3.2343 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^{15}}{x^7} \, dx\)

Optimal. Leaf size=72 \[ -\frac{b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{816 a^3 x^{16/3}}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{51 a^2 x^{17/3}}-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{6 a x^6} \]

[Out]

-(a + b*x^(1/3))^16/(6*a*x^6) + (b*(a + b*x^(1/3))^16)/(51*a^2*x^(17/3)) - (b^2*
(a + b*x^(1/3))^16)/(816*a^3*x^(16/3))

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Rubi [A]  time = 0.0791763, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{816 a^3 x^{16/3}}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{51 a^2 x^{17/3}}-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^15/x^7,x]

[Out]

-(a + b*x^(1/3))^16/(6*a*x^6) + (b*(a + b*x^(1/3))^16)/(51*a^2*x^(17/3)) - (b^2*
(a + b*x^(1/3))^16)/(816*a^3*x^(16/3))

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Rubi in Sympy [A]  time = 8.46751, size = 61, normalized size = 0.85 \[ - \frac{\left (a + b \sqrt [3]{x}\right )^{16}}{6 a x^{6}} + \frac{b \left (a + b \sqrt [3]{x}\right )^{16}}{51 a^{2} x^{\frac{17}{3}}} - \frac{b^{2} \left (a + b \sqrt [3]{x}\right )^{16}}{816 a^{3} x^{\frac{16}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/3))**15/x**7,x)

[Out]

-(a + b*x**(1/3))**16/(6*a*x**6) + b*(a + b*x**(1/3))**16/(51*a**2*x**(17/3)) -
b**2*(a + b*x**(1/3))**16/(816*a**3*x**(16/3))

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Mathematica [B]  time = 0.0594525, size = 189, normalized size = 2.62 \[ -\frac{136 a^{15}+2160 a^{14} b \sqrt [3]{x}+16065 a^{13} b^2 x^{2/3}+74256 a^{12} b^3 x+238680 a^{11} b^4 x^{4/3}+565488 a^{10} b^5 x^{5/3}+1021020 a^9 b^6 x^2+1432080 a^8 b^7 x^{7/3}+1575288 a^7 b^8 x^{8/3}+1361360 a^6 b^9 x^3+918918 a^5 b^{10} x^{10/3}+477360 a^4 b^{11} x^{11/3}+185640 a^3 b^{12} x^4+51408 a^2 b^{13} x^{13/3}+9180 a b^{14} x^{14/3}+816 b^{15} x^5}{816 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^15/x^7,x]

[Out]

-(136*a^15 + 2160*a^14*b*x^(1/3) + 16065*a^13*b^2*x^(2/3) + 74256*a^12*b^3*x + 2
38680*a^11*b^4*x^(4/3) + 565488*a^10*b^5*x^(5/3) + 1021020*a^9*b^6*x^2 + 1432080
*a^8*b^7*x^(7/3) + 1575288*a^7*b^8*x^(8/3) + 1361360*a^6*b^9*x^3 + 918918*a^5*b^
10*x^(10/3) + 477360*a^4*b^11*x^(11/3) + 185640*a^3*b^12*x^4 + 51408*a^2*b^13*x^
(13/3) + 9180*a*b^14*x^(14/3) + 816*b^15*x^5)/(816*x^6)

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Maple [B]  time = 0.011, size = 168, normalized size = 2.3 \[ -{\frac{5005\,{a}^{9}{b}^{6}}{4\,{x}^{4}}}-91\,{\frac{{a}^{12}{b}^{3}}{{x}^{5}}}-{\frac{315\,{a}^{13}{b}^{2}}{16}{x}^{-{\frac{16}{3}}}}-693\,{\frac{{a}^{10}{b}^{5}}{{x}^{13/3}}}-{\frac{455\,{a}^{3}{b}^{12}}{2\,{x}^{2}}}-{\frac{45\,a{b}^{14}}{4}{x}^{-{\frac{4}{3}}}}-{\frac{3861\,{a}^{7}{b}^{8}}{2}{x}^{-{\frac{10}{3}}}}-{\frac{45\,{a}^{14}b}{17}{x}^{-{\frac{17}{3}}}}-{\frac{9009\,{a}^{5}{b}^{10}}{8}{x}^{-{\frac{8}{3}}}}-1755\,{\frac{{a}^{8}{b}^{7}}{{x}^{11/3}}}-{\frac{585\,{a}^{11}{b}^{4}}{2}{x}^{-{\frac{14}{3}}}}-{\frac{5005\,{a}^{6}{b}^{9}}{3\,{x}^{3}}}-{\frac{{b}^{15}}{x}}-{\frac{{a}^{15}}{6\,{x}^{6}}}-63\,{\frac{{a}^{2}{b}^{13}}{{x}^{5/3}}}-585\,{\frac{{a}^{4}{b}^{11}}{{x}^{7/3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^15/x^7,x)

