\(\int \frac {1}{x^9 (a+b x^{3/2})^{2/3}} \, dx\) [165]

Optimal result

Integrand size = 17, antiderivative size = 158 \[ \int \frac {1}{x^9 \left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {\sqrt [3]{a+b x^{3/2}}}{8 a x^8}+\frac {15 b \sqrt [3]{a+b x^{3/2}}}{104 a^2 x^{13/2}}-\frac {9 b^2 \sqrt [3]{a+b x^{3/2}}}{52 a^3 x^5}+\frac {81 b^3 \sqrt [3]{a+b x^{3/2}}}{364 a^4 x^{7/2}}-\frac {243 b^4 \sqrt [3]{a+b x^{3/2}}}{728 a^5 x^2}+\frac {729 b^5 \sqrt [3]{a+b x^{3/2}}}{728 a^6 \sqrt {x}} \] Output:

-1/8*(a+b*x^(3/2))^(1/3)/a/x^8+15/104*b*(a+b*x^(3/2))^(1/3)/a^2/x^(13/2)-9 
/52*b^2*(a+b*x^(3/2))^(1/3)/a^3/x^5+81/364*b^3*(a+b*x^(3/2))^(1/3)/a^4/x^( 
7/2)-243/728*b^4*(a+b*x^(3/2))^(1/3)/a^5/x^2+729/728*b^5*(a+b*x^(3/2))^(1/ 
3)/a^6/x^(1/2)
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x^9 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {\sqrt [3]{a+b x^{3/2}} \left (-91 a^5+105 a^4 b x^{3/2}-126 a^3 b^2 x^3+162 a^2 b^3 x^{9/2}-243 a b^4 x^6+729 b^5 x^{15/2}\right )}{728 a^6 x^8} \] Input:

Integrate[1/(x^9*(a + b*x^(3/2))^(2/3)),x]
 

Output:

((a + b*x^(3/2))^(1/3)*(-91*a^5 + 105*a^4*b*x^(3/2) - 126*a^3*b^2*x^3 + 16 
2*a^2*b^3*x^(9/2) - 243*a*b^4*x^6 + 729*b^5*x^(15/2)))/(728*a^6*x^8)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {803, 803, 803, 803, 803, 796}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{x^9 \left (a+b x^{3/2}\right )^{2/3}} \, dx\)

\(\Big \downarrow \) 803

\(\displaystyle -\frac {15 b \int \frac {1}{x^{15/2} \left (b x^{3/2}+a\right )^{2/3}}dx}{16 a}-\frac {\sqrt [3]{a+b x^{3/2}}}{8 a x^8}\)

\(\Big \downarrow \) 803

\(\displaystyle -\frac {15 b \left (-\frac {12 b \int \frac {1}{x^6 \left (b x^{3/2}+a\right )^{2/3}}dx}{13 a}-\frac {2 \sqrt [3]{a+b x^{3/2}}}{13 a x^{13/2}}\right )}{16 a}-\frac {\sqrt [3]{a+b x^{3/2}}}{8 a x^8}\)

\(\Big \downarrow \) 803

\(\displaystyle -\frac {15 b \left (-\frac {12 b \left (-\frac {9 b \int \frac {1}{x^{9/2} \left (b x^{3/2}+a\right )^{2/3}}dx}{10 a}-\frac {\sqrt [3]{a+b x^{3/2}}}{5 a x^5}\right )}{13 a}-\frac {2 \sqrt [3]{a+b x^{3/2}}}{13 a x^{13/2}}\right )}{16 a}-\frac {\sqrt [3]{a+b x^{3/2}}}{8 a x^8}\)

\(\Big \downarrow \) 803

\(\displaystyle -\frac {15 b \left (-\frac {12 b \left (-\frac {9 b \left (-\frac {6 b \int \frac {1}{x^3 \left (b x^{3/2}+a\right )^{2/3}}dx}{7 a}-\frac {2 \sqrt [3]{a+b x^{3/2}}}{7 a x^{7/2}}\right )}{10 a}-\frac {\sqrt [3]{a+b x^{3/2}}}{5 a x^5}\right )}{13 a}-\frac {2 \sqrt [3]{a+b x^{3/2}}}{13 a x^{13/2}}\right )}{16 a}-\frac {\sqrt [3]{a+b x^{3/2}}}{8 a x^8}\)

