\(\int \frac {(c x)^{9/2}}{(a+b x^3)^{3/2}} \, dx\) [351]

Optimal result

Integrand size = 19, antiderivative size = 519 \[ \int \frac {(c x)^{9/2}}{\left (a+b x^3\right )^{3/2}} \, dx=-\frac {2 c^2 (c x)^{5/2}}{3 b \sqrt {a+b x^3}}+\frac {5 \left (1+\sqrt {3}\right ) c^4 \sqrt {c x} \sqrt {a+b x^3}}{3 b^{5/3} \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}-\frac {5 \sqrt [3]{a} c^4 \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3^{3/4} b^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} c^4 \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{6 \sqrt [4]{3} b^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:

-2/3*c^2*(c*x)^(5/2)/b/(b*x^3+a)^(1/2)+5/3*(1+3^(1/2))*c^4*(c*x)^(1/2)*(b* 
x^3+a)^(1/2)/b^(5/3)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)-5/3*a^(1/3)*c^4*(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)/b^(5/3)/(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)-5/18*(1-3^(1/2))*a^(1/3)*c^4*(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)/b^ 
(5/3)/(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 = 10.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.12 \[ \int \frac {(c x)^{9/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {c^2 (c x)^{5/2} \left (1-\sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {3}{2},\frac {11}{6},-\frac {b x^3}{a}\right )\right )}{b \sqrt {a+b x^3}} \] Input:

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

Output:

(c^2*(c*x)^(5/2)*(1 - Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[5/6, 3/2, 11/6 
, -((b*x^3)/a)]))/(b*Sqrt[a + b*x^3])
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 584, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {817, 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.

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

\(\Big \downarrow \) 817

\(\displaystyle \frac {5 c^3 \int \frac {(c x)^{3/2}}{\sqrt {b x^3+a}}dx}{3 b}-\frac {2 c^2 (c x)^{5/2}}{3 b \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {10 c^2 \int \frac {c^2 x^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{3 b}-\frac {2 c^2 (c x)^{5/2}}{3 b \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 837

\(\displaystyle \frac {10 c^2 \left (-\frac {\left (1-\sqrt {3}\right ) a^{2/3} c^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\int -\frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}\right )}{3 b}-\frac {2 c^2 (c x)^{5/2}}{3 b \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {10 c^2 \left (\frac {\int \frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) a^{2/3} c^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}\right )}{3 b}-\frac {2 c^2 (c x)^{5/2}}{3 b \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 766

\(\displaystyle \frac {10 c^2 \left (\frac {\int \frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}\right )}{3 b}-\frac {2 c^2 (c x)^{5/2}}{3 b \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 2420

\(\displaystyle \frac {10 c^2 \left (\frac {\frac {\left (1+\sqrt {3}\right ) c^3 \sqrt {c x} \sqrt {a+b x^3}}{\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x}-\frac {\sqrt [4]{3} \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}\right )}{3 b}-\frac {2 c^2 (c x)^{5/2}}{3 b \sqrt {a+b x^3}}\)

Input:

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

Output:

(-2*c^2*(c*x)^(5/2))/(3*b*Sqrt[a + b*x^3]) + (10*c^2*((((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])))/(3*b)
 

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 817

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( 
n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n 
*((m - n + 1)/(b*n*(p + 1)))   Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x 
] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  ! 
ILtQ[(m + n*(p + 1) + 1)/n, 0] && 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 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 = 1.89 (sec) , antiderivative size = 1122, normalized size of antiderivative = 2.16

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1122\)
default	\(\text {Expression too large to display}\)	\(2715\)

Input:

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

Output:

1/c/x*(c*x)^(1/2)/(b*x^3+a)^(1/2)*(c*x*(b*x^3+a))^(1/2)*(-2/3/b*c^5*x^3/(( 
x^3+a/b)*b*c*x)^(1/2)+5/3*c^5/b*(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))*EllipticE(((-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...
 

Fricas [F]

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

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

Output:

integral(sqrt(b*x^3 + a)*sqrt(c*x)*c^4*x^4/(b^2*x^6 + 2*a*b*x^3 + a^2), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 39.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.08 \[ \int \frac {(c x)^{9/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {c^{\frac {9}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {11}{6} \\ \frac {17}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (\frac {17}{6}\right )} \] Input:

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

Output:

c**(9/2)*x**(11/2)*gamma(11/6)*hyper((3/2, 11/6), (17/6,), b*x**3*exp_pola 
r(I*pi)/a)/(3*a**(3/2)*gamma(17/6))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(c x)^{9/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {\sqrt {c}\, c^{4} \left (2 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, x^{2}-5 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}\, x}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a^{2}-5 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}\, x}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a b \,x^{3}\right )}{2 b \left (b \,x^{3}+a \right )} \] Input:

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

Output:

(sqrt(c)*c**4*(2*sqrt(x)*sqrt(a + b*x**3)*x**2 - 5*int((sqrt(x)*sqrt(a + b 
*x**3)*x)/(a**2 + 2*a*b*x**3 + b**2*x**6),x)*a**2 - 5*int((sqrt(x)*sqrt(a 
+ b*x**3)*x)/(a**2 + 2*a*b*x**3 + b**2*x**6),x)*a*b*x**3))/(2*b*(a + b*x** 
3))