\(\int (c x)^{9/2} \sqrt {a+b x^3} \, dx\) [315]

Optimal result

Integrand size = 19, antiderivative size = 550 \[ \int (c x)^{9/2} \sqrt {a+b x^3} \, dx=\frac {3 a c^2 (c x)^{5/2} \sqrt {a+b x^3}}{56 b}+\frac {(c x)^{11/2} \sqrt {a+b x^3}}{7 c}-\frac {15 \left (1+\sqrt {3}\right ) a^2 c^4 \sqrt {c x} \sqrt {a+b x^3}}{112 b^{5/3} \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}+\frac {15 \sqrt [4]{3} a^{7/3} c^4 \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{112 b^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {5\ 3^{3/4} \left (1-\sqrt {3}\right ) a^{7/3} c^4 \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{224 b^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:

3/56*a*c^2*(c*x)^(5/2)*(b*x^3+a)^(1/2)/b+1/7*(c*x)^(11/2)*(b*x^3+a)^(1/2)/ 
c-15/112*(1+3^(1/2))*a^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)+15/112*3^(1/4)*a^(7/3)*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))/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)+5/224*3^(3/4)*(1-3^(1/2))*a^(7/3)*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))/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 = 10.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.16 \[ \int (c x)^{9/2} \sqrt {a+b x^3} \, dx=\frac {c^2 (c x)^{5/2} \sqrt {a+b x^3} \left (\left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}}-a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {5}{6},\frac {11}{6},-\frac {b x^3}{a}\right )\right )}{7 b \sqrt {1+\frac {b x^3}{a}}} \] Input:

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

Output:

(c^2*(c*x)^(5/2)*Sqrt[a + b*x^3]*((a + b*x^3)*Sqrt[1 + (b*x^3)/a] - a*Hype 
rgeometric2F1[-1/2, 5/6, 11/6, -((b*x^3)/a)]))/(7*b*Sqrt[1 + (b*x^3)/a])
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 616, 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 = {811, 843, 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.

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

\(\Big \downarrow \) 811

\(\displaystyle \frac {3}{14} a \int \frac {(c x)^{9/2}}{\sqrt {b x^3+a}}dx+\frac {(c x)^{11/2} \sqrt {a+b x^3}}{7 c}\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {3}{14} a \left (\frac {c^2 (c x)^{5/2} \sqrt {a+b x^3}}{4 b}-\frac {5 a c^3 \int \frac {(c x)^{3/2}}{\sqrt {b x^3+a}}dx}{8 b}\right )+\frac {(c x)^{11/2} \sqrt {a+b x^3}}{7 c}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {3}{14} a \left (\frac {c^2 (c x)^{5/2} \sqrt {a+b x^3}}{4 b}-\frac {5 a c^2 \int \frac {c^2 x^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{4 b}\right )+\frac {(c x)^{11/2} \sqrt {a+b x^3}}{7 c}\)

\(\Big \downarrow \) 837

\(\displaystyle \frac {3}{14} a \left (\frac {c^2 (c x)^{5/2} \sqrt {a+b x^3}}{4 b}-\frac {5 a c^2 \left (-\frac {\left (1-\sqrt {3}\right ) a^{2/3} c^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\int -\frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}\right )}{4 b}\right )+\frac {(c x)^{11/2} \sqrt {a+b x^3}}{7 c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3}{14} a \left (\frac {c^2 (c x)^{5/2} \sqrt {a+b x^3}}{4 b}-\frac {5 a c^2 \left (\frac {\int \frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) a^{2/3} c^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}\right )}{4 b}\right )+\frac {(c x)^{11/2} \sqrt {a+b x^3}}{7 c}\)

\(\Big \downarrow \) 766

\(\displaystyle \frac {3}{14} a \left (\frac {c^2 (c x)^{5/2} \sqrt {a+b x^3}}{4 b}-\frac {5 a c^2 \left (\frac {\int \frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}\right )}{4 b}\right )+\frac {(c x)^{11/2} \sqrt {a+b x^3}}{7 c}\)

\(\Big \downarrow \) 2420

\(\displaystyle \frac {3}{14} a \left (\frac {c^2 (c x)^{5/2} \sqrt {a+b x^3}}{4 b}-\frac {5 a c^2 \left (\frac {\frac {\left (1+\sqrt {3}\right ) c^3 \sqrt {c x} \sqrt {a+b x^3}}{\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x}-\frac {\sqrt [4]{3} \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}\right )}{4 b}\right )+\frac {(c x)^{11/2} \sqrt {a+b x^3}}{7 c}\)

Input:

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

Output:

((c*x)^(11/2)*Sqrt[a + b*x^3])/(7*c) + (3*a*((c^2*(c*x)^(5/2)*Sqrt[a + b*x 
^3])/(4*b) - (5*a*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 - Sqr 
t[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])))/(4*b)))/14
 

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 811

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* 
x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 
))   Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I 
GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 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 843

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*(m + n*p + 1))), x] - Simp[ 
a*c^n*((m - n + 1)/(b*(m + n*p + 1)))   Int[(c*x)^(m - n)*(a + b*x^n)^p, x] 
, x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* 
p + 1, 0] && 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 = 1127, normalized size of antiderivative = 2.05

method	result	size
risch	\(\text {Expression too large to display}\)	\(1127\)
elliptic	\(\text {Expression too large to display}\)	\(1145\)
default	\(\text {Expression too large to display}\)	\(2825\)

Input:

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

Output:

1/56*x^3*(8*b*x^3+3*a)*(b*x^3+a)^(1/2)/b*c^5/(c*x)^(1/2)-15/112/b*a^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))*Ell 
ipticE(((-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)))^(...
 

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 40.71 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.08 \[ \int (c x)^{9/2} \sqrt {a+b x^3} \, dx=\frac {\sqrt {a} c^{\frac {9}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{6} \\ \frac {17}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {17}{6}\right )} \] Input:

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

Output:

sqrt(a)*c**(9/2)*x**(11/2)*gamma(11/6)*hyper((-1/2, 11/6), (17/6,), b*x**3 
*exp_polar(I*pi)/a)/(3*gamma(17/6))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c x)^{9/2} \sqrt {a+b x^3} \, dx=\frac {\sqrt {c}\, c^{4} \left (6 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, a \,x^{2}+16 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, b \,x^{5}-15 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}\, x}{b \,x^{3}+a}d x \right ) a^{2}\right )}{112 b} \] Input:

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

Output:

(sqrt(c)*c**4*(6*sqrt(x)*sqrt(a + b*x**3)*a*x**2 + 16*sqrt(x)*sqrt(a + b*x 
**3)*b*x**5 - 15*int((sqrt(x)*sqrt(a + b*x**3)*x)/(a + b*x**3),x)*a**2))/( 
112*b)