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]

-(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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Rubi [A]  time = 0.0610956, 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, 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 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]]

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 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]

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 = 0.0165697, size = 80, normalized size = 0.54 \[ \frac{\left (a+b \sqrt [3]{x}\right )^{16} \left (-816 a^3 b^2 x^{2/3}+136 a^2 b^3 x+3876 a^4 b \sqrt [3]{x}-15504 a^5-16 a b^4 x^{4/3}+b^5 x^{5/3}\right )}{108528 a^6 x^7} \]

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)

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Maple [A]  time = 0.009, size = 168, normalized size = 1.1 \begin{align*} -{\frac{315\,{a}^{13}{b}^{2}}{19}{x}^{-{\frac{19}{3}}}}-{\frac{5005\,{a}^{6}{b}^{9}}{4\,{x}^{4}}}-1001\,{\frac{{a}^{9}{b}^{6}}{{x}^{5}}}-819\,{\frac{{a}^{5}{b}^{10}}{{x}^{11/3}}}-{\frac{4095\,{a}^{11}{b}^{4}}{17}{x}^{-{\frac{17}{3}}}}-{\frac{9\,{a}^{14}b}{4}{x}^{-{\frac{20}{3}}}}-{\frac{{a}^{15}}{7\,{x}^{7}}}-{\frac{19305\,{a}^{8}{b}^{7}}{14}{x}^{-{\frac{14}{3}}}}-{\frac{455\,{a}^{12}{b}^{3}}{6\,{x}^{6}}}-{\frac{819\,{a}^{4}{b}^{11}}{2}{x}^{-{\frac{10}{3}}}}-{\frac{315\,{a}^{2}{b}^{13}}{8}{x}^{-{\frac{8}{3}}}}-1485\,{\frac{{a}^{7}{b}^{8}}{{x}^{13/3}}}-{\frac{{b}^{15}}{2\,{x}^{2}}}-{\frac{45\,a{b}^{14}}{7}{x}^{-{\frac{7}{3}}}}-{\frac{455\,{a}^{3}{b}^{12}}{3\,{x}^{3}}}-{\frac{9009\,{a}^{10}{b}^{5}}{16}{x}^{-{\frac{16}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.985219, size = 225, normalized size = 1.52 \begin{align*} -\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}} \end{align*}

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

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Fricas [A]  time = 1.47907, size = 482, normalized size = 3.26 \begin{align*} -\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}} \end{align*}

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

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Sympy [A]  time = 17.5759, size = 216, normalized size = 1.46 \begin{align*} - \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}} \end{align*}

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)

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Giac [A]  time = 1.22611, size = 225, normalized size = 1.52 \begin{align*} -\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}} \end{align*}

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