\(\int \frac {(b x^2+c x^4)^{3/2}}{x^{12}} \, dx\) [190]

Optimal result

Integrand size = 19, antiderivative size = 137 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{12}} \, dx=-\frac {c \sqrt {b x^2+c x^4}}{16 x^7}-\frac {c^2 \sqrt {b x^2+c x^4}}{64 b x^5}+\frac {3 c^3 \sqrt {b x^2+c x^4}}{128 b^2 x^3}-\frac {\left (b x^2+c x^4\right )^{3/2}}{8 x^{11}}-\frac {3 c^4 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{128 b^{5/2}} \] Output:

-1/16*c*(c*x^4+b*x^2)^(1/2)/x^7-1/64*c^2*(c*x^4+b*x^2)^(1/2)/b/x^5+3/128*c 
^3*(c*x^4+b*x^2)^(1/2)/b^2/x^3-1/8*(c*x^4+b*x^2)^(3/2)/x^11-3/128*c^4*arct 
anh(b^(1/2)*x/(c*x^4+b*x^2)^(1/2))/b^(5/2)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.83 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{12}} \, dx=-\frac {\sqrt {x^2 \left (b+c x^2\right )} \left (\sqrt {b} \sqrt {b+c x^2} \left (16 b^3+24 b^2 c x^2+2 b c^2 x^4-3 c^3 x^6\right )+3 c^4 x^8 \text {arctanh}\left (\frac {\sqrt {b+c x^2}}{\sqrt {b}}\right )\right )}{128 b^{5/2} x^9 \sqrt {b+c x^2}} \] Input:

Integrate[(b*x^2 + c*x^4)^(3/2)/x^12,x]
 

Output:

-1/128*(Sqrt[x^2*(b + c*x^2)]*(Sqrt[b]*Sqrt[b + c*x^2]*(16*b^3 + 24*b^2*c* 
x^2 + 2*b*c^2*x^4 - 3*c^3*x^6) + 3*c^4*x^8*ArcTanh[Sqrt[b + c*x^2]/Sqrt[b] 
]))/(b^(5/2)*x^9*Sqrt[b + c*x^2])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1425, 1425, 1430, 1430, 1400, 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.

\(\displaystyle \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{12}} \, dx\)

\(\Big \downarrow \) 1425

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

\(\Big \downarrow \) 1425

\(\displaystyle \frac {3}{8} c \left (\frac {1}{6} c \int \frac {1}{x^4 \sqrt {c x^4+b x^2}}dx-\frac {\sqrt {b x^2+c x^4}}{6 x^7}\right )-\frac {\left (b x^2+c x^4\right )^{3/2}}{8 x^{11}}\)

\(\Big \downarrow \) 1430

\(\displaystyle \frac {3}{8} c \left (\frac {1}{6} c \left (-\frac {3 c \int \frac {1}{x^2 \sqrt {c x^4+b x^2}}dx}{4 b}-\frac {\sqrt {b x^2+c x^4}}{4 b x^5}\right )-\frac {\sqrt {b x^2+c x^4}}{6 x^7}\right )-\frac {\left (b x^2+c x^4\right )^{3/2}}{8 x^{11}}\)

\(\Big \downarrow \) 1430

\(\displaystyle \frac {3}{8} c \left (\frac {1}{6} c \left (-\frac {3 c \left (-\frac {c \int \frac {1}{\sqrt {c x^4+b x^2}}dx}{2 b}-\frac {\sqrt {b x^2+c x^4}}{2 b x^3}\right )}{4 b}-\frac {\sqrt {b x^2+c x^4}}{4 b x^5}\right )-\frac {\sqrt {b x^2+c x^4}}{6 x^7}\right )-\frac {\left (b x^2+c x^4\right )^{3/2}}{8 x^{11}}\)

\(\Big \downarrow \) 1400

\(\displaystyle \frac {3}{8} c \left (\frac {1}{6} c \left (-\frac {3 c \left (\frac {c \int \frac {1}{1-\frac {b x^2}{c x^4+b x^2}}d\frac {x}{\sqrt {c x^4+b x^2}}}{2 b}-\frac {\sqrt {b x^2+c x^4}}{2 b x^3}\right )}{4 b}-\frac {\sqrt {b x^2+c x^4}}{4 b x^5}\right )-\frac {\sqrt {b x^2+c x^4}}{6 x^7}\right )-\frac {\left (b x^2+c x^4\right )^{3/2}}{8 x^{11}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {3}{8} c \left (\frac {1}{6} c \left (-\frac {3 c \left (\frac {c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 b^{3/2}}-\frac {\sqrt {b x^2+c x^4}}{2 b x^3}\right )}{4 b}-\frac {\sqrt {b x^2+c x^4}}{4 b x^5}\right )-\frac {\sqrt {b x^2+c x^4}}{6 x^7}\right )-\frac {\left (b x^2+c x^4\right )^{3/2}}{8 x^{11}}\)

