3.3.37 \(\int x^2 (c (a+b x^2)^3)^{3/2} \, dx\) [237]

3.3.37.1 Optimal result

Integrand size = 19, antiderivative size = 253 \[ \int x^2 \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\frac {7}{128} a^3 c x^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {21 a^5 c x \sqrt {c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac {21 a^4 c x^3 \sqrt {c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}-\frac {21 a^{9/2} c \sqrt {c \left (a+b x^2\right )^3} \text {arcsinh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{1024 b^{3/2} \left (1+\frac {b x^2}{a}\right )^{3/2}} \]

output

7/128*a^3*c*x^3*(c*(b*x^2+a)^3)^(1/2)+21/1024*a^5*c*x*(c*(b*x^2+a)^3)^(1/2 
)/b/(b*x^2+a)+21/512*a^4*c*x^3*(c*(b*x^2+a)^3)^(1/2)/(b*x^2+a)+21/320*a^2* 
c*x^3*(b*x^2+a)*(c*(b*x^2+a)^3)^(1/2)+3/40*a*c*x^3*(b*x^2+a)^2*(c*(b*x^2+a 
)^3)^(1/2)+1/12*c*x^3*(b*x^2+a)^3*(c*(b*x^2+a)^3)^(1/2)-21/1024*a^(9/2)*c* 
arcsinh(x*b^(1/2)/a^(1/2))*(c*(b*x^2+a)^3)^(1/2)/b^(3/2)/(1+b*x^2/a)^(3/2)
 

3.3.37.2 Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.56 \[ \int x^2 \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\frac {\left (c \left (a+b x^2\right )^3\right )^{3/2} \left (\sqrt {b} x \sqrt {a+b x^2} \left (315 a^5+4910 a^4 b x^2+11432 a^3 b^2 x^4+12144 a^2 b^3 x^6+6272 a b^4 x^8+1280 b^5 x^{10}\right )+630 a^6 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}-\sqrt {a+b x^2}}\right )\right )}{15360 b^{3/2} \left (a+b x^2\right )^{9/2}} \]

input

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

output

((c*(a + b*x^2)^3)^(3/2)*(Sqrt[b]*x*Sqrt[a + b*x^2]*(315*a^5 + 4910*a^4*b* 
x^2 + 11432*a^3*b^2*x^4 + 12144*a^2*b^3*x^6 + 6272*a*b^4*x^8 + 1280*b^5*x^ 
10) + 630*a^6*ArcTanh[(Sqrt[b]*x)/(Sqrt[a] - Sqrt[a + b*x^2])]))/(15360*b^ 
(3/2)*(a + b*x^2)^(9/2))
 

3.3.37.3 Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.85, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2045, 248, 248, 248, 248, 248, 262, 222}

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.

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

\(\Big \downarrow \) 2045

\(\displaystyle \frac {a^3 c \sqrt {c \left (a+b x^2\right )^3} \int x^2 \left (\frac {b x^2}{a}+1\right )^{9/2}dx}{\left (\frac {b x^2}{a}+1\right )^{3/2}}\)

\(\Big \downarrow \) 248

\(\displaystyle \frac {a^3 c \sqrt {c \left (a+b x^2\right )^3} \left (\frac {3}{4} \int x^2 \left (\frac {b x^2}{a}+1\right )^{7/2}dx+\frac {1}{12} x^3 \left (\frac {b x^2}{a}+1\right )^{9/2}\right )}{\left (\frac {b x^2}{a}+1\right )^{3/2}}\)

\(\Big \downarrow \) 248

\(\displaystyle \frac {a^3 c \sqrt {c \left (a+b x^2\right )^3} \left (\frac {3}{4} \left (\frac {7}{10} \int x^2 \left (\frac {b x^2}{a}+1\right )^{5/2}dx+\frac {1}{10} x^3 \left (\frac {b x^2}{a}+1\right )^{7/2}\right )+\frac {1}{12} x^3 \left (\frac {b x^2}{a}+1\right )^{9/2}\right )}{\left (\frac {b x^2}{a}+1\right )^{3/2}}\)

