3.3.39 \(\int (c (a+b x^2)^3)^{3/2} \, dx\) [239]

3.3.39.1 Optimal result

Integrand size = 15, antiderivative size = 207 \[ \int \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\frac {21}{128} a^3 c x \sqrt {c \left (a+b x^2\right )^3}+\frac {63 a^4 c x \sqrt {c \left (a+b x^2\right )^3}}{256 \left (a+b x^2\right )}+\frac {21}{160} a^2 c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{80} a c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{10} c x \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {63 a^{7/2} c \sqrt {c \left (a+b x^2\right )^3} \text {arcsinh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 \sqrt {b} \left (1+\frac {b x^2}{a}\right )^{3/2}} \]

output

21/128*a^3*c*x*(c*(b*x^2+a)^3)^(1/2)+63/256*a^4*c*x*(c*(b*x^2+a)^3)^(1/2)/ 
(b*x^2+a)+21/160*a^2*c*x*(b*x^2+a)*(c*(b*x^2+a)^3)^(1/2)+9/80*a*c*x*(b*x^2 
+a)^2*(c*(b*x^2+a)^3)^(1/2)+1/10*c*x*(b*x^2+a)^3*(c*(b*x^2+a)^3)^(1/2)+63/ 
256*a^(7/2)*c*arcsinh(x*b^(1/2)/a^(1/2))*(c*(b*x^2+a)^3)^(1/2)/(1+b*x^2/a) 
^(3/2)/b^(1/2)
 

3.3.39.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.59 \[ \int \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\frac {\left (c \left (a+b x^2\right )^3\right )^{3/2} \left (\sqrt {b} x \sqrt {a+b x^2} \left (965 a^4+1490 a^3 b x^2+1368 a^2 b^2 x^4+656 a b^3 x^6+128 b^4 x^8\right )-315 a^5 \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )\right )}{1280 \sqrt {b} \left (a+b x^2\right )^{9/2}} \]

input

Integrate[(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]*(965*a^4 + 1490*a^3*b* 
x^2 + 1368*a^2*b^2*x^4 + 656*a*b^3*x^6 + 128*b^4*x^8) - 315*a^5*Log[-(Sqrt 
[b]*x) + Sqrt[a + b*x^2]]))/(1280*Sqrt[b]*(a + b*x^2)^(9/2))
 

3.3.39.3 Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.86, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {2045, 211, 211, 211, 211, 211, 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.

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

\(\Big \downarrow \) 2045

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 222

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

input

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

output

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

3.3.39.3.1 Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 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]
 

3.3.39.4 Maple [A] (verified)

Time = 1.54 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.67

method	result	size
risch	\(\frac {x \left (128 b^{4} x^{8}+656 a \,b^{3} x^{6}+1368 a^{2} b^{2} x^{4}+1490 a^{3} b \,x^{2}+965 a^{4}\right ) c \sqrt {c \left (b \,x^{2}+a \right )^{3}}}{1280 b \,x^{2}+1280 a}+\frac {63 a^{5} \ln \left (\frac {b c x}{\sqrt {b c}}+\sqrt {b c \,x^{2}+a c}\right ) c \sqrt {c \left (b \,x^{2}+a \right )^{3}}\, \sqrt {c \left (b \,x^{2}+a \right )}}{256 \sqrt {b c}\, \left (b \,x^{2}+a \right )^{2}}\)	\(138\)
default	\(\frac {{\left (c \left (b \,x^{2}+a \right )^{3}\right )}^{\frac {3}{2}} \left (128 x^{5} \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} b^{2} \sqrt {b c}+400 \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} \sqrt {b c}\, a b \,x^{3}+440 \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} \sqrt {b c}\, a^{2} x +210 \left (b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {b c}\, a^{3} c x +315 \sqrt {b c \,x^{2}+a c}\, \sqrt {b c}\, a^{4} c^{2} x +315 \ln \left (\frac {b c x +\sqrt {b c \,x^{2}+a c}\, \sqrt {b c}}{\sqrt {b c}}\right ) a^{5} c^{3}\right )}{1280 \left (b \,x^{2}+a \right )^{3} {\left (c \left (b \,x^{2}+a \right )\right )}^{\frac {3}{2}} c \sqrt {b c}}\)	\(205\)