\(\Big \downarrow \) 803

\(\displaystyle -\frac {15 b \left (-\frac {12 b \left (-\frac {9 b \left (-\frac {6 b \left (-\frac {3 b \int \frac {1}{x^{3/2} \left (b x^{3/2}+a\right )^{2/3}}dx}{4 a}-\frac {\sqrt [3]{a+b x^{3/2}}}{2 a x^2}\right )}{7 a}-\frac {2 \sqrt [3]{a+b x^{3/2}}}{7 a x^{7/2}}\right )}{10 a}-\frac {\sqrt [3]{a+b x^{3/2}}}{5 a x^5}\right )}{13 a}-\frac {2 \sqrt [3]{a+b x^{3/2}}}{13 a x^{13/2}}\right )}{16 a}-\frac {\sqrt [3]{a+b x^{3/2}}}{8 a x^8}\)

\(\Big \downarrow \) 796

\(\displaystyle -\frac {15 b \left (-\frac {12 b \left (-\frac {9 b \left (-\frac {6 b \left (\frac {3 b \sqrt [3]{a+b x^{3/2}}}{2 a^2 \sqrt {x}}-\frac {\sqrt [3]{a+b x^{3/2}}}{2 a x^2}\right )}{7 a}-\frac {2 \sqrt [3]{a+b x^{3/2}}}{7 a x^{7/2}}\right )}{10 a}-\frac {\sqrt [3]{a+b x^{3/2}}}{5 a x^5}\right )}{13 a}-\frac {2 \sqrt [3]{a+b x^{3/2}}}{13 a x^{13/2}}\right )}{16 a}-\frac {\sqrt [3]{a+b x^{3/2}}}{8 a x^8}\)

Input:

Int[1/(x^9*(a + b*x^(3/2))^(2/3)),x]
 

Output:

-1/8*(a + b*x^(3/2))^(1/3)/(a*x^8) - (15*b*((-2*(a + b*x^(3/2))^(1/3))/(13 
*a*x^(13/2)) - (12*b*(-1/5*(a + b*x^(3/2))^(1/3)/(a*x^5) - (9*b*((-2*(a + 
b*x^(3/2))^(1/3))/(7*a*x^(7/2)) - (6*b*(-1/2*(a + b*x^(3/2))^(1/3)/(a*x^2) 
 + (3*b*(a + b*x^(3/2))^(1/3))/(2*a^2*Sqrt[x])))/(7*a)))/(10*a)))/(13*a))) 
/(16*a)
 

Defintions of rubi rules used

rule 796

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

rule 803

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*(( 
a + b*x^n)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*(m + 1 
)))   Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x] && I 
LtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
 

Maple [F]

\[\int \frac {1}{x^{9} \left (a +b \,x^{\frac {3}{2}}\right )^{\frac {2}{3}}}d x\]

Input:

int(1/x^9/(a+b*x^(3/2))^(2/3),x)
 

Output:

int(1/x^9/(a+b*x^(3/2))^(2/3),x)
 

Fricas [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.47 \[ \int \frac {1}{x^9 \left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {{\left (243 \, a b^{4} x^{6} + 126 \, a^{3} b^{2} x^{3} + 91 \, a^{5} - 3 \, {\left (243 \, b^{5} x^{7} + 54 \, a^{2} b^{3} x^{4} + 35 \, a^{4} b x\right )} \sqrt {x}\right )} {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}}}{728 \, a^{6} x^{8}} \] Input:

integrate(1/x^9/(a+b*x^(3/2))^(2/3),x, algorithm="fricas")
 

Output:

-1/728*(243*a*b^4*x^6 + 126*a^3*b^2*x^3 + 91*a^5 - 3*(243*b^5*x^7 + 54*a^2 
*b^3*x^4 + 35*a^4*b*x)*sqrt(x))*(b*x^(3/2) + a)^(1/3)/(a^6*x^8)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1554 vs. \(2 (148) = 296\).

Time = 35.39 (sec) , antiderivative size = 1554, normalized size of antiderivative = 9.84 \[ \int \frac {1}{x^9 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\text {Too large to display} \] Input:

integrate(1/x**9/(a+b*x**(3/2))**(2/3),x)
 

Output:

