3.2351 \(\int \frac {(a+b \sqrt [3]{x})^{15}}{x^8} \, dx\)

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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {266, 45, 37} \[ \frac {b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{108528 a^6 x^{16/3}}-\frac {b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{6783 a^5 x^{17/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{798 a^4 x^6}-\frac {b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{133 a^3 x^{19/3}}+\frac {b \left (a+b \sqrt [3]{x}\right )^{16}}{28 a^2 x^{20/3}}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{7 a x^7} \]

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)/(108528*a^6*x^(16/3))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^8} \, dx &=3 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{22}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{7 a x^7}-\frac {(5 b) \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{21}} \, dx,x,\sqrt [3]{x}\right )}{7 a}\\ &=-\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^{20/3}}+\frac {b^2 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{20}} \, dx,x,\sqrt [3]{x}\right )}{7 a^2}\\ &=-\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^{20/3}}-\frac {b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{133 a^3 x^{19/3}}-\frac {\left (3 b^3\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{19}} \, dx,x,\sqrt [3]{x}\right )}{133 a^3}\\ &=-\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^{20/3}}-\frac {b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{133 a^3 x^{19/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{798 a^4 x^6}+\frac {b^4 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{18}} \, dx,x,\sqrt [3]{x}\right )}{399 a^4}\\ &=-\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^{20/3}}-\frac {b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{133 a^3 x^{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^{17/3}}-\frac {b^5 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{17}} \, dx,x,\sqrt [3]{x}\right )}{6783 a^5}\\ &=-\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^{20/3}}-\frac {b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{133 a^3 x^{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^{17/3}}+\frac {b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{108528 a^6 x^{16/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 80, normalized size = 0.54 \[ \frac {\left (a+b \sqrt [3]{x}\right )^{16} \left (-15504 a^5+3876 a^4 b \sqrt [3]{x}-816 a^3 b^2 x^{2/3}+136 a^2 b^3 x-16 a b^4 x^{4/3}+b^5 x^{5/3}\right )}{108528 a^6 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^15/x^8,x]

[Out]

((a + b*x^(1/3))^16*(-15504*a^5 + 3876*a^4*b*x^(1/3) - 816*a^3*b^2*x^(2/3) + 136*a^2*b^3*x - 16*a*b^4*x^(4/3)
+ b^5*x^(5/3)))/(108528*a^6*x^7)

________________________________________________________________________________________

fricas [A]  time = 0.72, size = 169, normalized size = 1.14 \[ -\frac {54264 \, b^{15} x^{5} + 16460080 \, a^{3} b^{12} x^{4} + 135795660 \, a^{6} b^{9} x^{3} + 108636528 \, a^{9} b^{6} x^{2} + 8230040 \, a^{12} b^{3} x + 15504 \, a^{15} + 459 \, {\left (1520 \, a b^{14} x^{4} + 96824 \, a^{4} b^{11} x^{3} + 351120 \, a^{7} b^{8} x^{2} + 133133 \, a^{10} b^{5} x + 3920 \, a^{13} b^{2}\right )} x^{\frac {2}{3}} + 1026 \, {\left (4165 \, a^{2} b^{13} x^{4} + 86632 \, a^{5} b^{10} x^{3} + 145860 \, a^{8} b^{7} x^{2} + 25480 \, a^{11} b^{4} x + 238 \, a^{14} b\right )} x^{\frac {1}{3}}}{108528 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^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 + 108636528*a^9*b^6*x^2 + 8230040*a^
12*b^3*x + 15504*a^15 + 459*(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 + 3
920*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

