\(\int x \sqrt {a+b (c x^2)^{3/2}} \, dx\) [46]

Optimal result

Integrand size = 19, antiderivative size = 642 \[ \int x \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx=\frac {2}{7} x^2 \sqrt {a+b \left (c x^2\right )^{3/2}}+\frac {6 a \sqrt {a+b \left (c x^2\right )^{3/2}}}{7 b^{2/3} c \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^2-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}\right )|-7-4 \sqrt {3}\right )}{7 b^{2/3} c \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}+\frac {2 \sqrt {2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^2-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}\right ),-7-4 \sqrt {3}\right )}{7 b^{2/3} c \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}} \] Output:

2/7*x^2*(a+b*(c*x^2)^(3/2))^(1/2)+6/7*a*(a+b*(c*x^2)^(3/2))^(1/2)/b^(2/3)/ 
c/((1+3^(1/2))*a^(1/3)+b^(1/3)*(c*x^2)^(1/2))-3/7*3^(1/4)*(1/2*6^(1/2)-1/2 
*2^(1/2))*a^(4/3)*(a^(1/3)+b^(1/3)*(c*x^2)^(1/2))*((a^(2/3)+b^(2/3)*c*x^2- 
a^(1/3)*b^(1/3)*(c*x^2)^(1/2))/((1+3^(1/2))*a^(1/3)+b^(1/3)*(c*x^2)^(1/2)) 
^2)^(1/2)*EllipticE(((1-3^(1/2))*a^(1/3)+b^(1/3)*(c*x^2)^(1/2))/((1+3^(1/2 
))*a^(1/3)+b^(1/3)*(c*x^2)^(1/2)),I*3^(1/2)+2*I)/b^(2/3)/c/(a^(1/3)*(a^(1/ 
3)+b^(1/3)*(c*x^2)^(1/2))/((1+3^(1/2))*a^(1/3)+b^(1/3)*(c*x^2)^(1/2))^2)^( 
1/2)/(a+b*(c*x^2)^(3/2))^(1/2)+2/7*2^(1/2)*3^(3/4)*a^(4/3)*(a^(1/3)+b^(1/3 
)*(c*x^2)^(1/2))*((a^(2/3)+b^(2/3)*c*x^2-a^(1/3)*b^(1/3)*(c*x^2)^(1/2))/(( 
1+3^(1/2))*a^(1/3)+b^(1/3)*(c*x^2)^(1/2))^2)^(1/2)*EllipticF(((1-3^(1/2))* 
a^(1/3)+b^(1/3)*(c*x^2)^(1/2))/((1+3^(1/2))*a^(1/3)+b^(1/3)*(c*x^2)^(1/2)) 
,I*3^(1/2)+2*I)/b^(2/3)/c/(a^(1/3)*(a^(1/3)+b^(1/3)*(c*x^2)^(1/2))/((1+3^( 
1/2))*a^(1/3)+b^(1/3)*(c*x^2)^(1/2))^2)^(1/2)/(a+b*(c*x^2)^(3/2))^(1/2)
 

Mathematica [C] (verified)

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

Time = 10.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.11 \[ \int x \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx=\frac {x^2 \sqrt {a+b \left (c x^2\right )^{3/2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {b \left (c x^2\right )^{3/2}}{a}\right )}{2 \sqrt {1+\frac {b \left (c x^2\right )^{3/2}}{a}}} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.89 (sec) , antiderivative size = 659, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {892, 811, 832, 759, 2416}

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.

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

\(\Big \downarrow \) 892

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

\(\Big \downarrow \) 811

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

\(\Big \downarrow \) 832

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

\(\Big \downarrow \) 759

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

\(\Big \downarrow \) 2416

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

Input:

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

Output:

((2*c*x^2*Sqrt[a + b*(c*x^2)^(3/2)])/7 + (3*a*(((2*Sqrt[a + b*(c*x^2)^(3/2 
)])/(b^(1/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])) - (3^(1/4)*Sqr 
t[2 - Sqrt[3]]*a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2])*Sqrt[(a^(2/3) + b^( 
2/3)*c*x^2 - a^(1/3)*b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3) 
*Sqrt[c*x^2])^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c* 
x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])], -7 - 4*Sqrt[3]])/(b^ 
(1/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2]))/((1 + Sqrt[3])*a^(1/3 
) + b^(1/3)*Sqrt[c*x^2])^2]*Sqrt[a + b*(c*x^2)^(3/2)]))/b^(1/3) - (2*(1 - 
Sqrt[3])*Sqrt[2 + Sqrt[3]]*a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2])*Sqrt[(a 
^(2/3) + b^(2/3)*c*x^2 - a^(1/3)*b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/ 
3) + b^(1/3)*Sqrt[c*x^2])^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^( 
1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])], -7 - 4*S 
qrt[3]])/(3^(1/4)*b^(2/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2]))/( 
(1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*Sqrt[a + b*(c*x^2)^(3/2)]) 
))/7)/c
 

Defintions of rubi rules used

rule 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[a]
 

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 832

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

rule 892

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

rule 2416

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S 
imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt 
[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) 
*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

Maple [A] (verified)