\(\Big \downarrow \) 248

\(\displaystyle \frac {a^3 c \sqrt {c \left (a+b x^2\right )^3} \left (\frac {3}{4} \left (\frac {7}{10} \left (\frac {5}{8} \int x^2 \left (\frac {b x^2}{a}+1\right )^{3/2}dx+\frac {1}{8} x^3 \left (\frac {b x^2}{a}+1\right )^{5/2}\right )+\frac {1}{10} x^3 \left (\frac {b x^2}{a}+1\right )^{7/2}\right )+\frac {1}{12} x^3 \left (\frac {b x^2}{a}+1\right )^{9/2}\right )}{\left (\frac {b x^2}{a}+1\right )^{3/2}}\)

\(\Big \downarrow \) 248

\(\displaystyle \frac {a^3 c \sqrt {c \left (a+b x^2\right )^3} \left (\frac {3}{4} \left (\frac {7}{10} \left (\frac {5}{8} \left (\frac {1}{2} \int x^2 \sqrt {\frac {b x^2}{a}+1}dx+\frac {1}{6} x^3 \left (\frac {b x^2}{a}+1\right )^{3/2}\right )+\frac {1}{8} x^3 \left (\frac {b x^2}{a}+1\right )^{5/2}\right )+\frac {1}{10} x^3 \left (\frac {b x^2}{a}+1\right )^{7/2}\right )+\frac {1}{12} x^3 \left (\frac {b x^2}{a}+1\right )^{9/2}\right )}{\left (\frac {b x^2}{a}+1\right )^{3/2}}\)

\(\Big \downarrow \) 248

\(\displaystyle \frac {a^3 c \sqrt {c \left (a+b x^2\right )^3} \left (\frac {3}{4} \left (\frac {7}{10} \left (\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \int \frac {x^2}{\sqrt {\frac {b x^2}{a}+1}}dx+\frac {1}{4} x^3 \sqrt {\frac {b x^2}{a}+1}\right )+\frac {1}{6} x^3 \left (\frac {b x^2}{a}+1\right )^{3/2}\right )+\frac {1}{8} x^3 \left (\frac {b x^2}{a}+1\right )^{5/2}\right )+\frac {1}{10} x^3 \left (\frac {b x^2}{a}+1\right )^{7/2}\right )+\frac {1}{12} x^3 \left (\frac {b x^2}{a}+1\right )^{9/2}\right )}{\left (\frac {b x^2}{a}+1\right )^{3/2}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {a^3 c \sqrt {c \left (a+b x^2\right )^3} \left (\frac {3}{4} \left (\frac {7}{10} \left (\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {a x \sqrt {\frac {b x^2}{a}+1}}{2 b}-\frac {a \int \frac {1}{\sqrt {\frac {b x^2}{a}+1}}dx}{2 b}\right )+\frac {1}{4} x^3 \sqrt {\frac {b x^2}{a}+1}\right )+\frac {1}{6} x^3 \left (\frac {b x^2}{a}+1\right )^{3/2}\right )+\frac {1}{8} x^3 \left (\frac {b x^2}{a}+1\right )^{5/2}\right )+\frac {1}{10} x^3 \left (\frac {b x^2}{a}+1\right )^{7/2}\right )+\frac {1}{12} x^3 \left (\frac {b x^2}{a}+1\right )^{9/2}\right )}{\left (\frac {b x^2}{a}+1\right )^{3/2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {a^3 c \left (\frac {3}{4} \left (\frac {7}{10} \left (\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {a x \sqrt {\frac {b x^2}{a}+1}}{2 b}-\frac {a^{3/2} \text {arcsinh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{3/2}}\right )+\frac {1}{4} x^3 \sqrt {\frac {b x^2}{a}+1}\right )+\frac {1}{6} x^3 \left (\frac {b x^2}{a}+1\right )^{3/2}\right )+\frac {1}{8} x^3 \left (\frac {b x^2}{a}+1\right )^{5/2}\right )+\frac {1}{10} x^3 \left (\frac {b x^2}{a}+1\right )^{7/2}\right )+\frac {1}{12} x^3 \left (\frac {b x^2}{a}+1\right )^{9/2}\right ) \sqrt {c \left (a+b x^2\right )^3}}{\left (\frac {b x^2}{a}+1\right )^{3/2}}\)