-7280*a**10*b**(76/3)*x**30*(a/(b*x**(3/2)) + 1)**(1/3)*gamma(-16/3)/(729* 
a**11*b**25*x**(75/2)*gamma(2/3) + 3645*a**10*b**26*x**39*gamma(2/3) + 729 
0*a**9*b**27*x**(81/2)*gamma(2/3) + 7290*a**8*b**28*x**42*gamma(2/3) + 364 
5*a**7*b**29*x**(87/2)*gamma(2/3) + 729*a**6*b**30*x**45*gamma(2/3)) - 280 
00*a**9*b**(79/3)*x**(63/2)*(a/(b*x**(3/2)) + 1)**(1/3)*gamma(-16/3)/(729* 
a**11*b**25*x**(75/2)*gamma(2/3) + 3645*a**10*b**26*x**39*gamma(2/3) + 729 
0*a**9*b**27*x**(81/2)*gamma(2/3) + 7290*a**8*b**28*x**42*gamma(2/3) + 364 
5*a**7*b**29*x**(87/2)*gamma(2/3) + 729*a**6*b**30*x**45*gamma(2/3)) - 408 
80*a**8*b**(82/3)*x**33*(a/(b*x**(3/2)) + 1)**(1/3)*gamma(-16/3)/(729*a**1 
1*b**25*x**(75/2)*gamma(2/3) + 3645*a**10*b**26*x**39*gamma(2/3) + 7290*a* 
*9*b**27*x**(81/2)*gamma(2/3) + 7290*a**8*b**28*x**42*gamma(2/3) + 3645*a* 
*7*b**29*x**(87/2)*gamma(2/3) + 729*a**6*b**30*x**45*gamma(2/3)) - 26240*a 
**7*b**(85/3)*x**(69/2)*(a/(b*x**(3/2)) + 1)**(1/3)*gamma(-16/3)/(729*a**1 
1*b**25*x**(75/2)*gamma(2/3) + 3645*a**10*b**26*x**39*gamma(2/3) + 7290*a* 
*9*b**27*x**(81/2)*gamma(2/3) + 7290*a**8*b**28*x**42*gamma(2/3) + 3645*a* 
*7*b**29*x**(87/2)*gamma(2/3) + 729*a**6*b**30*x**45*gamma(2/3)) - 7840*a* 
*6*b**(88/3)*x**36*(a/(b*x**(3/2)) + 1)**(1/3)*gamma(-16/3)/(729*a**11*b** 
25*x**(75/2)*gamma(2/3) + 3645*a**10*b**26*x**39*gamma(2/3) + 7290*a**9*b* 
*27*x**(81/2)*gamma(2/3) + 7290*a**8*b**28*x**42*gamma(2/3) + 3645*a**7*b* 
*29*x**(87/2)*gamma(2/3) + 729*a**6*b**30*x**45*gamma(2/3)) + 24640*a**...
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.65 \[ \int \frac {1}{x^9 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {\frac {1456 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} b^{5}}{\sqrt {x}} - \frac {1820 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {4}{3}} b^{4}}{x^{2}} + \frac {2080 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {7}{3}} b^{3}}{x^{\frac {7}{2}}} - \frac {1456 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {10}{3}} b^{2}}{x^{5}} + \frac {560 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {13}{3}} b}{x^{\frac {13}{2}}} - \frac {91 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {16}{3}}}{x^{8}}}{728 \, a^{6}} \] Input:

integrate(1/x^9/(a+b*x^(3/2))^(2/3),x, algorithm="maxima")
 

Output:

1/728*(1456*(b*x^(3/2) + a)^(1/3)*b^5/sqrt(x) - 1820*(b*x^(3/2) + a)^(4/3) 
*b^4/x^2 + 2080*(b*x^(3/2) + a)^(7/3)*b^3/x^(7/2) - 1456*(b*x^(3/2) + a)^( 
10/3)*b^2/x^5 + 560*(b*x^(3/2) + a)^(13/3)*b/x^(13/2) - 91*(b*x^(3/2) + a) 
^(16/3)/x^8)/a^6
 

Giac [F]

\[ \int \frac {1}{x^9 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\int { \frac {1}{{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {2}{3}} x^{9}} \,d x } \] Input:

integrate(1/x^9/(a+b*x^(3/2))^(2/3),x, algorithm="giac")
 

Output:

integrate(1/((b*x^(3/2) + a)^(2/3)*x^9), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^9 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\int \frac {1}{x^9\,{\left (a+b\,x^{3/2}\right )}^{2/3}} \,d x \] Input:

int(1/(x^9*(a + b*x^(3/2))^(2/3)),x)
 

Output:

int(1/(x^9*(a + b*x^(3/2))^(2/3)), x)
 

Reduce [F]

\[ \int \frac {1}{x^9 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\int \frac {1}{\left (\sqrt {x}\, b x +a \right )^{\frac {2}{3}} x^{9}}d x \] Input:

int(1/x^9/(a+b*x^(3/2))^(2/3),x)
 

Output:

int(1/((sqrt(x)*b*x + a)**(2/3)*x**9),x)