, x**(1/3)))

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Mathematica [A]  time = 0.0941067, size = 131, normalized size = 1. \[ -\frac{a^{10}}{3 x^3}-\frac{15 a^9 b}{4 x^{8/3}}-\frac{135 a^8 b^2}{7 x^{7/3}}-\frac{60 a^7 b^3}{x^2}-\frac{126 a^6 b^4}{x^{5/3}}-\frac{189 a^5 b^5}{x^{4/3}}-\frac{210 a^4 b^6}{x}-\frac{180 a^3 b^7}{x^{2/3}}-\frac{135 a^2 b^8}{\sqrt [3]{x}}+10 a b^9 \log (x)+3 b^{10} \sqrt [3]{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^10/x^4,x]

[Out]

-a^10/(3*x^3) - (15*a^9*b)/(4*x^(8/3)) - (135*a^8*b^2)/(7*x^(7/3)) - (60*a^7*b^3
)/x^2 - (126*a^6*b^4)/x^(5/3) - (189*a^5*b^5)/x^(4/3) - (210*a^4*b^6)/x - (180*a
^3*b^7)/x^(2/3) - (135*a^2*b^8)/x^(1/3) + 3*b^10*x^(1/3) + 10*a*b^9*Log[x]

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Maple [A]  time = 0.012, size = 112, normalized size = 0.9 \[ -{\frac{{a}^{10}}{3\,{x}^{3}}}-{\frac{15\,{a}^{9}b}{4}{x}^{-{\frac{8}{3}}}}-{\frac{135\,{a}^{8}{b}^{2}}{7}{x}^{-{\frac{7}{3}}}}-60\,{\frac{{a}^{7}{b}^{3}}{{x}^{2}}}-126\,{\frac{{a}^{6}{b}^{4}}{{x}^{5/3}}}-189\,{\frac{{a}^{5}{b}^{5}}{{x}^{4/3}}}-210\,{\frac{{a}^{4}{b}^{6}}{x}}-180\,{\frac{{a}^{3}{b}^{7}}{{x}^{2/3}}}-135\,{\frac{{a}^{2}{b}^{8}}{\sqrt [3]{x}}}+3\,{b}^{10}\sqrt [3]{x}+10\,a{b}^{9}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^10/x^4,x)

[Out]

-1/3*a^10/x^3-15/4*a^9*b/x^(8/3)-135/7*a^8*b^2/x^(7/3)-60*a^7*b^3/x^2-126*a^6*b^
4/x^(5/3)-189*a^5*b^5/x^(4/3)-210*a^4*b^6/x-180*a^3*b^7/x^(2/3)-135*a^2*b^8/x^(1
/3)+3*b^10*x^(1/3)+10*a*b^9*ln(x)

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Maxima [A]  time = 1.45017, size = 151, normalized size = 1.15 \[ 10 \, a b^{9} \log \left (x\right ) + 3 \, b^{10} x^{\frac{1}{3}} - \frac{11340 \, a^{2} b^{8} x^{\frac{8}{3}} + 15120 \, a^{3} b^{7} x^{\frac{7}{3}} + 17640 \, a^{4} b^{6} x^{2} + 15876 \, a^{5} b^{5} x^{\frac{5}{3}} + 10584 \, a^{6} b^{4} x^{\frac{4}{3}} + 5040 \, a^{7} b^{3} x + 1620 \, a^{8} b^{2} x^{\frac{2}{3}} + 315 \, a^{9} b x^{\frac{1}{3}} + 28 \, a^{10}}{84 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^10/x^4,x, algorithm="maxima")

[Out]

10*a*b^9*log(x) + 3*b^10*x^(1/3) - 1/84*(11340*a^2*b^8*x^(8/3) + 15120*a^3*b^7*x
^(7/3) + 17640*a^4*b^6*x^2 + 15876*a^5*b^5*x^(5/3) + 10584*a^6*b^4*x^(4/3) + 504
0*a^7*b^3*x + 1620*a^8*b^2*x^(2/3) + 315*a^9*b*x^(1/3) + 28*a^10)/x^3

