3.2344 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^{15}}{x^8} \, dx\)

Optimal. Leaf size=148 \[ \frac{b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{108528 a^6 x^{16/3}}-\frac{b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{6783 a^5 x^{17/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{798 a^4 x^6}-\frac{b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{133 a^3 x^{19/3}}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{28 a^2 x^{20/3}}-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{7 a x^7} \]

[Out]

-(a + b*x^(1/3))^16/(7*a*x^7) + (b*(a + b*x^(1/3))^16)/(28*a^2*x^(20/3)) - (b^2*
(a + b*x^(1/3))^16)/(133*a^3*x^(19/3)) + (b^3*(a + b*x^(1/3))^16)/(798*a^4*x^6)
- (b^4*(a + b*x^(1/3))^16)/(6783*a^5*x^(17/3)) + (b^5*(a + b*x^(1/3))^16)/(10852
8*a^6*x^(16/3))

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Rubi [A]  time = 0.175147, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{108528 a^6 x^{16/3}}-\frac{b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{6783 a^5 x^{17/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{798 a^4 x^6}-\frac{b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{133 a^3 x^{19/3}}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{28 a^2 x^{20/3}}-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^15/x^8,x]

[Out]

-(a + b*x^(1/3))^16/(7*a*x^7) + (b*(a + b*x^(1/3))^16)/(28*a^2*x^(20/3)) - (b^2*
(a + b*x^(1/3))^16)/(133*a^3*x^(19/3)) + (b^3*(a + b*x^(1/3))^16)/(798*a^4*x^6)
- (b^4*(a + b*x^(1/3))^16)/(6783*a^5*x^(17/3)) + (b^5*(a + b*x^(1/3))^16)/(10852
8*a^6*x^(16/3))

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Rubi in Sympy [A]  time = 21.9653, size = 131, normalized size = 0.89 \[ - \frac{\left (a + b \sqrt [3]{x}\right )^{16}}{7 a x^{7}} + \frac{b \left (a + b \sqrt [3]{x}\right )^{16}}{28 a^{2} x^{\frac{20}{3}}} - \frac{b^{2} \left (a + b \sqrt [3]{x}\right )^{16}}{133 a^{3} x^{\frac{19}{3}}} + \frac{b^{3} \left (a + b \sqrt [3]{x}\right )^{16}}{798 a^{4} x^{6}} - \frac{b^{4} \left (a + b \sqrt [3]{x}\right )^{16}}{6783 a^{5} x^{\frac{17}{3}}} + \frac{b^{5} \left (a + b \sqrt [3]{x}\right )^{16}}{108528 a^{6} x^{\frac{16}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/3))**15/x**8,x)

[Out]

-(a + b*x**(1/3))**16/(7*a*x**7) + b*(a + b*x**(1/3))**16/(28*a**2*x**(20/3)) -
b**2*(a + b*x**(1/3))**16/(133*a**3*x**(19/3)) + b**3*(a + b*x**(1/3))**16/(798*
a**4*x**6) - b**4*(a + b*x**(1/3))**16/(6783*a**5*x**(17/3)) + b**5*(a + b*x**(1
/3))**16/(108528*a**6*x**(16/3))

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Mathematica [A]  time = 0.0713744, size = 213, normalized size = 1.44 \[ -\frac{a^{15}}{7 x^7}-\frac{9 a^{14} b}{4 x^{20/3}}-\frac{315 a^{13} b^2}{19 x^{19/3}}-\frac{455 a^{12} b^3}{6 x^6}-\frac{4095 a^{11} b^4}{17 x^{17/3}}-\frac{9009 a^{10} b^5}{16 x^{16/3}}-\frac{1001 a^9 b^6}{x^5}-\frac{19305 a^8 b^7}{14 x^{14/3}}-\frac{1485 a^7 b^8}{x^{13/3}}-\frac{5005 a^6 b^9}{4 x^4}-\frac{819 a^5 b^{10}}{x^{11/3}}-\frac{819 a^4 b^{11}}{2 x^{10/3}}-\frac{455 a^3 b^{12}}{3 x^3}-\frac{315 a^2 b^{13}}{8 x^{8/3}}-\frac{45 a b^{14}}{7 x^{7/3}}-\frac{b^{15}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^15/x^8,x]