Time = 1.44 (sec) , antiderivative size = 495, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(\frac {\frac {4 c \,x^{2} \sqrt {a +\left (c \,x^{2}\right )^{\frac {3}{2}} b}}{7}-\frac {4 i a \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}} \sqrt {\frac {i \left (\sqrt {c \,x^{2}}+\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-b^{2} a \right )^{\frac {1}{3}}}}\, \sqrt {\frac {\sqrt {c \,x^{2}}-\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (\sqrt {c \,x^{2}}+\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-b^{2} a \right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (\sqrt {c \,x^{2}}+\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-b^{2} a \right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right )}}\right )+\frac {\left (-b^{2} a \right )^{\frac {1}{3}} \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (\sqrt {c \,x^{2}}+\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-b^{2} a \right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{b}\right )}{7 b \sqrt {a +\left (c \,x^{2}\right )^{\frac {3}{2}} b}}}{2 c}\)	\(495\)
default	\(\frac {\frac {4 c \,x^{2} \sqrt {a +\left (c \,x^{2}\right )^{\frac {3}{2}} b}}{7}-\frac {4 i a \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}} \sqrt {\frac {i \left (\sqrt {c \,x^{2}}+\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-b^{2} a \right )^{\frac {1}{3}}}}\, \sqrt {\frac {\sqrt {c \,x^{2}}-\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (\sqrt {c \,x^{2}}+\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-b^{2} a \right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (\sqrt {c \,x^{2}}+\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-b^{2} a \right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right )}}\right )+\frac {\left (-b^{2} a \right )^{\frac {1}{3}} \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (\sqrt {c \,x^{2}}+\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-b^{2} a \right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{b}\right )}{7 b \sqrt {a +\left (c \,x^{2}\right )^{\frac {3}{2}} b}}}{2 c}\)	\(495\)

Input:

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

Output:

1/2/c*(4/7*c*x^2*(a+(c*x^2)^(3/2)*b)^(1/2)-4/7*I*a*3^(1/2)/b*(-b^2*a)^(1/3 
)*(I*((c*x^2)^(1/2)+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3 
^(1/2)*b/(-b^2*a)^(1/3))^(1/2)*(((c*x^2)^(1/2)-1/b*(-b^2*a)^(1/3))/(-3/2/b 
*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2)*(-I*((c*x^2)^(1/2)+ 
1/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1 
/3))^(1/2)/(a+(c*x^2)^(3/2)*b)^(1/2)*((-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2) 
/b*(-b^2*a)^(1/3))*EllipticE(1/3*3^(1/2)*(I*((c*x^2)^(1/2)+1/2/b*(-b^2*a)^ 
(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2),(I*3 
^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1 
/3)))^(1/2))+1/b*(-b^2*a)^(1/3)*EllipticF(1/3*3^(1/2)*(I*((c*x^2)^(1/2)+1/ 
2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3 
))^(1/2),(I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/ 
b*(-b^2*a)^(1/3)))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.16 \[ \int x \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (\sqrt {\sqrt {c x^{2}} b c x^{2} + a} b c^{2} x^{3} - 3 \, \sqrt {c x^{2}} \sqrt {\frac {\sqrt {c x^{2}} b c}{x}} a {\rm weierstrassZeta}\left (0, -\frac {4 \, \sqrt {c x^{2}} a}{b c^{2} x}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, \sqrt {c x^{2}} a}{b c^{2} x}, x\right )\right )\right )}}{7 \, b c^{2} x} \] Input:

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

Output:

2/7*(sqrt(sqrt(c*x^2)*b*c*x^2 + a)*b*c^2*x^3 - 3*sqrt(c*x^2)*sqrt(sqrt(c*x 
^2)*b*c/x)*a*weierstrassZeta(0, -4*sqrt(c*x^2)*a/(b*c^2*x), weierstrassPIn 
verse(0, -4*sqrt(c*x^2)*a/(b*c^2*x), x)))/(b*c^2*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 23.79 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.09 \[ \int x \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx=\frac {x^2\,\sqrt {a+b\,c^{3/2}\,\sqrt {x^6}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {2}{3};\ \frac {5}{3};\ -\frac {b\,c^{3/2}\,\sqrt {x^6}}{a}\right )}{2\,\sqrt {\frac {b\,c^{3/2}\,\sqrt {x^6}}{a}+1}} \] Input:

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

Output:

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

Reduce [F]

\[ \int x \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {\sqrt {c}\, b c \,x^{3}+a}\, x^{2}}{7}-\frac {3 \sqrt {c}\, \left (\int \frac {\sqrt {\sqrt {c}\, b c \,x^{3}+a}\, x^{4}}{-b^{2} c^{3} x^{6}+a^{2}}d x \right ) a b c}{7}+\frac {3 \left (\int \frac {\sqrt {\sqrt {c}\, b c \,x^{3}+a}\, x}{-b^{2} c^{3} x^{6}+a^{2}}d x \right ) a^{2}}{7} \] Input:

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

Output:

(2*sqrt(sqrt(c)*b*c*x**3 + a)*x**2 - 3*sqrt(c)*int((sqrt(sqrt(c)*b*c*x**3 
+ a)*x**4)/(a**2 - b**2*c**3*x**6),x)*a*b*c + 3*int((sqrt(sqrt(c)*b*c*x**3 
 + a)*x)/(a**2 - b**2*c**3*x**6),x)*a**2)/7