\(\int \frac {(a+b \sqrt [3]{x})^{15}}{x^9} \, dx\) [2352]

Optimal result

Integrand size = 15, antiderivative size = 224 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^9} \, dx=-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}+\frac {b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac {7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}-\frac {b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{1012 a^5 x^{20/3}}+\frac {b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{4807 a^6 x^{19/3}}-\frac {b^6 \left (a+b \sqrt [3]{x}\right )^{16}}{28842 a^7 x^6}+\frac {b^7 \left (a+b \sqrt [3]{x}\right )^{16}}{245157 a^8 x^{17/3}}-\frac {b^8 \left (a+b \sqrt [3]{x}\right )^{16}}{3922512 a^9 x^{16/3}} \]

[Out]

-1/8*(a+b*x^(1/3))^16/a/x^8+1/23*b*(a+b*x^(1/3))^16/a^2/x^(23/3)-7/506*b^2*(a+b*x^(1/3))^16/a^3/x^(22/3)+1/253
*b^3*(a+b*x^(1/3))^16/a^4/x^7-1/1012*b^4*(a+b*x^(1/3))^16/a^5/x^(20/3)+1/4807*b^5*(a+b*x^(1/3))^16/a^6/x^(19/3
)-1/28842*b^6*(a+b*x^(1/3))^16/a^7/x^6+1/245157*b^7*(a+b*x^(1/3))^16/a^8/x^(17/3)-1/3922512*b^8*(a+b*x^(1/3))^
16/a^9/x^(16/3)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 47, 37} \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^9} \, dx=-\frac {b^8 \left (a+b \sqrt [3]{x}\right )^{16}}{3922512 a^9 x^{16/3}}+\frac {b^7 \left (a+b \sqrt [3]{x}\right )^{16}}{245157 a^8 x^{17/3}}-\frac {b^6 \left (a+b \sqrt [3]{x}\right )^{16}}{28842 a^7 x^6}+\frac {b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{4807 a^6 x^{19/3}}-\frac {b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{1012 a^5 x^{20/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}-\frac {7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac {b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8} \]

[In]

Int[(a + b*x^(1/3))^15/x^9,x]

[Out]

-1/8*(a + b*x^(1/3))^16/(a*x^8) + (b*(a + b*x^(1/3))^16)/(23*a^2*x^(23/3)) - (7*b^2*(a + b*x^(1/3))^16)/(506*a
^3*x^(22/3)) + (b^3*(a + b*x^(1/3))^16)/(253*a^4*x^7) - (b^4*(a + b*x^(1/3))^16)/(1012*a^5*x^(20/3)) + (b^5*(a
+ b*x^(1/3))^16)/(4807*a^6*x^(19/3)) - (b^6*(a + b*x^(1/3))^16)/(28842*a^7*x^6) + (b^7*(a + b*x^(1/3))^16)/(2
45157*a^8*x^(17/3)) - (b^8*(a + b*x^(1/3))^16)/(3922512*a^9*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 47