input

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

output

(a^3*c*Sqrt[c*(a + b*x^2)^3]*((x^3*(1 + (b*x^2)/a)^(9/2))/12 + (3*((x^3*(1 
 + (b*x^2)/a)^(7/2))/10 + (7*((x^3*(1 + (b*x^2)/a)^(5/2))/8 + (5*((x^3*(1 
+ (b*x^2)/a)^(3/2))/6 + ((x^3*Sqrt[1 + (b*x^2)/a])/4 + ((a*x*Sqrt[1 + (b*x 
^2)/a])/(2*b) - (a^(3/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(3/2)))/4)/2)) 
/8))/10))/4))/(1 + (b*x^2)/a)^(3/2)
 

3.3.37.3.1 Defintions of rubi rules used

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 248

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

rule 262

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

rule 2045

Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Simp[Si 
mp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/a))^(p*q)]   Int[u*(1 + b*(x^n/a))^(p*q) 
, x], x] /; FreeQ[{a, b, c, n, p, q}, x] &&  !GeQ[a, 0]
 

3.3.37.4 Maple [A] (verified)

Time = 2.04 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.61

method	result	size
risch	\(\frac {x \left (1280 x^{10} b^{5}+6272 a \,x^{8} b^{4}+12144 a^{2} x^{6} b^{3}+11432 a^{3} x^{4} b^{2}+4910 x^{2} a^{4} b +315 a^{5}\right ) c \sqrt {c \left (b \,x^{2}+a \right )^{3}}}{15360 \left (b \,x^{2}+a \right ) b}-\frac {21 a^{6} \ln \left (\frac {b c x}{\sqrt {b c}}+\sqrt {b c \,x^{2}+a c}\right ) c \sqrt {c \left (b \,x^{2}+a \right )^{3}}\, \sqrt {c \left (b \,x^{2}+a \right )}}{1024 b \sqrt {b c}\, \left (b \,x^{2}+a \right )^{2}}\)	\(155\)
default	\(-\frac {{\left (c \left (b \,x^{2}+a \right )^{3}\right )}^{\frac {3}{2}} \left (-1280 x^{7} \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} b^{3} \sqrt {b c}-3712 \sqrt {b c}\, \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} a \,b^{2} x^{5}-3440 \sqrt {b c}\, \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} a^{2} b \,x^{3}+315 \ln \left (\frac {b c x +\sqrt {b c \,x^{2}+a c}\, \sqrt {b c}}{\sqrt {b c}}\right ) a^{6} c^{3}-840 \sqrt {b c}\, \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} a^{3} x +210 \sqrt {b c}\, \left (b c \,x^{2}+a c \right )^{\frac {3}{2}} a^{4} c x +315 \sqrt {b c}\, \sqrt {b c \,x^{2}+a c}\, a^{5} c^{2} x \right )}{15360 b \left (b \,x^{2}+a \right )^{3} {\left (c \left (b \,x^{2}+a \right )\right )}^{\frac {3}{2}} c \sqrt {b c}}\)	\(236\)

input

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

output

1/15360*x*(1280*b^5*x^10+6272*a*b^4*x^8+12144*a^2*b^3*x^6+11432*a^3*b^2*x^ 
4+4910*a^4*b*x^2+315*a^5)/(b*x^2+a)/b*c*(c*(b*x^2+a)^3)^(1/2)-21/1024*a^6/ 
b*ln(b*c*x/(b*c)^(1/2)+(b*c*x^2+a*c)^(1/2))/(b*c)^(1/2)*c/(b*x^2+a)^2*(c*( 
b*x^2+a)^3)^(1/2)*(c*(b*x^2+a))^(1/2)
 