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Fricas [A]  time = 0.22203, size = 159, normalized size = 1.21 \[ \frac{2520 \, a b^{9} x^{3} \log \left (x^{\frac{1}{3}}\right ) - 17640 \, a^{4} b^{6} x^{2} - 5040 \, a^{7} b^{3} x - 28 \, a^{10} - 324 \,{\left (35 \, a^{2} b^{8} x^{2} + 49 \, a^{5} b^{5} x + 5 \, a^{8} b^{2}\right )} x^{\frac{2}{3}} + 63 \,{\left (4 \, b^{10} x^{3} - 240 \, a^{3} b^{7} x^{2} - 168 \, a^{6} b^{4} x - 5 \, a^{9} b\right )} x^{\frac{1}{3}}}{84 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^10/x^4,x, algorithm="fricas")

[Out]

1/84*(2520*a*b^9*x^3*log(x^(1/3)) - 17640*a^4*b^6*x^2 - 5040*a^7*b^3*x - 28*a^10
- 324*(35*a^2*b^8*x^2 + 49*a^5*b^5*x + 5*a^8*b^2)*x^(2/3) + 63*(4*b^10*x^3 - 24
0*a^3*b^7*x^2 - 168*a^6*b^4*x - 5*a^9*b)*x^(1/3))/x^3

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Sympy [A]  time = 8.70147, size = 133, normalized size = 1.02 \[ - \frac{a^{10}}{3 x^{3}} - \frac{15 a^{9} b}{4 x^{\frac{8}{3}}} - \frac{135 a^{8} b^{2}}{7 x^{\frac{7}{3}}} - \frac{60 a^{7} b^{3}}{x^{2}} - \frac{126 a^{6} b^{4}}{x^{\frac{5}{3}}} - \frac{189 a^{5} b^{5}}{x^{\frac{4}{3}}} - \frac{210 a^{4} b^{6}}{x} - \frac{180 a^{3} b^{7}}{x^{\frac{2}{3}}} - \frac{135 a^{2} b^{8}}{\sqrt [3]{x}} + 10 a b^{9} \log{\left (x \right )} + 3 b^{10} \sqrt [3]{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**10/x**4,x)

[Out]

-a**10/(3*x**3) - 15*a**9*b/(4*x**(8/3)) - 135*a**8*b**2/(7*x**(7/3)) - 60*a**7*
b**3/x**2 - 126*a**6*b**4/x**(5/3) - 189*a**5*b**5/x**(4/3) - 210*a**4*b**6/x -
180*a**3*b**7/x**(2/3) - 135*a**2*b**8/x**(1/3) + 10*a*b**9*log(x) + 3*b**10*x**
(1/3)

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GIAC/XCAS [A]  time = 0.268407, size = 153, normalized size = 1.17 \[ 10 \, a b^{9}{\rm ln}\left ({\left | x \right |}\right ) + 3 \, b^{10} x^{\frac{1}{3}} - \frac{11340 \, a^{2} b^{8} x^{\frac{8}{3}} + 15120 \, a^{3} b^{7} x^{\frac{7}{3}} + 17640 \, a^{4} b^{6} x^{2} + 15876 \, a^{5} b^{5} x^{\frac{5}{3}} + 10584 \, a^{6} b^{4} x^{\frac{4}{3}} + 5040 \, a^{7} b^{3} x + 1620 \, a^{8} b^{2} x^{\frac{2}{3}} + 315 \, a^{9} b x^{\frac{1}{3}} + 28 \, a^{10}}{84 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^10/x^4,x, algorithm="giac")

[Out]

10*a*b^9*ln(abs(x)) + 3*b^10*x^(1/3) - 1/84*(11340*a^2*b^8*x^(8/3) + 15120*a^3*b
^7*x^(7/3) + 17640*a^4*b^6*x^2 + 15876*a^5*b^5*x^(5/3) + 10584*a^6*b^4*x^(4/3) +
5040*a^7*b^3*x + 1620*a^8*b^2*x^(2/3) + 315*a^9*b*x^(1/3) + 28*a^10)/x^3