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 272

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*} \text {integral}& = 3 \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{25}} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}-\frac {b \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{24}} \, dx,x,\sqrt [3]{x}\right )}{a} \\ & = -\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}+\frac {b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}+\frac {\left (7 b^2\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{23}} \, dx,x,\sqrt [3]{x}\right )}{23 a^2} \\ & = -\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}+\frac {b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac {7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}-\frac {\left (21 b^3\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{22}} \, dx,x,\sqrt [3]{x}\right )}{253 a^3} \\ & = -\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}+\frac {b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac {7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}+\frac {\left (5 b^4\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{21}} \, dx,x,\sqrt [3]{x}\right )}{253 a^4} \\ & = -\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}+\frac {b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac {7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}-\frac {b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{1012 a^5 x^{20/3}}-\frac {b^5 \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{20}} \, dx,x,\sqrt [3]{x}\right )}{253 a^5} \\ & = -\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}+\frac {b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac {7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}-\frac {b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{1012 a^5 x^{20/3}}+\frac {b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{4807 a^6 x^{19/3}}+\frac {\left (3 b^6\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{19}} \, dx,x,\sqrt [3]{x}\right )}{4807 a^6} \\ & = -\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}+\frac {b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac {7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}-\frac {b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{1012 a^5 x^{20/3}}+\frac {b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{4807 a^6 x^{19/3}}-\frac {b^6 \left (a+b \sqrt [3]{x}\right )^{16}}{28842 a^7 x^6}-\frac {b^7 \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{18}} \, dx,x,\sqrt [3]{x}\right )}{14421 a^7} \\ & = -\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}+\frac {b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac {7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}-\frac {b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{1012 a^5 x^{20/3}}+\frac {b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{4807 a^6 x^{19/3}}-\frac {b^6 \left (a+b \sqrt [3]{x}\right )^{16}}{28842 a^7 x^6}+\frac {b^7 \left (a+b \sqrt [3]{x}\right )^{16}}{245157 a^8 x^{17/3}}+\frac {b^8 \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{17}} \, dx,x,\sqrt [3]{x}\right )}{245157 a^8} \\ & = -\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}+\frac {b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac {7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}-\frac {b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{1012 a^5 x^{20/3}}+\frac {b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{4807 a^6 x^{19/3}}-\frac {b^6 \left (a+b \sqrt [3]{x}\right )^{16}}{28842 a^7 x^6}+\frac {b^7 \left (a+b \sqrt [3]{x}\right )^{16}}{245157 a^8 x^{17/3}}-\frac {b^8 \left (a+b \sqrt [3]{x}\right )^{16}}{3922512 a^9 x^{16/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^9} \, dx=\frac {-490314 a^{15}-7674480 a^{14} b \sqrt [3]{x}-56163240 a^{13} b^2 x^{2/3}-254963280 a^{12} b^3 x-803134332 a^{11} b^4 x^{4/3}-1859890032 a^{10} b^5 x^{5/3}-3272028760 a^9 b^6 x^2-4454358480 a^8 b^7 x^{7/3}-4732755885 a^7 b^8 x^{8/3}-3926434512 a^6 b^9 x^3-2524136472 a^5 b^{10} x^{10/3}-1235591280 a^4 b^{11} x^{11/3}-446185740 a^3 b^{12} x^4-112326480 a^2 b^{13} x^{13/3}-17651304 a b^{14} x^{14/3}-1307504 b^{15} x^5}{3922512 x^8} \]

[In]

Integrate[(a + b*x^(1/3))^15/x^9,x]

[Out]

(-490314*a^15 - 7674480*a^14*b*x^(1/3) - 56163240*a^13*b^2*x^(2/3) - 254963280*a^12*b^3*x - 803134332*a^11*b^4
*x^(4/3) - 1859890032*a^10*b^5*x^(5/3) - 3272028760*a^9*b^6*x^2 - 4454358480*a^8*b^7*x^(7/3) - 4732755885*a^7*
b^8*x^(8/3) - 3926434512*a^6*b^9*x^3 - 2524136472*a^5*b^10*x^(10/3) - 1235591280*a^4*b^11*x^(11/3) - 446185740
*a^3*b^12*x^4 - 112326480*a^2*b^13*x^(13/3) - 17651304*a*b^14*x^(14/3) - 1307504*b^15*x^5)/(3922512*x^8)

Maple [A] (verified)