[Out]

-a^15/(7*x^7) - (9*a^14*b)/(4*x^(20/3)) - (315*a^13*b^2)/(19*x^(19/3)) - (455*a^
12*b^3)/(6*x^6) - (4095*a^11*b^4)/(17*x^(17/3)) - (9009*a^10*b^5)/(16*x^(16/3))
- (1001*a^9*b^6)/x^5 - (19305*a^8*b^7)/(14*x^(14/3)) - (1485*a^7*b^8)/x^(13/3) -
(5005*a^6*b^9)/(4*x^4) - (819*a^5*b^10)/x^(11/3) - (819*a^4*b^11)/(2*x^(10/3))
- (455*a^3*b^12)/(3*x^3) - (315*a^2*b^13)/(8*x^(8/3)) - (45*a*b^14)/(7*x^(7/3))
- b^15/(2*x^2)

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Maple [A]  time = 0.011, size = 168, normalized size = 1.1 \[ -{\frac{9\,{a}^{14}b}{4}{x}^{-{\frac{20}{3}}}}-{\frac{5005\,{a}^{6}{b}^{9}}{4\,{x}^{4}}}-1001\,{\frac{{a}^{9}{b}^{6}}{{x}^{5}}}-{\frac{9009\,{a}^{10}{b}^{5}}{16}{x}^{-{\frac{16}{3}}}}-{\frac{315\,{a}^{13}{b}^{2}}{19}{x}^{-{\frac{19}{3}}}}-1485\,{\frac{{a}^{7}{b}^{8}}{{x}^{13/3}}}-{\frac{{b}^{15}}{2\,{x}^{2}}}-{\frac{819\,{a}^{4}{b}^{11}}{2}{x}^{-{\frac{10}{3}}}}-{\frac{4095\,{a}^{11}{b}^{4}}{17}{x}^{-{\frac{17}{3}}}}-{\frac{315\,{a}^{2}{b}^{13}}{8}{x}^{-{\frac{8}{3}}}}-819\,{\frac{{a}^{5}{b}^{10}}{{x}^{11/3}}}-{\frac{19305\,{a}^{8}{b}^{7}}{14}{x}^{-{\frac{14}{3}}}}-{\frac{455\,{a}^{3}{b}^{12}}{3\,{x}^{3}}}-{\frac{455\,{a}^{12}{b}^{3}}{6\,{x}^{6}}}-{\frac{{a}^{15}}{7\,{x}^{7}}}-{\frac{45\,a{b}^{14}}{7}{x}^{-{\frac{7}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^15/x^8,x)

[Out]

-9/4*a^14*b/x^(20/3)-5005/4*a^6*b^9/x^4-1001*a^9*b^6/x^5-9009/16*a^10*b^5/x^(16/
3)-315/19*a^13*b^2/x^(19/3)-1485*a^7*b^8/x^(13/3)-1/2*b^15/x^2-819/2*a^4*b^11/x^
(10/3)-4095/17*a^11*b^4/x^(17/3)-315/8*a^2*b^13/x^(8/3)-819*a^5*b^10/x^(11/3)-19
305/14*a^8*b^7/x^(14/3)-455/3*a^3*b^12/x^3-455/6*a^12*b^3/x^6-1/7*a^15/x^7-45/7*
a*b^14/x^(7/3)