________________________________________________________________________________________

giac [A]  time = 0.18, size = 167, normalized size = 1.13 \[ -\frac {54264 \, b^{15} x^{5} + 697680 \, a b^{14} x^{\frac {14}{3}} + 4273290 \, a^{2} b^{13} x^{\frac {13}{3}} + 16460080 \, a^{3} b^{12} x^{4} + 44442216 \, a^{4} b^{11} x^{\frac {11}{3}} + 88884432 \, a^{5} b^{10} x^{\frac {10}{3}} + 135795660 \, a^{6} b^{9} x^{3} + 161164080 \, a^{7} b^{8} x^{\frac {8}{3}} + 149652360 \, a^{8} b^{7} x^{\frac {7}{3}} + 108636528 \, a^{9} b^{6} x^{2} + 61108047 \, a^{10} b^{5} x^{\frac {5}{3}} + 26142480 \, a^{11} b^{4} x^{\frac {4}{3}} + 8230040 \, a^{12} b^{3} x + 1799280 \, a^{13} b^{2} x^{\frac {2}{3}} + 244188 \, a^{14} b x^{\frac {1}{3}} + 15504 \, a^{15}}{108528 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^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 + 44442
216*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) + 14965
2360*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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 168, normalized size = 1.14 \[ -\frac {b^{15}}{2 x^{2}}-\frac {45 a \,b^{14}}{7 x^{\frac {7}{3}}}-\frac {315 a^{2} b^{13}}{8 x^{\frac {8}{3}}}-\frac {455 a^{3} b^{12}}{3 x^{3}}-\frac {819 a^{4} b^{11}}{2 x^{\frac {10}{3}}}-\frac {819 a^{5} b^{10}}{x^{\frac {11}{3}}}-\frac {5005 a^{6} b^{9}}{4 x^{4}}-\frac {1485 a^{7} b^{8}}{x^{\frac {13}{3}}}-\frac {19305 a^{8} b^{7}}{14 x^{\frac {14}{3}}}-\frac {1001 a^{9} b^{6}}{x^{5}}-\frac {9009 a^{10} b^{5}}{16 x^{\frac {16}{3}}}-\frac {4095 a^{11} b^{4}}{17 x^{\frac {17}{3}}}-\frac {455 a^{12} b^{3}}{6 x^{6}}-\frac {315 a^{13} b^{2}}{19 x^{\frac {19}{3}}}-\frac {9 a^{14} b}{4 x^{\frac {20}{3}}}-\frac {a^{15}}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^15/x^8,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.87, size = 167, normalized size = 1.13 \[ -\frac {54264 \, b^{15} x^{5} + 697680 \, a b^{14} x^{\frac {14}{3}} + 4273290 \, a^{2} b^{13} x^{\frac {13}{3}} + 16460080 \, a^{3} b^{12} x^{4} + 44442216 \, a^{4} b^{11} x^{\frac {11}{3}} + 88884432 \, a^{5} b^{10} x^{\frac {10}{3}} + 135795660 \, a^{6} b^{9} x^{3} + 161164080 \, a^{7} b^{8} x^{\frac {8}{3}} + 149652360 \, a^{8} b^{7} x^{\frac {7}{3}} + 108636528 \, a^{9} b^{6} x^{2} + 61108047 \, a^{10} b^{5} x^{\frac {5}{3}} + 26142480 \, a^{11} b^{4} x^{\frac {4}{3}} + 8230040 \, a^{12} b^{3} x + 1799280 \, a^{13} b^{2} x^{\frac {2}{3}} + 244188 \, a^{14} b x^{\frac {1}{3}} + 15504 \, a^{15}}{108528 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^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 + 44442
216*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) + 14965
2360*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

________________________________________________________________________________________

mupad [B]  time = 1.34, size = 167, normalized size = 1.13 \[ -\frac {\frac {a^{15}}{7}+\frac {b^{15}\,x^5}{2}+\frac {455\,a^{12}\,b^3\,x}{6}+\frac {9\,a^{14}\,b\,x^{1/3}}{4}+\frac {45\,a\,b^{14}\,x^{14/3}}{7}+1001\,a^9\,b^6\,x^2+\frac {5005\,a^6\,b^9\,x^3}{4}+\frac {455\,a^3\,b^{12}\,x^4}{3}+\frac {315\,a^{13}\,b^2\,x^{2/3}}{19}+\frac {4095\,a^{11}\,b^4\,x^{4/3}}{17}+\frac {9009\,a^{10}\,b^5\,x^{5/3}}{16}+\frac {19305\,a^8\,b^7\,x^{7/3}}{14}+1485\,a^7\,b^8\,x^{8/3}+819\,a^5\,b^{10}\,x^{10/3}+\frac {819\,a^4\,b^{11}\,x^{11/3}}{2}+\frac {315\,a^2\,b^{13}\,x^{13/3}}{8}}{x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/3))^15/x^8,x)

[Out]

-(a^15/7 + (b^15*x^5)/2 + (455*a^12*b^3*x)/6 + (9*a^14*b*x^(1/3))/4 + (45*a*b^14*x^(14/3))/7 + 1001*a^9*b^6*x^
2 + (5005*a^6*b^9*x^3)/4 + (455*a^3*b^12*x^4)/3 + (315*a^13*b^2*x^(2/3))/19 + (4095*a^11*b^4*x^(4/3))/17 + (90
09*a^10*b^5*x^(5/3))/16 + (19305*a^8*b^7*x^(7/3))/14 + 1485*a^7*b^8*x^(8/3) + 819*a^5*b^10*x^(10/3) + (819*a^4
*b^11*x^(11/3))/2 + (315*a^2*b^13*x^(13/3))/8)/x^7

________________________________________________________________________________________

sympy [A]  time = 15.18, 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**(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)

________________________________________________________________________________________