Time = 3.65 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.75

method	result	size
derivativedivides	\(-\frac {65 a^{12} b^{3}}{x^{7}}-\frac {9 a \,b^{14}}{2 x^{\frac {10}{3}}}-\frac {315 a^{2} b^{13}}{11 x^{\frac {11}{3}}}-\frac {315 a^{13} b^{2}}{22 x^{\frac {22}{3}}}-\frac {9009 a^{10} b^{5}}{19 x^{\frac {19}{3}}}-\frac {45 a^{14} b}{23 x^{\frac {23}{3}}}-\frac {819 a^{11} b^{4}}{4 x^{\frac {20}{3}}}-\frac {19305 a^{7} b^{8}}{16 x^{\frac {16}{3}}}-\frac {b^{15}}{3 x^{3}}-\frac {1287 a^{5} b^{10}}{2 x^{\frac {14}{3}}}-\frac {455 a^{3} b^{12}}{4 x^{4}}-\frac {5005 a^{9} b^{6}}{6 x^{6}}-\frac {19305 a^{8} b^{7}}{17 x^{\frac {17}{3}}}-\frac {1001 a^{6} b^{9}}{x^{5}}-\frac {315 a^{4} b^{11}}{x^{\frac {13}{3}}}-\frac {a^{15}}{8 x^{8}}\)	\(168\)
default	\(-\frac {65 a^{12} b^{3}}{x^{7}}-\frac {9 a \,b^{14}}{2 x^{\frac {10}{3}}}-\frac {315 a^{2} b^{13}}{11 x^{\frac {11}{3}}}-\frac {315 a^{13} b^{2}}{22 x^{\frac {22}{3}}}-\frac {9009 a^{10} b^{5}}{19 x^{\frac {19}{3}}}-\frac {45 a^{14} b}{23 x^{\frac {23}{3}}}-\frac {819 a^{11} b^{4}}{4 x^{\frac {20}{3}}}-\frac {19305 a^{7} b^{8}}{16 x^{\frac {16}{3}}}-\frac {b^{15}}{3 x^{3}}-\frac {1287 a^{5} b^{10}}{2 x^{\frac {14}{3}}}-\frac {455 a^{3} b^{12}}{4 x^{4}}-\frac {5005 a^{9} b^{6}}{6 x^{6}}-\frac {19305 a^{8} b^{7}}{17 x^{\frac {17}{3}}}-\frac {1001 a^{6} b^{9}}{x^{5}}-\frac {315 a^{4} b^{11}}{x^{\frac {13}{3}}}-\frac {a^{15}}{8 x^{8}}\)	\(168\)
trager	\(\frac {\left (-1+x \right ) \left (3 a^{15} x^{7}+1560 a^{12} b^{3} x^{7}+20020 a^{9} b^{6} x^{7}+24024 a^{6} b^{9} x^{7}+2730 a^{3} b^{12} x^{7}+8 b^{15} x^{7}+3 a^{15} x^{6}+1560 a^{12} b^{3} x^{6}+20020 a^{9} b^{6} x^{6}+24024 a^{6} b^{9} x^{6}+2730 a^{3} b^{12} x^{6}+8 b^{15} x^{6}+3 a^{15} x^{5}+1560 a^{12} b^{3} x^{5}+20020 a^{9} b^{6} x^{5}+24024 a^{6} b^{9} x^{5}+2730 a^{3} b^{12} x^{5}+8 b^{15} x^{5}+3 x^{4} a^{15}+1560 a^{12} b^{3} x^{4}+20020 a^{9} b^{6} x^{4}+24024 a^{6} b^{9} x^{4}+2730 a^{3} b^{12} x^{4}+3 x^{3} a^{15}+1560 a^{12} b^{3} x^{3}+20020 a^{9} b^{6} x^{3}+24024 a^{6} b^{9} x^{3}+3 x^{2} a^{15}+1560 a^{12} b^{3} x^{2}+20020 a^{9} b^{6} x^{2}+3 x \,a^{15}+1560 a^{12} b^{3} x +3 a^{15}\right )}{24 x^{8}}-\frac {9 \left (54740 b^{12} x^{4}+1230086 a^{3} b^{9} x^{3}+2170740 a^{6} b^{6} x^{2}+391391 a^{9} b^{3} x +3740 a^{12}\right ) a^{2} b}{17204 x^{\frac {23}{3}}}-\frac {9 \left (1672 b^{12} x^{4}+117040 a^{3} b^{9} x^{3}+448305 a^{6} b^{6} x^{2}+176176 a^{9} b^{3} x +5320 a^{12}\right ) a \,b^{2}}{3344 x^{\frac {22}{3}}}\)	\(442\)

[In]

int((a+b*x^(1/3))^15/x^9,x,method=_RETURNVERBOSE)

[Out]

-65/x^7*a^12*b^3-9/2*a*b^14/x^(10/3)-315/11*a^2*b^13/x^(11/3)-315/22*a^13*b^2/x^(22/3)-9009/19*a^10*b^5/x^(19/
3)-45/23*a^14*b/x^(23/3)-819/4*a^11*b^4/x^(20/3)-19305/16*a^7*b^8/x^(16/3)-1/3/x^3*b^15-1287/2*a^5*b^10/x^(14/
3)-455/4*a^3*b^12/x^4-5005/6*a^9*b^6/x^6-19305/17*a^8*b^7/x^(17/3)-1001/x^5*a^6*b^9-315*a^4*b^11/x^(13/3)-1/8*
a^15/x^8

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^9} \, dx=-\frac {1307504 \, b^{15} x^{5} + 446185740 \, a^{3} b^{12} x^{4} + 3926434512 \, a^{6} b^{9} x^{3} + 3272028760 \, a^{9} b^{6} x^{2} + 254963280 \, a^{12} b^{3} x + 490314 \, a^{15} + 10557 \, {\left (1672 \, a b^{14} x^{4} + 117040 \, a^{4} b^{11} x^{3} + 448305 \, a^{7} b^{8} x^{2} + 176176 \, a^{10} b^{5} x + 5320 \, a^{13} b^{2}\right )} x^{\frac {2}{3}} + 2052 \, {\left (54740 \, a^{2} b^{13} x^{4} + 1230086 \, a^{5} b^{10} x^{3} + 2170740 \, a^{8} b^{7} x^{2} + 391391 \, a^{11} b^{4} x + 3740 \, a^{14} b\right )} x^{\frac {1}{3}}}{3922512 \, x^{8}} \]

