\(\int x^2 (a+b x^3+c x^6)^{3/2} \, dx\) [190]

Optimal result

Integrand size = 20, antiderivative size = 124 \[ \int x^2 \left (a+b x^3+c x^6\right )^{3/2} \, dx=-\frac {\left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{64 c^2}+\frac {\left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 c}+\frac {\left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{128 c^{5/2}} \] Output:

-1/64*(-4*a*c+b^2)*(2*c*x^3+b)*(c*x^6+b*x^3+a)^(1/2)/c^2+1/24*(2*c*x^3+b)* 
(c*x^6+b*x^3+a)^(3/2)/c+1/128*(-4*a*c+b^2)^2*arctanh(1/2*(2*c*x^3+b)/c^(1/ 
2)/(c*x^6+b*x^3+a)^(1/2))/c^(5/2)
 

Mathematica [A] (verified)

Time = 1.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.92 \[ \int x^2 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {\left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6} \left (-3 b^2+20 a c+8 b c x^3+8 c^2 x^6\right )}{192 c^2}+\frac {\left (-b^2+4 a c\right )^2 \text {arctanh}\left (\frac {\sqrt {c} x^3}{-\sqrt {a}+\sqrt {a+b x^3+c x^6}}\right )}{64 c^{5/2}} \] Input:

Integrate[x^2*(a + b*x^3 + c*x^6)^(3/2),x]
 

Output:

((b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6]*(-3*b^2 + 20*a*c + 8*b*c*x^3 + 8*c^ 
2*x^6))/(192*c^2) + ((-b^2 + 4*a*c)^2*ArcTanh[(Sqrt[c]*x^3)/(-Sqrt[a] + Sq 
rt[a + b*x^3 + c*x^6])])/(64*c^(5/2))
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1690, 1087, 1087, 1092, 219}

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.

\(\displaystyle \int x^2 \left (a+b x^3+c x^6\right )^{3/2} \, dx\)

\(\Big \downarrow \) 1690

\(\displaystyle \frac {1}{3} \int \left (c x^6+b x^3+a\right )^{3/2}dx^3\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{3} \left (\frac {\left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^6+b x^3+a}dx^3}{16 c}\right )\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{3} \left (\frac {\left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^6+b x^3+a}}dx^3}{8 c}\right )}{16 c}\right )\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{3} \left (\frac {\left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-x^6}d\frac {2 c x^3+b}{\sqrt {c x^6+b x^3+a}}}{4 c}\right )}{16 c}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{3} \left (\frac {\left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{8 c^{3/2}}\right )}{16 c}\right )\)

Input:

Int[x^2*(a + b*x^3 + c*x^6)^(3/2),x]
 

Output:

(((b + 2*c*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(8*c) - (3*(b^2 - 4*a*c)*(((b + 
 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c 
*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(8*c^(3/2))))/(16*c))/3
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1690

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, 
 b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]
 

Maple [F]

\[\int x^{2} \left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}d x\]

Input:

int(x^2*(c*x^6+b*x^3+a)^(3/2),x)
 

Output:

int(x^2*(c*x^6+b*x^3+a)^(3/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 297, normalized size of antiderivative = 2.40 \[ \int x^2 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\left [\frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (16 \, c^{4} x^{9} + 24 \, b c^{3} x^{6} - 3 \, b^{3} c + 20 \, a b c^{2} + 2 \, {\left (b^{2} c^{2} + 20 \, a c^{3}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{768 \, c^{3}}, -\frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) - 2 \, {\left (16 \, c^{4} x^{9} + 24 \, b c^{3} x^{6} - 3 \, b^{3} c + 20 \, a b c^{2} + 2 \, {\left (b^{2} c^{2} + 20 \, a c^{3}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{384 \, c^{3}}\right ] \] Input:

integrate(x^2*(c*x^6+b*x^3+a)^(3/2),x, algorithm="fricas")
 

Output:

[1/768*(3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(c)*log(-8*c^2*x^6 - 8*b*c*x^ 
3 - b^2 - 4*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)*sqrt(c) - 4*a*c) + 4*(16 
*c^4*x^9 + 24*b*c^3*x^6 - 3*b^3*c + 20*a*b*c^2 + 2*(b^2*c^2 + 20*a*c^3)*x^ 
3)*sqrt(c*x^6 + b*x^3 + a))/c^3, -1/384*(3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)* 
sqrt(-c)*arctan(1/2*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)*sqrt(-c)/(c^2*x^ 
6 + b*c*x^3 + a*c)) - 2*(16*c^4*x^9 + 24*b*c^3*x^6 - 3*b^3*c + 20*a*b*c^2 
+ 2*(b^2*c^2 + 20*a*c^3)*x^3)*sqrt(c*x^6 + b*x^3 + a))/c^3]
 

Sympy [F]

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

integrate(x**2*(c*x**6+b*x**3+a)**(3/2),x)
 

Output:

Integral(x**2*(a + b*x**3 + c*x**6)**(3/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int x^2 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^2*(c*x^6+b*x^3+a)^(3/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.07 \[ \int x^2 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {1}{192} \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, {\left (4 \, {\left (2 \, c x^{3} + 3 \, b\right )} x^{3} + \frac {b^{2} c^{2} + 20 \, a c^{3}}{c^{3}}\right )} x^{3} - \frac {3 \, b^{3} c - 20 \, a b c^{2}}{c^{3}}\right )} - \frac {{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x^{3} - \sqrt {c x^{6} + b x^{3} + a}\right )} \sqrt {c} + b \right |}\right )}{128 \, c^{\frac {5}{2}}} \] Input:

integrate(x^2*(c*x^6+b*x^3+a)^(3/2),x, algorithm="giac")
 

Output:

1/192*sqrt(c*x^6 + b*x^3 + a)*(2*(4*(2*c*x^3 + 3*b)*x^3 + (b^2*c^2 + 20*a* 
c^3)/c^3)*x^3 - (3*b^3*c - 20*a*b*c^2)/c^3) - 1/128*(b^4 - 8*a*b^2*c + 16* 
a^2*c^2)*log(abs(2*(sqrt(c)*x^3 - sqrt(c*x^6 + b*x^3 + a))*sqrt(c) + b))/c 
^(5/2)
 

Mupad [B] (verification not implemented)

Time = 21.23 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.93 \[ \int x^2 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {\left (c\,x^3+\frac {b}{2}\right )\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{12\,c}+\frac {\left (3\,a\,c-\frac {3\,b^2}{4}\right )\,\left (\left (\frac {b}{4\,c}+\frac {x^3}{2}\right )\,\sqrt {c\,x^6+b\,x^3+a}+\frac {\ln \left (\sqrt {c\,x^6+b\,x^3+a}+\frac {c\,x^3+\frac {b}{2}}{\sqrt {c}}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{12\,c} \] Input:

int(x^2*(a + b*x^3 + c*x^6)^(3/2),x)
 

Output:

((b/2 + c*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(12*c) + ((3*a*c - (3*b^2)/4)*(( 
b/(4*c) + x^3/2)*(a + b*x^3 + c*x^6)^(1/2) + (log((a + b*x^3 + c*x^6)^(1/2 
) + (b/2 + c*x^3)/c^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(12*c)
 

Reduce [F]

