\(\int \frac {1}{(c x)^{9/2} (a+b x^3)^{3/2}} \, dx\) [354]

Optimal result

Integrand size = 19, antiderivative size = 581 \[ \int \frac {1}{(c x)^{9/2} \left (a+b x^3\right )^{3/2}} \, dx=\frac {2}{3 a c (c x)^{7/2} \sqrt {a+b x^3}}-\frac {20 \sqrt {a+b x^3}}{21 a^2 c (c x)^{7/2}}+\frac {80 b \sqrt {a+b x^3}}{21 a^3 c^4 \sqrt {c x}}-\frac {80 \left (1+\sqrt {3}\right ) b^{4/3} \sqrt {c x} \sqrt {a+b x^3}}{21 a^3 c^5 \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}+\frac {80 b^{4/3} \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7\ 3^{3/4} a^{8/3} c^5 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {40 \left (1-\sqrt {3}\right ) b^{4/3} \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{21 \sqrt [4]{3} a^{8/3} c^5 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:

2/3/a/c/(c*x)^(7/2)/(b*x^3+a)^(1/2)-20/21*(b*x^3+a)^(1/2)/a^2/c/(c*x)^(7/2 
)+80/21*b*(b*x^3+a)^(1/2)/a^3/c^4/(c*x)^(1/2)-80/21*(1+3^(1/2))*b^(4/3)*(c 
*x)^(1/2)*(b*x^3+a)^(1/2)/a^3/c^5/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)+80/21*b^ 
(4/3)*(c*x)^(1/2)*(a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)* 
x^2)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2)*EllipticE((1-(a^(1/3)+(1-3^( 
1/2))*b^(1/3)*x)^2/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2),1/4*6^(1/2)+1/ 
4*2^(1/2))*3^(1/4)/a^(8/3)/c^5/(b^(1/3)*x*(a^(1/3)+b^(1/3)*x)/(a^(1/3)+(1+ 
3^(1/2))*b^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)+40/63*(1-3^(1/2))*b^(4/3)*(c* 
x)^(1/2)*(a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^( 
1/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2)*InverseJacobiAM(arccos((a^(1/3)+(1-3^ 
(1/2))*b^(1/3)*x)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)),1/4*6^(1/2)+1/4*2^(1/2) 
)*3^(3/4)/a^(8/3)/c^5/(b^(1/3)*x*(a^(1/3)+b^(1/3)*x)/(a^(1/3)+(1+3^(1/2))* 
b^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.10 \[ \int \frac {1}{(c x)^{9/2} \left (a+b x^3\right )^{3/2}} \, dx=-\frac {2 x \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {7}{6},\frac {3}{2},-\frac {1}{6},-\frac {b x^3}{a}\right )}{7 a (c x)^{9/2} \sqrt {a+b x^3}} \] Input:

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

Output:

(-2*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-7/6, 3/2, -1/6, -((b*x^3)/a)] 
)/(7*a*(c*x)^(9/2)*Sqrt[a + b*x^3])
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 657, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {819, 847, 847, 851, 837, 25, 766, 2420}

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.

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

\(\Big \downarrow \) 819

\(\displaystyle \frac {10 \int \frac {1}{(c x)^{9/2} \sqrt {b x^3+a}}dx}{3 a}+\frac {2}{3 a c (c x)^{7/2} \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 847

\(\displaystyle \frac {10 \left (-\frac {4 b \int \frac {1}{(c x)^{3/2} \sqrt {b x^3+a}}dx}{7 a c^3}-\frac {2 \sqrt {a+b x^3}}{7 a c (c x)^{7/2}}\right )}{3 a}+\frac {2}{3 a c (c x)^{7/2} \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 847

\(\displaystyle \frac {10 \left (-\frac {4 b \left (\frac {2 b \int \frac {(c x)^{3/2}}{\sqrt {b x^3+a}}dx}{a c^3}-\frac {2 \sqrt {a+b x^3}}{a c \sqrt {c x}}\right )}{7 a c^3}-\frac {2 \sqrt {a+b x^3}}{7 a c (c x)^{7/2}}\right )}{3 a}+\frac {2}{3 a c (c x)^{7/2} \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {10 \left (-\frac {4 b \left (\frac {4 b \int \frac {c^2 x^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{a c^4}-\frac {2 \sqrt {a+b x^3}}{a c \sqrt {c x}}\right )}{7 a c^3}-\frac {2 \sqrt {a+b x^3}}{7 a c (c x)^{7/2}}\right )}{3 a}+\frac {2}{3 a c (c x)^{7/2} \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 837