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Maxima [A]  time = 1.43709, size = 225, normalized size = 1.52 \[ -\frac{54264 \, b^{15} x^{5} + 697680 \, a b^{14} x^{\frac{14}{3}} + 4273290 \, a^{2} b^{13} x^{\frac{13}{3}} + 16460080 \, a^{3} b^{12} x^{4} + 44442216 \, a^{4} b^{11} x^{\frac{11}{3}} + 88884432 \, a^{5} b^{10} x^{\frac{10}{3}} + 135795660 \, a^{6} b^{9} x^{3} + 161164080 \, a^{7} b^{8} x^{\frac{8}{3}} + 149652360 \, a^{8} b^{7} x^{\frac{7}{3}} + 108636528 \, a^{9} b^{6} x^{2} + 61108047 \, a^{10} b^{5} x^{\frac{5}{3}} + 26142480 \, a^{11} b^{4} x^{\frac{4}{3}} + 8230040 \, a^{12} b^{3} x + 1799280 \, a^{13} b^{2} x^{\frac{2}{3}} + 244188 \, a^{14} b x^{\frac{1}{3}} + 15504 \, a^{15}}{108528 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^15/x^8,x, algorithm="maxima")

[Out]

-1/108528*(54264*b^15*x^5 + 697680*a*b^14*x^(14/3) + 4273290*a^2*b^13*x^(13/3) +
16460080*a^3*b^12*x^4 + 44442216*a^4*b^11*x^(11/3) + 88884432*a^5*b^10*x^(10/3)
+ 135795660*a^6*b^9*x^3 + 161164080*a^7*b^8*x^(8/3) + 149652360*a^8*b^7*x^(7/3)
+ 108636528*a^9*b^6*x^2 + 61108047*a^10*b^5*x^(5/3) + 26142480*a^11*b^4*x^(4/3)
+ 8230040*a^12*b^3*x + 1799280*a^13*b^2*x^(2/3) + 244188*a^14*b*x^(1/3) + 15504
*a^15)/x^7

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Fricas [A]  time = 0.217849, size = 228, normalized size = 1.54 \[ -\frac{54264 \, b^{15} x^{5} + 16460080 \, a^{3} b^{12} x^{4} + 135795660 \, a^{6} b^{9} x^{3} + 108636528 \, a^{9} b^{6} x^{2} + 8230040 \, a^{12} b^{3} x + 15504 \, a^{15} + 459 \,{\left (1520 \, a b^{14} x^{4} + 96824 \, a^{4} b^{11} x^{3} + 351120 \, a^{7} b^{8} x^{2} + 133133 \, a^{10} b^{5} x + 3920 \, a^{13} b^{2}\right )} x^{\frac{2}{3}} + 1026 \,{\left (4165 \, a^{2} b^{13} x^{4} + 86632 \, a^{5} b^{10} x^{3} + 145860 \, a^{8} b^{7} x^{2} + 25480 \, a^{11} b^{4} x + 238 \, a^{14} b\right )} x^{\frac{1}{3}}}{108528 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^15/x^8,x, algorithm="fricas")

[Out]

-1/108528*(54264*b^15*x^5 + 16460080*a^3*b^12*x^4 + 135795660*a^6*b^9*x^3 + 1086
36528*a^9*b^6*x^2 + 8230040*a^12*b^3*x + 15504*a^15 + 459*(1520*a*b^14*x^4 + 968
24*a^4*b^11*x^3 + 351120*a^7*b^8*x^2 + 133133*a^10*b^5*x + 3920*a^13*b^2)*x^(2/3
) + 1026*(4165*a^2*b^13*x^4 + 86632*a^5*b^10*x^3 + 145860*a^8*b^7*x^2 + 25480*a^
11*b^4*x + 238*a^14*b)*x^(1/3))/x^7