input

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

output

1/1280*x*(128*b^4*x^8+656*a*b^3*x^6+1368*a^2*b^2*x^4+1490*a^3*b*x^2+965*a^ 
4)/(b*x^2+a)*c*(c*(b*x^2+a)^3)^(1/2)+63/256*a^5*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)
 

3.3.39.5 Fricas [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.94 \[ \int \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\left [\frac {315 \, {\left (a^{5} b c x^{2} + a^{6} c\right )} \sqrt {\frac {c}{b}} \log \left (-\frac {2 \, b^{2} c x^{4} + 3 \, a b c x^{2} + a^{2} c + 2 \, \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} b x \sqrt {\frac {c}{b}}}{b x^{2} + a}\right ) + 2 \, {\left (128 \, b^{4} c x^{9} + 656 \, a b^{3} c x^{7} + 1368 \, a^{2} b^{2} c x^{5} + 1490 \, a^{3} b c x^{3} + 965 \, a^{4} c x\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{2560 \, {\left (b x^{2} + a\right )}}, -\frac {315 \, {\left (a^{5} b c x^{2} + a^{6} c\right )} \sqrt {-\frac {c}{b}} \arctan \left (\frac {\sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} b x \sqrt {-\frac {c}{b}}}{b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}\right ) - {\left (128 \, b^{4} c x^{9} + 656 \, a b^{3} c x^{7} + 1368 \, a^{2} b^{2} c x^{5} + 1490 \, a^{3} b c x^{3} + 965 \, a^{4} c x\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{1280 \, {\left (b x^{2} + a\right )}}\right ] \]

input

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

output

[1/2560*(315*(a^5*b*c*x^2 + a^6*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*(128*b^4*c*x^9 + 656*a*b^3*c*x^7 + 1368*a^2*b 
^2*c*x^5 + 1490*a^3*b*c*x^3 + 965*a^4*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*x^2 + a), -1/1280*(315*(a^5*b*c*x^2 + a^6*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)) - (128*b^4*c*x^9 + 656*a 
*b^3*c*x^7 + 1368*a^2*b^2*c*x^5 + 1490*a^3*b*c*x^3 + 965*a^4*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*x^2 + a)]
 

3.3.39.6 Sympy [F]

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

input

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

output

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

3.3.39.7 Maxima [F]

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

input

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

output

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

3.3.39.8 Giac [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.74 \[ \int \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=-\frac {1}{1280} \, {\left (\frac {315 \, a^{5} c \log \left ({\left | -\sqrt {b c} x + \sqrt {b c x^{2} + a c} \right |}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {b c}} - {\left (965 \, a^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, {\left (745 \, a^{3} b \mathrm {sgn}\left (b x^{2} + a\right ) + 4 \, {\left (171 \, a^{2} b^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, {\left (8 \, b^{4} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 41 \, a b^{3} \mathrm {sgn}\left (b x^{2} + a\right )\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt {b c x^{2} + a c} x\right )} c \]

input

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

output

-1/1280*(315*a^5*c*log(abs(-sqrt(b*c)*x + sqrt(b*c*x^2 + a*c)))*sgn(b*x^2 
+ a)/sqrt(b*c) - (965*a^4*sgn(b*x^2 + a) + 2*(745*a^3*b*sgn(b*x^2 + a) + 4 
*(171*a^2*b^2*sgn(b*x^2 + a) + 2*(8*b^4*x^2*sgn(b*x^2 + a) + 41*a*b^3*sgn( 
b*x^2 + a))*x^2)*x^2)*x^2)*sqrt(b*c*x^2 + a*c)*x)*c
 

3.3.39.9 Mupad [F(-1)]

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

input

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

output

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