[In]

integrate((a+b*x^(1/3))^15/x^9,x, algorithm="fricas")

[Out]

-1/3922512*(1307504*b^15*x^5 + 446185740*a^3*b^12*x^4 + 3926434512*a^6*b^9*x^3 + 3272028760*a^9*b^6*x^2 + 2549
63280*a^12*b^3*x + 490314*a^15 + 10557*(1672*a*b^14*x^4 + 117040*a^4*b^11*x^3 + 448305*a^7*b^8*x^2 + 176176*a^
10*b^5*x + 5320*a^13*b^2)*x^(2/3) + 2052*(54740*a^2*b^13*x^4 + 1230086*a^5*b^10*x^3 + 2170740*a^8*b^7*x^2 + 39
1391*a^11*b^4*x + 3740*a^14*b)*x^(1/3))/x^8

Sympy [A] (verification not implemented)

Time = 1.42 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^9} \, dx=- \frac {a^{15}}{8 x^{8}} - \frac {45 a^{14} b}{23 x^{\frac {23}{3}}} - \frac {315 a^{13} b^{2}}{22 x^{\frac {22}{3}}} - \frac {65 a^{12} b^{3}}{x^{7}} - \frac {819 a^{11} b^{4}}{4 x^{\frac {20}{3}}} - \frac {9009 a^{10} b^{5}}{19 x^{\frac {19}{3}}} - \frac {5005 a^{9} b^{6}}{6 x^{6}} - \frac {19305 a^{8} b^{7}}{17 x^{\frac {17}{3}}} - \frac {19305 a^{7} b^{8}}{16 x^{\frac {16}{3}}} - \frac {1001 a^{6} b^{9}}{x^{5}} - \frac {1287 a^{5} b^{10}}{2 x^{\frac {14}{3}}} - \frac {315 a^{4} b^{11}}{x^{\frac {13}{3}}} - \frac {455 a^{3} b^{12}}{4 x^{4}} - \frac {315 a^{2} b^{13}}{11 x^{\frac {11}{3}}} - \frac {9 a b^{14}}{2 x^{\frac {10}{3}}} - \frac {b^{15}}{3 x^{3}} \]

[In]

integrate((a+b*x**(1/3))**15/x**9,x)

[Out]

-a**15/(8*x**8) - 45*a**14*b/(23*x**(23/3)) - 315*a**13*b**2/(22*x**(22/3)) - 65*a**12*b**3/x**7 - 819*a**11*b
**4/(4*x**(20/3)) - 9009*a**10*b**5/(19*x**(19/3)) - 5005*a**9*b**6/(6*x**6) - 19305*a**8*b**7/(17*x**(17/3))
- 19305*a**7*b**8/(16*x**(16/3)) - 1001*a**6*b**9/x**5 - 1287*a**5*b**10/(2*x**(14/3)) - 315*a**4*b**11/x**(13
/3) - 455*a**3*b**12/(4*x**4) - 315*a**2*b**13/(11*x**(11/3)) - 9*a*b**14/(2*x**(10/3)) - b**15/(3*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^9} \, dx=-\frac {1307504 \, b^{15} x^{5} + 17651304 \, a b^{14} x^{\frac {14}{3}} + 112326480 \, a^{2} b^{13} x^{\frac {13}{3}} + 446185740 \, a^{3} b^{12} x^{4} + 1235591280 \, a^{4} b^{11} x^{\frac {11}{3}} + 2524136472 \, a^{5} b^{10} x^{\frac {10}{3}} + 3926434512 \, a^{6} b^{9} x^{3} + 4732755885 \, a^{7} b^{8} x^{\frac {8}{3}} + 4454358480 \, a^{8} b^{7} x^{\frac {7}{3}} + 3272028760 \, a^{9} b^{6} x^{2} + 1859890032 \, a^{10} b^{5} x^{\frac {5}{3}} + 803134332 \, a^{11} b^{4} x^{\frac {4}{3}} + 254963280 \, a^{12} b^{3} x + 56163240 \, a^{13} b^{2} x^{\frac {2}{3}} + 7674480 \, a^{14} b x^{\frac {1}{3}} + 490314 \, a^{15}}{3922512 \, x^{8}} \]

[In]

integrate((a+b*x^(1/3))^15/x^9,x, algorithm="maxima")

[Out]