[Out]

-5005/4*a^9*b^6/x^4-91*a^12*b^3/x^5-315/16*a^13*b^2/x^(16/3)-693*a^10*b^5/x^(13/
3)-455/2*a^3*b^12/x^2-45/4*a*b^14/x^(4/3)-3861/2*a^7*b^8/x^(10/3)-45/17*a^14*b/x
^(17/3)-9009/8*a^5*b^10/x^(8/3)-1755*a^8*b^7/x^(11/3)-585/2*a^11*b^4/x^(14/3)-50
05/3*a^6*b^9/x^3-b^15/x-1/6*a^15/x^6-63*a^2*b^13/x^(5/3)-585*a^4*b^11/x^(7/3)

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Maxima [A]  time = 1.44051, size = 225, normalized size = 3.12 \[ -\frac{816 \, b^{15} x^{5} + 9180 \, a b^{14} x^{\frac{14}{3}} + 51408 \, a^{2} b^{13} x^{\frac{13}{3}} + 185640 \, a^{3} b^{12} x^{4} + 477360 \, a^{4} b^{11} x^{\frac{11}{3}} + 918918 \, a^{5} b^{10} x^{\frac{10}{3}} + 1361360 \, a^{6} b^{9} x^{3} + 1575288 \, a^{7} b^{8} x^{\frac{8}{3}} + 1432080 \, a^{8} b^{7} x^{\frac{7}{3}} + 1021020 \, a^{9} b^{6} x^{2} + 565488 \, a^{10} b^{5} x^{\frac{5}{3}} + 238680 \, a^{11} b^{4} x^{\frac{4}{3}} + 74256 \, a^{12} b^{3} x + 16065 \, a^{13} b^{2} x^{\frac{2}{3}} + 2160 \, a^{14} b x^{\frac{1}{3}} + 136 \, a^{15}}{816 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^15/x^7,x, algorithm="maxima")

[Out]

-1/816*(816*b^15*x^5 + 9180*a*b^14*x^(14/3) + 51408*a^2*b^13*x^(13/3) + 185640*a
^3*b^12*x^4 + 477360*a^4*b^11*x^(11/3) + 918918*a^5*b^10*x^(10/3) + 1361360*a^6*
b^9*x^3 + 1575288*a^7*b^8*x^(8/3) + 1432080*a^8*b^7*x^(7/3) + 1021020*a^9*b^6*x^
2 + 565488*a^10*b^5*x^(5/3) + 238680*a^11*b^4*x^(4/3) + 74256*a^12*b^3*x + 16065
*a^13*b^2*x^(2/3) + 2160*a^14*b*x^(1/3) + 136*a^15)/x^6

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Fricas [A]  time = 0.216964, size = 228, normalized size = 3.17 \[ -\frac{816 \, b^{15} x^{5} + 185640 \, a^{3} b^{12} x^{4} + 1361360 \, a^{6} b^{9} x^{3} + 1021020 \, a^{9} b^{6} x^{2} + 74256 \, a^{12} b^{3} x + 136 \, a^{15} + 459 \,{\left (20 \, a b^{14} x^{4} + 1040 \, a^{4} b^{11} x^{3} + 3432 \, a^{7} b^{8} x^{2} + 1232 \, a^{10} b^{5} x + 35 \, a^{13} b^{2}\right )} x^{\frac{2}{3}} + 54 \,{\left (952 \, a^{2} b^{13} x^{4} + 17017 \, a^{5} b^{10} x^{3} + 26520 \, a^{8} b^{7} x^{2} + 4420 \, a^{11} b^{4} x + 40 \, a^{14} b\right )} x^{\frac{1}{3}}}{816 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^15/x^7,x, algorithm="fricas")

[Out]