\(\displaystyle \frac {10 \left (-\frac {4 b \left (\frac {4 b \left (-\frac {\left (1-\sqrt {3}\right ) a^{2/3} c^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\int -\frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}\right )}{a c^4}-\frac {2 \sqrt {a+b x^3}}{a c \sqrt {c x}}\right )}{7 a c^3}-\frac {2 \sqrt {a+b x^3}}{7 a c (c x)^{7/2}}\right )}{3 a}+\frac {2}{3 a c (c x)^{7/2} \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {10 \left (-\frac {4 b \left (\frac {4 b \left (\frac {\int \frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) a^{2/3} c^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}\right )}{a c^4}-\frac {2 \sqrt {a+b x^3}}{a c \sqrt {c x}}\right )}{7 a c^3}-\frac {2 \sqrt {a+b x^3}}{7 a c (c x)^{7/2}}\right )}{3 a}+\frac {2}{3 a c (c x)^{7/2} \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 766

\(\displaystyle \frac {10 \left (-\frac {4 b \left (\frac {4 b \left (\frac {\int \frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}\right )}{a c^4}-\frac {2 \sqrt {a+b x^3}}{a c \sqrt {c x}}\right )}{7 a c^3}-\frac {2 \sqrt {a+b x^3}}{7 a c (c x)^{7/2}}\right )}{3 a}+\frac {2}{3 a c (c x)^{7/2} \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 2420

\(\displaystyle \frac {10 \left (-\frac {4 b \left (\frac {4 b \left (\frac {\frac {\left (1+\sqrt {3}\right ) c^3 \sqrt {c x} \sqrt {a+b x^3}}{\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x}-\frac {\sqrt [4]{3} \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}\right )}{a c^4}-\frac {2 \sqrt {a+b x^3}}{a c \sqrt {c x}}\right )}{7 a c^3}-\frac {2 \sqrt {a+b x^3}}{7 a c (c x)^{7/2}}\right )}{3 a}+\frac {2}{3 a c (c x)^{7/2} \sqrt {a+b x^3}}\)

Input:

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

Output:

2/(3*a*c*(c*x)^(7/2)*Sqrt[a + b*x^3]) + (10*((-2*Sqrt[a + b*x^3])/(7*a*c*( 
c*x)^(7/2)) - (4*b*((-2*Sqrt[a + b*x^3])/(a*c*Sqrt[c*x]) + (4*b*((((1 + Sq 
rt[3])*c^3*Sqrt[c*x]*Sqrt[a + b*x^3])/(a^(1/3)*c + (1 + Sqrt[3])*b^(1/3)*c 
*x) - (3^(1/4)*a^(1/3)*c*Sqrt[c*x]*(a^(1/3)*c + b^(1/3)*c*x)*Sqrt[(a^(2/3) 
*c^2 - a^(1/3)*b^(1/3)*c^2*x + b^(2/3)*c^2*x^2)/(a^(1/3)*c + (1 + Sqrt[3]) 
*b^(1/3)*c*x)^2]*EllipticE[ArcCos[(a^(1/3)*c + (1 - Sqrt[3])*b^(1/3)*c*x)/ 
(a^(1/3)*c + (1 + Sqrt[3])*b^(1/3)*c*x)], (2 + Sqrt[3])/4])/(Sqrt[(b^(1/3) 
*c*x*(a^(1/3)*c + b^(1/3)*c*x))/(a^(1/3)*c + (1 + Sqrt[3])*b^(1/3)*c*x)^2] 
*Sqrt[a + b*x^3]))/(2*b^(2/3)) - ((1 - Sqrt[3])*a^(1/3)*c*Sqrt[c*x]*(a^(1/ 
3)*c + b^(1/3)*c*x)*Sqrt[(a^(2/3)*c^2 - a^(1/3)*b^(1/3)*c^2*x + b^(2/3)*c^ 
2*x^2)/(a^(1/3)*c + (1 + Sqrt[3])*b^(1/3)*c*x)^2]*EllipticF[ArcCos[(a^(1/3 
)*c + (1 - Sqrt[3])*b^(1/3)*c*x)/(a^(1/3)*c + (1 + Sqrt[3])*b^(1/3)*c*x)], 
 (2 + Sqrt[3])/4])/(4*3^(1/4)*b^(2/3)*Sqrt[(b^(1/3)*c*x*(a^(1/3)*c + b^(1/ 
3)*c*x))/(a^(1/3)*c + (1 + Sqrt[3])*b^(1/3)*c*x)^2]*Sqrt[a + b*x^3])))/(a* 
c^4)))/(7*a*c^3)))/(3*a)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 766

