3.2350 \(\int \frac {(a+b \sqrt [3]{x})^{15}}{x^7} \, dx\)

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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {266, 45, 37} \[ -\frac {b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{816 a^3 x^{16/3}}+\frac {b \left (a+b \sqrt [3]{x}\right )^{16}}{51 a^2 x^{17/3}}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{6 a x^6} \]

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))

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^7} \, dx &=3 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{19}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{6 a x^6}-\frac {b \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{18}} \, dx,x,\sqrt [3]{x}\right )}{3 a}\\ &=-\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^{17/3}}+\frac {b^2 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{17}} \, dx,x,\sqrt [3]{x}\right )}{51 a^2}\\ &=-\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^{17/3}}-\frac {b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{816 a^3 x^{16/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 45, normalized size = 0.62 \[ -\frac {\left (a+b \sqrt [3]{x}\right )^{16} \left (136 a^2-16 a b \sqrt [3]{x}+b^2 x^{2/3}\right )}{816 a^3 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^15/x^7,x]

[Out]

-1/816*((a + b*x^(1/3))^16*(136*a^2 - 16*a*b*x^(1/3) + b^2*x^(2/3)))/(a^3*x^6)

________________________________________________________________________________________

fricas [B]  time = 0.82, size = 169, normalized size = 2.35 \[ -\frac {816 \, b^{15} x^{5} + 185640 \, a^{3} b^{12} x^{4} + 1361360 \, a^{6} b^{9} x^{3} + 1021020 \, a^{9} b^{6} x^{2} + 74256 \, a^{12} b^{3} x + 136 \, a^{15} + 459 \, {\left (20 \, a b^{14} x^{4} + 1040 \, a^{4} b^{11} x^{3} + 3432 \, a^{7} b^{8} x^{2} + 1232 \, a^{10} b^{5} x + 35 \, a^{13} b^{2}\right )} x^{\frac {2}{3}} + 54 \, {\left (952 \, a^{2} b^{13} x^{4} + 17017 \, a^{5} b^{10} x^{3} + 26520 \, a^{8} b^{7} x^{2} + 4420 \, a^{11} b^{4} x + 40 \, a^{14} b\right )} x^{\frac {1}{3}}}{816 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^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 + 13
6*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

________________________________________________________________________________________

giac [B]  time = 0.18, size = 167, normalized size = 2.32 \[ -\frac {816 \, b^{15} x^{5} + 9180 \, a b^{14} x^{\frac {14}{3}} + 51408 \, a^{2} b^{13} x^{\frac {13}{3}} + 185640 \, a^{3} b^{12} x^{4} + 477360 \, a^{4} b^{11} x^{\frac {11}{3}} + 918918 \, a^{5} b^{10} x^{\frac {10}{3}} + 1361360 \, a^{6} b^{9} x^{3} + 1575288 \, a^{7} b^{8} x^{\frac {8}{3}} + 1432080 \, a^{8} b^{7} x^{\frac {7}{3}} + 1021020 \, a^{9} b^{6} x^{2} + 565488 \, a^{10} b^{5} x^{\frac {5}{3}} + 238680 \, a^{11} b^{4} x^{\frac {4}{3}} + 74256 \, a^{12} b^{3} x + 16065 \, a^{13} b^{2} x^{\frac {2}{3}} + 2160 \, a^{14} b x^{\frac {1}{3}} + 136 \, a^{15}}{816 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^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