Input:

Int[(b*x^2 + c*x^4)^(3/2)/x^12,x]
 

Output:

-1/8*(b*x^2 + c*x^4)^(3/2)/x^11 + (3*c*(-1/6*Sqrt[b*x^2 + c*x^4]/x^7 + (c* 
(-1/4*Sqrt[b*x^2 + c*x^4]/(b*x^5) - (3*c*(-1/2*Sqrt[b*x^2 + c*x^4]/(b*x^3) 
 + (c*ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]])/(2*b^(3/2))))/(4*b)))/6))/ 
8
 

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 1400

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

rule 1425

Int[((d_.)*(x_))^(m_)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp 
[(d*x)^(m + 1)*((b*x^2 + c*x^4)^p/(d*(m + 2*p + 1))), x] - Simp[2*c*(p/(d^4 
*(m + 2*p + 1)))   Int[(d*x)^(m + 4)*(b*x^2 + c*x^4)^(p - 1), x], x] /; Fre 
eQ[{b, c, d, m, p}, x] &&  !IntegerQ[p] && GtQ[p, 0] && LtQ[m + 2*p + 1, 0]
 

rule 1430

Int[((d_.)*(x_))^(m_)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp 
[d*(d*x)^(m - 1)*((b*x^2 + c*x^4)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[c*( 
(m + 4*p + 3)/(b*d^2*(m + 2*p + 1)))   Int[(d*x)^(m + 2)*(b*x^2 + c*x^4)^p, 
 x], x] /; FreeQ[{b, c, d, m, p}, x] &&  !IntegerQ[p] && LtQ[m + 2*p + 1, 0 
]
 

Maple [A] (verified)

Time = 1.48 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.81

method	result	size
risch	\(-\frac {\left (-3 c^{3} x^{6}+2 b \,c^{2} x^{4}+24 b^{2} c \,x^{2}+16 b^{3}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{128 x^{9} b^{2}}-\frac {3 c^{4} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {c \,x^{2}+b}}{x}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{128 b^{\frac {5}{2}} x \sqrt {c \,x^{2}+b}}\)	\(111\)
default	\(-\frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (3 b^{\frac {3}{2}} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {c \,x^{2}+b}}{x}\right ) c^{4} x^{8}-\left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{4} x^{8}+\left (c \,x^{2}+b \right )^{\frac {5}{2}} c^{3} x^{6}-3 \sqrt {c \,x^{2}+b}\, b \,c^{4} x^{8}+2 \left (c \,x^{2}+b \right )^{\frac {5}{2}} b \,c^{2} x^{4}-8 \left (c \,x^{2}+b \right )^{\frac {5}{2}} b^{2} c \,x^{2}+16 \left (c \,x^{2}+b \right )^{\frac {5}{2}} b^{3}\right )}{128 x^{11} \left (c \,x^{2}+b \right )^{\frac {3}{2}} b^{4}}\)	\(165\)

Input:

int((c*x^4+b*x^2)^(3/2)/x^12,x,method=_RETURNVERBOSE)
 

Output:

-1/128*(-3*c^3*x^6+2*b*c^2*x^4+24*b^2*c*x^2+16*b^3)/x^9/b^2*(x^2*(c*x^2+b) 
)^(1/2)-3/128*c^4/b^(5/2)*ln((2*b+2*b^(1/2)*(c*x^2+b)^(1/2))/x)*(x^2*(c*x^ 
2+b))^(1/2)/x/(c*x^2+b)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.47 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{12}} \, dx=\left [\frac {3 \, \sqrt {b} c^{4} x^{9} \log \left (-\frac {c x^{3} + 2 \, b x - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) + 2 \, {\left (3 \, b c^{3} x^{6} - 2 \, b^{2} c^{2} x^{4} - 24 \, b^{3} c x^{2} - 16 \, b^{4}\right )} \sqrt {c x^{4} + b x^{2}}}{256 \, b^{3} x^{9}}, \frac {3 \, \sqrt {-b} c^{4} x^{9} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{b x}\right ) + {\left (3 \, b c^{3} x^{6} - 2 \, b^{2} c^{2} x^{4} - 24 \, b^{3} c x^{2} - 16 \, b^{4}\right )} \sqrt {c x^{4} + b x^{2}}}{128 \, b^{3} x^{9}}\right ] \] Input:

integrate((c*x^4+b*x^2)^(3/2)/x^12,x, algorithm="fricas")
 