-1/3922512*(1307504*b^15*x^5 + 17651304*a*b^14*x^(14/3) + 112326480*a^2*b^13*x^(13/3) + 446185740*a^3*b^12*x^4
+ 1235591280*a^4*b^11*x^(11/3) + 2524136472*a^5*b^10*x^(10/3) + 3926434512*a^6*b^9*x^3 + 4732755885*a^7*b^8*x
^(8/3) + 4454358480*a^8*b^7*x^(7/3) + 3272028760*a^9*b^6*x^2 + 1859890032*a^10*b^5*x^(5/3) + 803134332*a^11*b^
4*x^(4/3) + 254963280*a^12*b^3*x + 56163240*a^13*b^2*x^(2/3) + 7674480*a^14*b*x^(1/3) + 490314*a^15)/x^8

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^9} \, dx=-\frac {1307504 \, b^{15} x^{5} + 17651304 \, a b^{14} x^{\frac {14}{3}} + 112326480 \, a^{2} b^{13} x^{\frac {13}{3}} + 446185740 \, a^{3} b^{12} x^{4} + 1235591280 \, a^{4} b^{11} x^{\frac {11}{3}} + 2524136472 \, a^{5} b^{10} x^{\frac {10}{3}} + 3926434512 \, a^{6} b^{9} x^{3} + 4732755885 \, a^{7} b^{8} x^{\frac {8}{3}} + 4454358480 \, a^{8} b^{7} x^{\frac {7}{3}} + 3272028760 \, a^{9} b^{6} x^{2} + 1859890032 \, a^{10} b^{5} x^{\frac {5}{3}} + 803134332 \, a^{11} b^{4} x^{\frac {4}{3}} + 254963280 \, a^{12} b^{3} x + 56163240 \, a^{13} b^{2} x^{\frac {2}{3}} + 7674480 \, a^{14} b x^{\frac {1}{3}} + 490314 \, a^{15}}{3922512 \, x^{8}} \]

[In]

integrate((a+b*x^(1/3))^15/x^9,x, algorithm="giac")

[Out]

-1/3922512*(1307504*b^15*x^5 + 17651304*a*b^14*x^(14/3) + 112326480*a^2*b^13*x^(13/3) + 446185740*a^3*b^12*x^4
+ 1235591280*a^4*b^11*x^(11/3) + 2524136472*a^5*b^10*x^(10/3) + 3926434512*a^6*b^9*x^3 + 4732755885*a^7*b^8*x
^(8/3) + 4454358480*a^8*b^7*x^(7/3) + 3272028760*a^9*b^6*x^2 + 1859890032*a^10*b^5*x^(5/3) + 803134332*a^11*b^
4*x^(4/3) + 254963280*a^12*b^3*x + 56163240*a^13*b^2*x^(2/3) + 7674480*a^14*b*x^(1/3) + 490314*a^15)/x^8

Mupad [B] (verification not implemented)

Time = 6.05 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^9} \, dx=-\frac {\frac {a^{15}}{8}+\frac {b^{15}\,x^5}{3}+65\,a^{12}\,b^3\,x+\frac {45\,a^{14}\,b\,x^{1/3}}{23}+\frac {9\,a\,b^{14}\,x^{14/3}}{2}+\frac {5005\,a^9\,b^6\,x^2}{6}+1001\,a^6\,b^9\,x^3+\frac {455\,a^3\,b^{12}\,x^4}{4}+\frac {315\,a^{13}\,b^2\,x^{2/3}}{22}+\frac {819\,a^{11}\,b^4\,x^{4/3}}{4}+\frac {9009\,a^{10}\,b^5\,x^{5/3}}{19}+\frac {19305\,a^8\,b^7\,x^{7/3}}{17}+\frac {19305\,a^7\,b^8\,x^{8/3}}{16}+\frac {1287\,a^5\,b^{10}\,x^{10/3}}{2}+315\,a^4\,b^{11}\,x^{11/3}+\frac {315\,a^2\,b^{13}\,x^{13/3}}{11}}{x^8} \]

[In]

int((a + b*x^(1/3))^15/x^9,x)

[Out]

-(a^15/8 + (b^15*x^5)/3 + 65*a^12*b^3*x + (45*a^14*b*x^(1/3))/23 + (9*a*b^14*x^(14/3))/2 + (5005*a^9*b^6*x^2)/
6 + 1001*a^6*b^9*x^3 + (455*a^3*b^12*x^4)/4 + (315*a^13*b^2*x^(2/3))/22 + (819*a^11*b^4*x^(4/3))/4 + (9009*a^1
0*b^5*x^(5/3))/19 + (19305*a^8*b^7*x^(7/3))/17 + (19305*a^7*b^8*x^(8/3))/16 + (1287*a^5*b^10*x^(10/3))/2 + 315
*a^4*b^11*x^(11/3) + (315*a^2*b^13*x^(13/3))/11)/x^8