________________________________________________________________________________________

maple [B]  time = 0.01, size = 168, normalized size = 2.33 \[ -\frac {b^{15}}{x}-\frac {45 a \,b^{14}}{4 x^{\frac {4}{3}}}-\frac {63 a^{2} b^{13}}{x^{\frac {5}{3}}}-\frac {455 a^{3} b^{12}}{2 x^{2}}-\frac {585 a^{4} b^{11}}{x^{\frac {7}{3}}}-\frac {9009 a^{5} b^{10}}{8 x^{\frac {8}{3}}}-\frac {5005 a^{6} b^{9}}{3 x^{3}}-\frac {3861 a^{7} b^{8}}{2 x^{\frac {10}{3}}}-\frac {1755 a^{8} b^{7}}{x^{\frac {11}{3}}}-\frac {5005 a^{9} b^{6}}{4 x^{4}}-\frac {693 a^{10} b^{5}}{x^{\frac {13}{3}}}-\frac {585 a^{11} b^{4}}{2 x^{\frac {14}{3}}}-\frac {91 a^{12} b^{3}}{x^{5}}-\frac {315 a^{13} b^{2}}{16 x^{\frac {16}{3}}}-\frac {45 a^{14} b}{17 x^{\frac {17}{3}}}-\frac {a^{15}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^15/x^7,x)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.92, size = 167, normalized size = 2.32 \[ -\frac {816 \, b^{15} x^{5} + 9180 \, a b^{14} x^{\frac {14}{3}} + 51408 \, a^{2} b^{13} x^{\frac {13}{3}} + 185640 \, a^{3} b^{12} x^{4} + 477360 \, a^{4} b^{11} x^{\frac {11}{3}} + 918918 \, a^{5} b^{10} x^{\frac {10}{3}} + 1361360 \, a^{6} b^{9} x^{3} + 1575288 \, a^{7} b^{8} x^{\frac {8}{3}} + 1432080 \, a^{8} b^{7} x^{\frac {7}{3}} + 1021020 \, a^{9} b^{6} x^{2} + 565488 \, a^{10} b^{5} x^{\frac {5}{3}} + 238680 \, a^{11} b^{4} x^{\frac {4}{3}} + 74256 \, a^{12} b^{3} x + 16065 \, a^{13} b^{2} x^{\frac {2}{3}} + 2160 \, a^{14} b x^{\frac {1}{3}} + 136 \, a^{15}}{816 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^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

________________________________________________________________________________________

mupad [B]  time = 0.19, size = 166, normalized size = 2.31 \[ -\frac {\frac {a^{15}}{6}+b^{15}\,x^5+91\,a^{12}\,b^3\,x+\frac {45\,a^{14}\,b\,x^{1/3}}{17}+\frac {45\,a\,b^{14}\,x^{14/3}}{4}+\frac {5005\,a^9\,b^6\,x^2}{4}+\frac {5005\,a^6\,b^9\,x^3}{3}+\frac {455\,a^3\,b^{12}\,x^4}{2}+\frac {315\,a^{13}\,b^2\,x^{2/3}}{16}+\frac {585\,a^{11}\,b^4\,x^{4/3}}{2}+693\,a^{10}\,b^5\,x^{5/3}+1755\,a^8\,b^7\,x^{7/3}+\frac {3861\,a^7\,b^8\,x^{8/3}}{2}+\frac {9009\,a^5\,b^{10}\,x^{10/3}}{8}+585\,a^4\,b^{11}\,x^{11/3}+63\,a^2\,b^{13}\,x^{13/3}}{x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/3))^15/x^7,x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 10.49, size = 209, normalized size = 2.90 \[ - \frac {a^{15}}{6 x^{6}} - \frac {45 a^{14} b}{17 x^{\frac {17}{3}}} - \frac {315 a^{13} b^{2}}{16 x^{\frac {16}{3}}} - \frac {91 a^{12} b^{3}}{x^{5}} - \frac {585 a^{11} b^{4}}{2 x^{\frac {14}{3}}} - \frac {693 a^{10} b^{5}}{x^{\frac {13}{3}}} - \frac {5005 a^{9} b^{6}}{4 x^{4}} - \frac {1755 a^{8} b^{7}}{x^{\frac {11}{3}}} - \frac {3861 a^{7} b^{8}}{2 x^{\frac {10}{3}}} - \frac {5005 a^{6} b^{9}}{3 x^{3}} - \frac {9009 a^{5} b^{10}}{8 x^{\frac {8}{3}}} - \frac {585 a^{4} b^{11}}{x^{\frac {7}{3}}} - \frac {455 a^{3} b^{12}}{2 x^{2}} - \frac {63 a^{2} b^{13}}{x^{\frac {5}{3}}} - \frac {45 a b^{14}}{4 x^{\frac {4}{3}}} - \frac {b^{15}}{x} \]

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) - 5005*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

________________________________________________________________________________________