3.3.37.5 Fricas [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.71 \[ \int x^2 \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\left [\frac {315 \, {\left (a^{6} b c x^{2} + a^{7} c\right )} \sqrt {\frac {c}{b}} \log \left (-\frac {2 \, b^{2} c x^{4} + 3 \, a b c x^{2} + a^{2} c - 2 \, \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} b x \sqrt {\frac {c}{b}}}{b x^{2} + a}\right ) + 2 \, {\left (1280 \, b^{5} c x^{11} + 6272 \, a b^{4} c x^{9} + 12144 \, a^{2} b^{3} c x^{7} + 11432 \, a^{3} b^{2} c x^{5} + 4910 \, a^{4} b c x^{3} + 315 \, a^{5} c x\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{30720 \, {\left (b^{2} x^{2} + a b\right )}}, \frac {315 \, {\left (a^{6} b c x^{2} + a^{7} c\right )} \sqrt {-\frac {c}{b}} \arctan \left (\frac {\sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} b x \sqrt {-\frac {c}{b}}}{b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}\right ) + {\left (1280 \, b^{5} c x^{11} + 6272 \, a b^{4} c x^{9} + 12144 \, a^{2} b^{3} c x^{7} + 11432 \, a^{3} b^{2} c x^{5} + 4910 \, a^{4} b c x^{3} + 315 \, a^{5} c x\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{15360 \, {\left (b^{2} x^{2} + a b\right )}}\right ] \]

input

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

output

[1/30720*(315*(a^6*b*c*x^2 + a^7*c)*sqrt(c/b)*log(-(2*b^2*c*x^4 + 3*a*b*c* 
x^2 + a^2*c - 2*sqrt(b^3*c*x^6 + 3*a*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c)*b* 
x*sqrt(c/b))/(b*x^2 + a)) + 2*(1280*b^5*c*x^11 + 6272*a*b^4*c*x^9 + 12144* 
a^2*b^3*c*x^7 + 11432*a^3*b^2*c*x^5 + 4910*a^4*b*c*x^3 + 315*a^5*c*x)*sqrt 
(b^3*c*x^6 + 3*a*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c))/(b^2*x^2 + a*b), 1/15 
360*(315*(a^6*b*c*x^2 + a^7*c)*sqrt(-c/b)*arctan(sqrt(b^3*c*x^6 + 3*a*b^2* 
c*x^4 + 3*a^2*b*c*x^2 + a^3*c)*b*x*sqrt(-c/b)/(b^2*c*x^4 + 2*a*b*c*x^2 + a 
^2*c)) + (1280*b^5*c*x^11 + 6272*a*b^4*c*x^9 + 12144*a^2*b^3*c*x^7 + 11432 
*a^3*b^2*c*x^5 + 4910*a^4*b*c*x^3 + 315*a^5*c*x)*sqrt(b^3*c*x^6 + 3*a*b^2* 
c*x^4 + 3*a^2*b*c*x^2 + a^3*c))/(b^2*x^2 + a*b)]
 

3.3.37.6 Sympy [F]

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

input

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

output

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

3.3.37.7 Maxima [F]

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

input

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

output

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

3.3.37.8 Giac [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.70 \[ \int x^2 \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\frac {1}{15360} \, {\left (\frac {315 \, a^{6} c \log \left ({\left | -\sqrt {b c} x + \sqrt {b c x^{2} + a c} \right |}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {b c} b} + {\left (\frac {315 \, a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{b} + 2 \, {\left (2455 \, a^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 4 \, {\left (1429 \, a^{3} b \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, {\left (759 \, a^{2} b^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 8 \, {\left (10 \, b^{4} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 49 \, a b^{3} \mathrm {sgn}\left (b x^{2} + a\right )\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt {b c x^{2} + a c} x\right )} c \]

input

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

output

1/15360*(315*a^6*c*log(abs(-sqrt(b*c)*x + sqrt(b*c*x^2 + a*c)))*sgn(b*x^2 
+ a)/(sqrt(b*c)*b) + (315*a^5*sgn(b*x^2 + a)/b + 2*(2455*a^4*sgn(b*x^2 + a 
) + 4*(1429*a^3*b*sgn(b*x^2 + a) + 2*(759*a^2*b^2*sgn(b*x^2 + a) + 8*(10*b 
^4*x^2*sgn(b*x^2 + a) + 49*a*b^3*sgn(b*x^2 + a))*x^2)*x^2)*x^2)*x^2)*sqrt( 
b*c*x^2 + a*c)*x)*c
 

3.3.37.9 Mupad [F(-1)]

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

input

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

output

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