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Sympy [A]  time = 62.9059, size = 216, normalized size = 1.46 \[ - \frac{a^{15}}{7 x^{7}} - \frac{9 a^{14} b}{4 x^{\frac{20}{3}}} - \frac{315 a^{13} b^{2}}{19 x^{\frac{19}{3}}} - \frac{455 a^{12} b^{3}}{6 x^{6}} - \frac{4095 a^{11} b^{4}}{17 x^{\frac{17}{3}}} - \frac{9009 a^{10} b^{5}}{16 x^{\frac{16}{3}}} - \frac{1001 a^{9} b^{6}}{x^{5}} - \frac{19305 a^{8} b^{7}}{14 x^{\frac{14}{3}}} - \frac{1485 a^{7} b^{8}}{x^{\frac{13}{3}}} - \frac{5005 a^{6} b^{9}}{4 x^{4}} - \frac{819 a^{5} b^{10}}{x^{\frac{11}{3}}} - \frac{819 a^{4} b^{11}}{2 x^{\frac{10}{3}}} - \frac{455 a^{3} b^{12}}{3 x^{3}} - \frac{315 a^{2} b^{13}}{8 x^{\frac{8}{3}}} - \frac{45 a b^{14}}{7 x^{\frac{7}{3}}} - \frac{b^{15}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**15/x**8,x)

[Out]

-a**15/(7*x**7) - 9*a**14*b/(4*x**(20/3)) - 315*a**13*b**2/(19*x**(19/3)) - 455*
a**12*b**3/(6*x**6) - 4095*a**11*b**4/(17*x**(17/3)) - 9009*a**10*b**5/(16*x**(1
6/3)) - 1001*a**9*b**6/x**5 - 19305*a**8*b**7/(14*x**(14/3)) - 1485*a**7*b**8/x*
*(13/3) - 5005*a**6*b**9/(4*x**4) - 819*a**5*b**10/x**(11/3) - 819*a**4*b**11/(2
*x**(10/3)) - 455*a**3*b**12/(3*x**3) - 315*a**2*b**13/(8*x**(8/3)) - 45*a*b**14
/(7*x**(7/3)) - b**15/(2*x**2)

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GIAC/XCAS [A]  time = 0.222169, size = 225, normalized size = 1.52 \[ -\frac{54264 \, b^{15} x^{5} + 697680 \, a b^{14} x^{\frac{14}{3}} + 4273290 \, a^{2} b^{13} x^{\frac{13}{3}} + 16460080 \, a^{3} b^{12} x^{4} + 44442216 \, a^{4} b^{11} x^{\frac{11}{3}} + 88884432 \, a^{5} b^{10} x^{\frac{10}{3}} + 135795660 \, a^{6} b^{9} x^{3} + 161164080 \, a^{7} b^{8} x^{\frac{8}{3}} + 149652360 \, a^{8} b^{7} x^{\frac{7}{3}} + 108636528 \, a^{9} b^{6} x^{2} + 61108047 \, a^{10} b^{5} x^{\frac{5}{3}} + 26142480 \, a^{11} b^{4} x^{\frac{4}{3}} + 8230040 \, a^{12} b^{3} x + 1799280 \, a^{13} b^{2} x^{\frac{2}{3}} + 244188 \, a^{14} b x^{\frac{1}{3}} + 15504 \, a^{15}}{108528 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^15/x^8,x, algorithm="giac")

[Out]

-1/108528*(54264*b^15*x^5 + 697680*a*b^14*x^(14/3) + 4273290*a^2*b^13*x^(13/3) +
16460080*a^3*b^12*x^4 + 44442216*a^4*b^11*x^(11/3) + 88884432*a^5*b^10*x^(10/3)
+ 135795660*a^6*b^9*x^3 + 161164080*a^7*b^8*x^(8/3) + 149652360*a^8*b^7*x^(7/3)
+ 108636528*a^9*b^6*x^2 + 61108047*a^10*b^5*x^(5/3) + 26142480*a^11*b^4*x^(4/3)
+ 8230040*a^12*b^3*x + 1799280*a^13*b^2*x^(2/3) + 244188*a^14*b*x^(1/3) + 15504
*a^15)/x^7