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ 
(s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x 
]
 

rule 819

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

rule 837

Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 
3]], s = Denom[Rt[b/a, 3]]}, Simp[(Sqrt[3] - 1)*(s^2/(2*r^2))   Int[1/Sqrt[ 
a + b*x^6], x], x] - Simp[1/(2*r^2)   Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/S 
qrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, 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 851

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = 
 Denominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ 
n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && 
FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 2420

Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = 
 Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqr 
t[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d*s*x* 
(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2 
*r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]) 
)*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 
 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 
- Sqrt[3])*d, 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.56 (sec) , antiderivative size = 1204, normalized size of antiderivative = 2.07

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1204\)
risch	\(\text {Expression too large to display}\)	\(2213\)
default	\(\text {Expression too large to display}\)	\(3074\)

Input:

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

Output:

(c*x*(b*x^3+a))^(1/2)/(c*x)^(1/2)/(b*x^3+a)^(1/2)*(-2/7/a^2/c^5*(b*c*x^4+a 
*c*x)^(1/2)/x^4+8/21*b/a^3/c^5/x*(b*c*x^4+a*c*x)^(1/2)+58/21*(b*c*x^3+a*c) 
*b/a^3/c^5/(x*(b*c*x^3+a*c))^(1/2)+2/3*b^2/c^4*x^3/a^3/((x^3+a/b)*b*c*x)^( 
1/2)-80/21*b^2/a^3/c^4*(x*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2) 
^(1/3))*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))+(1/2/b*(-a 
*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*((-3/2/b*(-a*b^2)^(1/3)+1/2*I* 
3^(1/2)/b*(-a*b^2)^(1/3))*x/(-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2 
)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)*(x-1/b*(-a*b^2)^(1/3))^2*(1/b*(-a*b 
^2)^(1/3)*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(-1/2/b* 
(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/ 
2)*(1/b*(-a*b^2)^(1/3)*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1 
/3))/(-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2 
)^(1/3)))^(1/2)*(((-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/b 
*(-a*b^2)^(1/3)+1/b^2*(-a*b^2)^(2/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2) 
/b*(-a*b^2)^(1/3))*b/(-a*b^2)^(1/3)*EllipticF(((-3/2/b*(-a*b^2)^(1/3)+1/2* 
I*3^(1/2)/b*(-a*b^2)^(1/3))*x/(-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b 
^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2),((3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1 
/2)/b*(-a*b^2)^(1/3))*(1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3) 
)/(1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(3/2/b*(-a*b^2)^(1 
/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))+(1/2/b*(-a*b^2)^(1/3)+1/2*I...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.16 \[ \int \frac {1}{(c x)^{9/2} \left (a+b x^3\right )^{3/2}} \, dx=-\frac {2 \, {\left (40 \, {\left (b^{2} x^{7} + a b x^{4}\right )} \sqrt {a c} {\rm weierstrassZeta}\left (0, -\frac {4 \, b}{a}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, b}{a}, \frac {1}{x}\right )\right ) + {\left (10 \, a b x^{3} + 3 \, a^{2}\right )} \sqrt {b x^{3} + a} \sqrt {c x}\right )}}{21 \, {\left (a^{3} b c^{5} x^{7} + a^{4} c^{5} x^{4}\right )}} \] Input:

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

Output:

-2/21*(40*(b^2*x^7 + a*b*x^4)*sqrt(a*c)*weierstrassZeta(0, -4*b/a, weierst 
rassPInverse(0, -4*b/a, 1/x)) + (10*a*b*x^3 + 3*a^2)*sqrt(b*x^3 + a)*sqrt( 
c*x))/(a^3*b*c^5*x^7 + a^4*c^5*x^4)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 47.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.09 \[ \int \frac {1}{(c x)^{9/2} \left (a+b x^3\right )^{3/2}} \, dx=\frac {\Gamma \left (- \frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{6}, \frac {3}{2} \\ - \frac {1}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} c^{\frac {9}{2}} x^{\frac {7}{2}} \Gamma \left (- \frac {1}{6}\right )} \] Input:

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

Output:

gamma(-7/6)*hyper((-7/6, 3/2), (-1/6,), b*x**3*exp_polar(I*pi)/a)/(3*a**(3 
/2)*c**(9/2)*x**(7/2)*gamma(-1/6))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(c x)^{9/2} \left (a+b x^3\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}}{b^{2} x^{11}+2 a b \,x^{8}+a^{2} x^{5}}d x \right )}{c^{5}} \] Input:

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

Output:

(sqrt(c)*int((sqrt(x)*sqrt(a + b*x**3))/(a**2*x**5 + 2*a*b*x**8 + b**2*x** 
11),x))/c**5