Output:

[1/256*(3*sqrt(b)*c^4*x^9*log(-(c*x^3 + 2*b*x - 2*sqrt(c*x^4 + b*x^2)*sqrt 
(b))/x^3) + 2*(3*b*c^3*x^6 - 2*b^2*c^2*x^4 - 24*b^3*c*x^2 - 16*b^4)*sqrt(c 
*x^4 + b*x^2))/(b^3*x^9), 1/128*(3*sqrt(-b)*c^4*x^9*arctan(sqrt(c*x^4 + b* 
x^2)*sqrt(-b)/(b*x)) + (3*b*c^3*x^6 - 2*b^2*c^2*x^4 - 24*b^3*c*x^2 - 16*b^ 
4)*sqrt(c*x^4 + b*x^2))/(b^3*x^9)]
 

Sympy [F]

\[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{12}} \, dx=\int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{12}}\, dx \] Input:

integrate((c*x**4+b*x**2)**(3/2)/x**12,x)
 

Output:

Integral((x**2*(b + c*x**2))**(3/2)/x**12, x)
 

Maxima [F]

\[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{12}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{x^{12}} \,d x } \] Input:

integrate((c*x^4+b*x^2)^(3/2)/x^12,x, algorithm="maxima")
 

Output:

integrate((c*x^4 + b*x^2)^(3/2)/x^12, x)
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.87 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{12}} \, dx=\frac {\frac {3 \, c^{5} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b} b^{2}} + \frac {3 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} c^{5} \mathrm {sgn}\left (x\right ) - 11 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} b c^{5} \mathrm {sgn}\left (x\right ) - 11 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} b^{2} c^{5} \mathrm {sgn}\left (x\right ) + 3 \, \sqrt {c x^{2} + b} b^{3} c^{5} \mathrm {sgn}\left (x\right )}{b^{2} c^{4} x^{8}}}{128 \, c} \] Input:

integrate((c*x^4+b*x^2)^(3/2)/x^12,x, algorithm="giac")
 

Output:

1/128*(3*c^5*arctan(sqrt(c*x^2 + b)/sqrt(-b))*sgn(x)/(sqrt(-b)*b^2) + (3*( 
c*x^2 + b)^(7/2)*c^5*sgn(x) - 11*(c*x^2 + b)^(5/2)*b*c^5*sgn(x) - 11*(c*x^ 
2 + b)^(3/2)*b^2*c^5*sgn(x) + 3*sqrt(c*x^2 + b)*b^3*c^5*sgn(x))/(b^2*c^4*x 
^8))/c
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{12}} \, dx=\int \frac {{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^{12}} \,d x \] Input:

int((b*x^2 + c*x^4)^(3/2)/x^12,x)
 

Output:

int((b*x^2 + c*x^4)^(3/2)/x^12, x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.01 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{12}} \, dx=\frac {-16 \sqrt {c \,x^{2}+b}\, b^{4}-24 \sqrt {c \,x^{2}+b}\, b^{3} c \,x^{2}-2 \sqrt {c \,x^{2}+b}\, b^{2} c^{2} x^{4}+3 \sqrt {c \,x^{2}+b}\, b \,c^{3} x^{6}+3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+b}-\sqrt {b}+\sqrt {c}\, x}{\sqrt {b}}\right ) c^{4} x^{8}-3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+b}+\sqrt {b}+\sqrt {c}\, x}{\sqrt {b}}\right ) c^{4} x^{8}}{128 b^{3} x^{8}} \] Input:

int((c*x^4+b*x^2)^(3/2)/x^12,x)
 

Output:

( - 16*sqrt(b + c*x**2)*b**4 - 24*sqrt(b + c*x**2)*b**3*c*x**2 - 2*sqrt(b 
+ c*x**2)*b**2*c**2*x**4 + 3*sqrt(b + c*x**2)*b*c**3*x**6 + 3*sqrt(b)*log( 
(sqrt(b + c*x**2) - sqrt(b) + sqrt(c)*x)/sqrt(b))*c**4*x**8 - 3*sqrt(b)*lo 
g((sqrt(b + c*x**2) + sqrt(b) + sqrt(c)*x)/sqrt(b))*c**4*x**8)/(128*b**3*x 
**8)