\[ \int x^2 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {-96 \sqrt {c \,x^{6}+b \,x^{3}+a}\, a^{2} c^{3}+88 \sqrt {c \,x^{6}+b \,x^{3}+a}\, a \,b^{2} c^{2}+80 \sqrt {c \,x^{6}+b \,x^{3}+a}\, a b \,c^{3} x^{3}-12 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b^{4} c +4 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b^{3} c^{2} x^{3}+48 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b^{2} c^{3} x^{6}+32 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b \,c^{4} x^{9}-48 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{6}+b \,x^{3}+a}-\sqrt {c}\, x^{3}\right ) a^{2} b \,c^{2}+24 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{6}+b \,x^{3}+a}-\sqrt {c}\, x^{3}\right ) a \,b^{3} c -3 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{6}+b \,x^{3}+a}-\sqrt {c}\, x^{3}\right ) b^{5}+48 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{6}+b \,x^{3}+a}+\sqrt {c}\, x^{3}\right ) a^{2} b \,c^{2}-24 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{6}+b \,x^{3}+a}+\sqrt {c}\, x^{3}\right ) a \,b^{3} c +3 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{6}+b \,x^{3}+a}+\sqrt {c}\, x^{3}\right ) b^{5}+288 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{8}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a^{2} b \,c^{4}-144 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{8}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a \,b^{3} c^{3}+18 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{8}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) b^{5} c^{2}+288 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{5}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a^{3} c^{4}-54 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{5}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a \,b^{4} c^{2}+9 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{5}}{b c \,x^{9}+a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) b^{6} c}{384 b \,c^{3}} \] Input:

int(x^2*(c*x^6+b*x^3+a)^(3/2),x)
 

Output:

( - 96*sqrt(a + b*x**3 + c*x**6)*a**2*c**3 + 88*sqrt(a + b*x**3 + c*x**6)* 
a*b**2*c**2 + 80*sqrt(a + b*x**3 + c*x**6)*a*b*c**3*x**3 - 12*sqrt(a + b*x 
**3 + c*x**6)*b**4*c + 4*sqrt(a + b*x**3 + c*x**6)*b**3*c**2*x**3 + 48*sqr 
t(a + b*x**3 + c*x**6)*b**2*c**3*x**6 + 32*sqrt(a + b*x**3 + c*x**6)*b*c** 
4*x**9 - 48*sqrt(c)*log(sqrt(a + b*x**3 + c*x**6) - sqrt(c)*x**3)*a**2*b*c 
**2 + 24*sqrt(c)*log(sqrt(a + b*x**3 + c*x**6) - sqrt(c)*x**3)*a*b**3*c - 
3*sqrt(c)*log(sqrt(a + b*x**3 + c*x**6) - sqrt(c)*x**3)*b**5 + 48*sqrt(c)* 
log(sqrt(a + b*x**3 + c*x**6) + sqrt(c)*x**3)*a**2*b*c**2 - 24*sqrt(c)*log 
(sqrt(a + b*x**3 + c*x**6) + sqrt(c)*x**3)*a*b**3*c + 3*sqrt(c)*log(sqrt(a 
 + b*x**3 + c*x**6) + sqrt(c)*x**3)*b**5 + 288*int((sqrt(a + b*x**3 + c*x* 
*6)*x**8)/(a**2 + 2*a*b*x**3 + a*c*x**6 + b**2*x**6 + b*c*x**9),x)*a**2*b* 
c**4 - 144*int((sqrt(a + b*x**3 + c*x**6)*x**8)/(a**2 + 2*a*b*x**3 + a*c*x 
**6 + b**2*x**6 + b*c*x**9),x)*a*b**3*c**3 + 18*int((sqrt(a + b*x**3 + c*x 
**6)*x**8)/(a**2 + 2*a*b*x**3 + a*c*x**6 + b**2*x**6 + b*c*x**9),x)*b**5*c 
**2 + 288*int((sqrt(a + b*x**3 + c*x**6)*x**5)/(a**2 + 2*a*b*x**3 + a*c*x* 
*6 + b**2*x**6 + b*c*x**9),x)*a**3*c**4 - 54*int((sqrt(a + b*x**3 + c*x**6 
)*x**5)/(a**2 + 2*a*b*x**3 + a*c*x**6 + b**2*x**6 + b*c*x**9),x)*a*b**4*c* 
*2 + 9*int((sqrt(a + b*x**3 + c*x**6)*x**5)/(a**2 + 2*a*b*x**3 + a*c*x**6 
+ b**2*x**6 + b*c*x**9),x)*b**6*c)/(384*b*c**3)