\(\int \frac {\sqrt {a+b x^3}}{(c x)^{9/2}} \, dx\) [318]

Optimal result

Integrand size = 19, antiderivative size = 550 \[ \int \frac {\sqrt {a+b x^3}}{(c x)^{9/2}} \, dx=-\frac {2 \sqrt {a+b x^3}}{7 c (c x)^{7/2}}-\frac {6 b \sqrt {a+b x^3}}{7 a c^4 \sqrt {c x}}+\frac {6 \left (1+\sqrt {3}\right ) b^{4/3} \sqrt {c x} \sqrt {a+b x^3}}{7 a c^5 \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}-\frac {6 \sqrt [4]{3} b^{4/3} \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7 a^{2/3} c^5 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3^{3/4} \left (1-\sqrt {3}\right ) b^{4/3} \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7 a^{2/3} c^5 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:

-2/7*(b*x^3+a)^(1/2)/c/(c*x)^(7/2)-6/7*b*(b*x^3+a)^(1/2)/a/c^4/(c*x)^(1/2) 
+6/7*(1+3^(1/2))*b^(4/3)*(c*x)^(1/2)*(b*x^3+a)^(1/2)/a/c^5/(a^(1/3)+(1+3^( 
1/2))*b^(1/3)*x)-6/7*3^(1/4)*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))/a^(2/3)/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)-1/7*3^ 
(3/4)*(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)*InverseJ 
acobiAM(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))/a^(2/3)/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 = 56, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt {a+b x^3}}{(c x)^{9/2}} \, dx=-\frac {2 x \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {7}{6},-\frac {1}{2},-\frac {1}{6},-\frac {b x^3}{a}\right )}{7 (c x)^{9/2} \sqrt {1+\frac {b x^3}{a}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {809, 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.

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

\(\Big \downarrow \) 809

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

\(\Big \downarrow \) 847

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

\(\Big \downarrow \) 851

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

\(\Big \downarrow \) 837

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 766

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

\(\Big \downarrow \) 2420

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

Input:

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

Output:

(-2*Sqrt[a + b*x^3])/(7*c*(c*x)^(7/2)) + (3*b*((-2*Sqrt[a + b*x^3])/(a*c*S 
qrt[c*x]) + (4*b*((((1 + Sqrt[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*c^3)
 

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 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 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 = 0.95 (sec) , antiderivative size = 1125, normalized size of antiderivative = 2.05

method	result	size
risch	\(\text {Expression too large to display}\)	\(1125\)
elliptic	\(\text {Expression too large to display}\)	\(1172\)
default	\(\text {Expression too large to display}\)	\(3074\)

Input:

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

Output:

-2/7*(b*x^3+a)^(1/2)*(3*b*x^3+a)/x^3/a/c^4/(c*x)^(1/2)+6/7/a*b^2*(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*3^(1/2)/b*(-a*b^2)^(1/3))*Elliptic 
E(((-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),...
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt {a+b x^3}}{(c x)^{9/2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {a c} b x^{4} {\rm weierstrassZeta}\left (0, -\frac {4 \, b}{a}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, b}{a}, \frac {1}{x}\right )\right ) - \sqrt {b x^{3} + a} \sqrt {c x} a\right )}}{7 \, a c^{5} x^{4}} \] Input:

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

Output:

2/7*(3*sqrt(a*c)*b*x^4*weierstrassZeta(0, -4*b/a, weierstrassPInverse(0, - 
4*b/a, 1/x)) - sqrt(b*x^3 + a)*sqrt(c*x)*a)/(a*c^5*x^4)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 15.61 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt {a+b x^3}}{(c x)^{9/2}} \, dx=\frac {\sqrt {a} \Gamma \left (- \frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{6}, - \frac {1}{2} \\ - \frac {1}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 c^{\frac {9}{2}} x^{\frac {7}{2}} \Gamma \left (- \frac {1}{6}\right )} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {a+b x^3}}{(c x)^{9/2}} \, dx=\frac {\sqrt {c}\, \left (-2 \sqrt {b \,x^{3}+a}-3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}}{b \,x^{8}+a \,x^{5}}d x \right ) a \,x^{3}\right )}{4 \sqrt {x}\, c^{5} x^{3}} \] Input:

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

Output:

(sqrt(c)*( - 2*sqrt(a + b*x**3) - 3*sqrt(x)*int((sqrt(x)*sqrt(a + b*x**3)) 
/(a*x**5 + b*x**8),x)*a*x**3))/(4*sqrt(x)*c**5*x**3)