\(\int \frac {\sqrt {a+b (c x^3)^{3/2}}}{x^9} \, dx\) [75]

Optimal result

Integrand size = 21, antiderivative size = 69 \[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx=-\frac {\sqrt {a+b \left (c x^3\right )^{3/2}} \operatorname {Hypergeometric2F1}\left (-\frac {16}{9},-\frac {1}{2},-\frac {7}{9},-\frac {b \left (c x^3\right )^{3/2}}{a}\right )}{8 x^8 \sqrt {1+\frac {b \left (c x^3\right )^{3/2}}{a}}} \] Output:

-1/8*(a+b*(c*x^3)^(3/2))^(1/2)*hypergeom([-16/9, -1/2],[-7/9],-b*(c*x^3)^( 
3/2)/a)/x^8/(1+b*(c*x^3)^(3/2)/a)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [F]

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

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(185\) vs. \(2(69)=138\).

Time = 0.29 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.68, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {893, 864, 809, 847, 889, 888}

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.

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

\(\Big \downarrow \) 893

\(\displaystyle \int \frac {\sqrt {a+b c^{3/2} x^{9/2}}}{x^9}dx\)

\(\Big \downarrow \) 864

\(\displaystyle 2 \int \frac {c^{17/2} x^{17} \sqrt {\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}+a}}{\left (c x^3\right )^{17/2}}d\frac {\sqrt {c x^3}}{\sqrt {c} x}\)

\(\Big \downarrow \) 809

\(\displaystyle 2 \left (\frac {9}{32} b c^{3/2} \int \frac {1}{x^4 \sqrt {\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}+a}}d\frac {\sqrt {c x^3}}{\sqrt {c} x}-\frac {\sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}}}{16 x^8}\right )\)

\(\Big \downarrow \) 847

\(\displaystyle 2 \left (\frac {9}{32} b c^{3/2} \left (-\frac {5 b c^{3/2} \int \frac {\sqrt {c x^3}}{\sqrt {c} x \sqrt {\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}+a}}d\frac {\sqrt {c x^3}}{\sqrt {c} x}}{14 a}-\frac {c^{7/2} x^7 \sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}}}{7 a \left (c x^3\right )^{7/2}}\right )-\frac {\sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}}}{16 x^8}\right )\)

\(\Big \downarrow \) 889

\(\displaystyle 2 \left (\frac {9}{32} b c^{3/2} \left (-\frac {5 b c^{3/2} \sqrt {\frac {b \left (c x^3\right )^{9/2}}{a c^3 x^9}+1} \int \frac {\sqrt {c x^3}}{\sqrt {c} x \sqrt {\frac {b \left (c x^3\right )^{9/2}}{a c^3 x^9}+1}}d\frac {\sqrt {c x^3}}{\sqrt {c} x}}{14 a \sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}}}-\frac {c^{7/2} x^7 \sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}}}{7 a \left (c x^3\right )^{7/2}}\right )-\frac {\sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}}}{16 x^8}\right )\)

\(\Big \downarrow \) 888

\(\displaystyle 2 \left (\frac {9}{32} b c^{3/2} \left (-\frac {5 b c^{3/2} x \sqrt {\frac {b \left (c x^3\right )^{9/2}}{a c^3 x^9}+1} \operatorname {Hypergeometric2F1}\left (\frac {2}{9},\frac {1}{2},\frac {11}{9},-\frac {b \left (c x^3\right )^{9/2}}{a c^3 x^9}\right )}{28 a \sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}}}-\frac {c^{7/2} x^7 \sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}}}{7 a \left (c x^3\right )^{7/2}}\right )-\frac {\sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}}}{16 x^8}\right )\)

Input:

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

Output:

2*(-1/16*Sqrt[a + (b*(c*x^3)^(9/2))/(c^3*x^9)]/x^8 + (9*b*c^(3/2)*(-1/7*(c 
^(7/2)*x^7*Sqrt[a + (b*(c*x^3)^(9/2))/(c^3*x^9)])/(a*(c*x^3)^(7/2)) - (5*b 
*c^(3/2)*x*Sqrt[1 + (b*(c*x^3)^(9/2))/(a*c^3*x^9)]*Hypergeometric2F1[2/9, 
1/2, 11/9, -((b*(c*x^3)^(9/2))/(a*c^3*x^9))])/(28*a*Sqrt[a + (b*(c*x^3)^(9 
/2))/(c^3*x^9)])))/32)
 

Defintions of rubi rules used

rule 809

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* 
x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1)))   I 
nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ 
[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntB 
inomialQ[a, b, c, n, m, p, x]
 

rule 847

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] - Simp[b*((m + n*(p + 1) 
+ 1)/(a*c^n*(m + 1)))   Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a 
, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 864

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi 
nator[n]}, Simp[k   Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x 
^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]
 

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 889

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I 
ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(c*x) 
^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0 
] &&  !(ILtQ[p, 0] || GtQ[a, 0])
 

rule 893

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo 
l] :> With[{k = Denominator[n]}, Subst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x 
], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b, c, 
d, m, p, q}, x] && FractionQ[n]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral(sqrt(a + b*(c*x**3)**(3/2))/x**9, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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