-1/816*(816*b^15*x^5 + 185640*a^3*b^12*x^4 + 1361360*a^6*b^9*x^3 + 1021020*a^9*b
^6*x^2 + 74256*a^12*b^3*x + 136*a^15 + 459*(20*a*b^14*x^4 + 1040*a^4*b^11*x^3 +
3432*a^7*b^8*x^2 + 1232*a^10*b^5*x + 35*a^13*b^2)*x^(2/3) + 54*(952*a^2*b^13*x^4
+ 17017*a^5*b^10*x^3 + 26520*a^8*b^7*x^2 + 4420*a^11*b^4*x + 40*a^14*b)*x^(1/3)
)/x^6

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Sympy [A]  time = 41.2952, size = 209, normalized size = 2.9 \[ - \frac{a^{15}}{6 x^{6}} - \frac{45 a^{14} b}{17 x^{\frac{17}{3}}} - \frac{315 a^{13} b^{2}}{16 x^{\frac{16}{3}}} - \frac{91 a^{12} b^{3}}{x^{5}} - \frac{585 a^{11} b^{4}}{2 x^{\frac{14}{3}}} - \frac{693 a^{10} b^{5}}{x^{\frac{13}{3}}} - \frac{5005 a^{9} b^{6}}{4 x^{4}} - \frac{1755 a^{8} b^{7}}{x^{\frac{11}{3}}} - \frac{3861 a^{7} b^{8}}{2 x^{\frac{10}{3}}} - \frac{5005 a^{6} b^{9}}{3 x^{3}} - \frac{9009 a^{5} b^{10}}{8 x^{\frac{8}{3}}} - \frac{585 a^{4} b^{11}}{x^{\frac{7}{3}}} - \frac{455 a^{3} b^{12}}{2 x^{2}} - \frac{63 a^{2} b^{13}}{x^{\frac{5}{3}}} - \frac{45 a b^{14}}{4 x^{\frac{4}{3}}} - \frac{b^{15}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**15/x**7,x)

[Out]

-a**15/(6*x**6) - 45*a**14*b/(17*x**(17/3)) - 315*a**13*b**2/(16*x**(16/3)) - 91
*a**12*b**3/x**5 - 585*a**11*b**4/(2*x**(14/3)) - 693*a**10*b**5/x**(13/3) - 500
5*a**9*b**6/(4*x**4) - 1755*a**8*b**7/x**(11/3) - 3861*a**7*b**8/(2*x**(10/3)) -
5005*a**6*b**9/(3*x**3) - 9009*a**5*b**10/(8*x**(8/3)) - 585*a**4*b**11/x**(7/3
) - 455*a**3*b**12/(2*x**2) - 63*a**2*b**13/x**(5/3) - 45*a*b**14/(4*x**(4/3)) -
b**15/x

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GIAC/XCAS [A]  time = 0.220504, size = 225, normalized size = 3.12 \[ -\frac{816 \, b^{15} x^{5} + 9180 \, a b^{14} x^{\frac{14}{3}} + 51408 \, a^{2} b^{13} x^{\frac{13}{3}} + 185640 \, a^{3} b^{12} x^{4} + 477360 \, a^{4} b^{11} x^{\frac{11}{3}} + 918918 \, a^{5} b^{10} x^{\frac{10}{3}} + 1361360 \, a^{6} b^{9} x^{3} + 1575288 \, a^{7} b^{8} x^{\frac{8}{3}} + 1432080 \, a^{8} b^{7} x^{\frac{7}{3}} + 1021020 \, a^{9} b^{6} x^{2} + 565488 \, a^{10} b^{5} x^{\frac{5}{3}} + 238680 \, a^{11} b^{4} x^{\frac{4}{3}} + 74256 \, a^{12} b^{3} x + 16065 \, a^{13} b^{2} x^{\frac{2}{3}} + 2160 \, a^{14} b x^{\frac{1}{3}} + 136 \, a^{15}}{816 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^15/x^7,x, algorithm="giac")

[Out]

-1/816*(816*b^15*x^5 + 9180*a*b^14*x^(14/3) + 51408*a^2*b^13*x^(13/3) + 185640*a
^3*b^12*x^4 + 477360*a^4*b^11*x^(11/3) + 918918*a^5*b^10*x^(10/3) + 1361360*a^6*
b^9*x^3 + 1575288*a^7*b^8*x^(8/3) + 1432080*a^8*b^7*x^(7/3) + 1021020*a^9*b^6*x^
2 + 565488*a^10*b^5*x^(5/3) + 238680*a^11*b^4*x^(4/3) + 74256*a^12*b^3*x + 16065
*a^13*b^2*x^(2/3) + 2160*a^14*b*x^(1/3) + 136*a^15)/x^6