Integrand size = 20, antiderivative size = 163 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{10}} \, dx=-\frac {\left (2 a b+\left (b^2+8 a c\right ) x^3\right ) \sqrt {a+b x^3+c x^6}}{24 a x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{9 x^9}+\frac {b \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{48 a^{3/2}}+\frac {1}{3} c^{3/2} \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right ) \]
-1/9*(c*x^6+b*x^3+a)^(3/2)/x^9+1/48*b*(-12*a*c+b^2)*arctanh(1/2*(b*x^3+2*a )/a^(1/2)/(c*x^6+b*x^3+a)^(1/2))/a^(3/2)+1/3*c^(3/2)*arctanh(1/2*(2*c*x^3+ b)/c^(1/2)/(c*x^6+b*x^3+a)^(1/2))-1/24*(2*a*b+(8*a*c+b^2)*x^3)*(c*x^6+b*x^ 3+a)^(1/2)/a/x^6
Time = 0.67 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{10}} \, dx=\frac {\sqrt {a+b x^3+c x^6} \left (-8 a^2-14 a b x^3-3 b^2 x^6-32 a c x^6\right )}{72 a x^9}+\frac {\left (b^3-12 a b c\right ) \text {arctanh}\left (\frac {-\sqrt {c} x^3+\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )}{24 a^{3/2}}-\frac {1}{3} c^{3/2} \log \left (b+2 c x^3-2 \sqrt {c} \sqrt {a+b x^3+c x^6}\right ) \]
(Sqrt[a + b*x^3 + c*x^6]*(-8*a^2 - 14*a*b*x^3 - 3*b^2*x^6 - 32*a*c*x^6))/( 72*a*x^9) + ((b^3 - 12*a*b*c)*ArcTanh[(-(Sqrt[c]*x^3) + Sqrt[a + b*x^3 + c *x^6])/Sqrt[a]])/(24*a^(3/2)) - (c^(3/2)*Log[b + 2*c*x^3 - 2*Sqrt[c]*Sqrt[ a + b*x^3 + c*x^6]])/3
Time = 0.38 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {1693, 1161, 1229, 27, 1269, 1092, 219, 1154, 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.
| \(\displaystyle \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{10}} \, dx\) |
| \(\Big \downarrow \) 1693 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (c x^6+b x^3+a\right )^{3/2}}{x^{12}}dx^3\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \frac {\left (2 c x^3+b\right ) \sqrt {c x^6+b x^3+a}}{x^9}dx^3-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (-\frac {\int \frac {b \left (b^2-12 a c\right )-16 a c^2 x^3}{2 x^3 \sqrt {c x^6+b x^3+a}}dx^3}{4 a}-\frac {\sqrt {a+b x^3+c x^6} \left (x^3 \left (8 a c+b^2\right )+2 a b\right )}{4 a x^6}\right )-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (-\frac {\int \frac {b \left (b^2-12 a c\right )-16 a c^2 x^3}{x^3 \sqrt {c x^6+b x^3+a}}dx^3}{8 a}-\frac {\sqrt {a+b x^3+c x^6} \left (x^3 \left (8 a c+b^2\right )+2 a b\right )}{4 a x^6}\right )-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (-\frac {b \left (b^2-12 a c\right ) \int \frac {1}{x^3 \sqrt {c x^6+b x^3+a}}dx^3-16 a c^2 \int \frac {1}{\sqrt {c x^6+b x^3+a}}dx^3}{8 a}-\frac {\sqrt {a+b x^3+c x^6} \left (x^3 \left (8 a c+b^2\right )+2 a b\right )}{4 a x^6}\right )-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (-\frac {b \left (b^2-12 a c\right ) \int \frac {1}{x^3 \sqrt {c x^6+b x^3+a}}dx^3-32 a c^2 \int \frac {1}{4 c-x^6}d\frac {2 c x^3+b}{\sqrt {c x^6+b x^3+a}}}{8 a}-\frac {\sqrt {a+b x^3+c x^6} \left (x^3 \left (8 a c+b^2\right )+2 a b\right )}{4 a x^6}\right )-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (-\frac {b \left (b^2-12 a c\right ) \int \frac {1}{x^3 \sqrt {c x^6+b x^3+a}}dx^3-16 a c^{3/2} \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{8 a}-\frac {\sqrt {a+b x^3+c x^6} \left (x^3 \left (8 a c+b^2\right )+2 a b\right )}{4 a x^6}\right )-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (-\frac {-2 b \left (b^2-12 a c\right ) \int \frac {1}{4 a-x^6}d\frac {b x^3+2 a}{\sqrt {c x^6+b x^3+a}}-16 a c^{3/2} \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{8 a}-\frac {\sqrt {a+b x^3+c x^6} \left (x^3 \left (8 a c+b^2\right )+2 a b\right )}{4 a x^6}\right )-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (-\frac {-\frac {b \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{\sqrt {a}}-16 a c^{3/2} \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{8 a}-\frac {\sqrt {a+b x^3+c x^6} \left (x^3 \left (8 a c+b^2\right )+2 a b\right )}{4 a x^6}\right )-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^9}\right )\) |
(-1/3*(a + b*x^3 + c*x^6)^(3/2)/x^9 + (-1/4*((2*a*b + (b^2 + 8*a*c)*x^3)*S qrt[a + b*x^3 + c*x^6])/(a*x^6) - (-((b*(b^2 - 12*a*c)*ArcTanh[(2*a + b*x^ 3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/Sqrt[a]) - 16*a*c^(3/2)*ArcTanh[( b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(8*a))/2)/3
3.3.9.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si mp[p/(e*(m + 1)) Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && IntegerQ [Simplify[(m + 1)/n]]
\[\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}{x^{10}}d x\]
Time = 0.33 (sec) , antiderivative size = 771, normalized size of antiderivative = 4.73 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{10}} \, dx=\left [\frac {48 \, a^{2} c^{\frac {3}{2}} x^{9} \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 ) - 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {a} x^{9} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 4 \, {\left ({\left (3 \, a b^{2} + 32 \, a^{2} c\right )} x^{6} + 14 \, a^{2} b x^{3} + 8 \, a^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{288 \, a^{2} x^{9}}, -\frac {96 \, a^{2} \sqrt {-c} c x^{9} \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 ) + 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {a} x^{9} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 4 \, {\left ({\left (3 \, a b^{2} + 32 \, a^{2} c\right )} x^{6} + 14 \, a^{2} b x^{3} + 8 \, a^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{288 \, a^{2} x^{9}}, \frac {24 \, a^{2} c^{\frac {3}{2}} x^{9} \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 ) - 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {-a} x^{9} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) - 2 \, {\left ({\left (3 \, a b^{2} + 32 \, a^{2} c\right )} x^{6} + 14 \, a^{2} b x^{3} + 8 \, a^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{144 \, a^{2} x^{9}}, -\frac {48 \, a^{2} \sqrt {-c} c x^{9} \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 ) + 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {-a} x^{9} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 2 \, {\left ({\left (3 \, a b^{2} + 32 \, a^{2} c\right )} x^{6} + 14 \, a^{2} b x^{3} + 8 \, a^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{144 \, a^{2} x^{9}}\right ] \]
[1/288*(48*a^2*c^(3/2)*x^9*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) - 3*(b^3 - 12*a*b*c)*sqrt(a)* x^9*log(-((b^2 + 4*a*c)*x^6 + 8*a*b*x^3 - 4*sqrt(c*x^6 + b*x^3 + a)*(b*x^3 + 2*a)*sqrt(a) + 8*a^2)/x^6) - 4*((3*a*b^2 + 32*a^2*c)*x^6 + 14*a^2*b*x^3 + 8*a^3)*sqrt(c*x^6 + b*x^3 + a))/(a^2*x^9), -1/288*(96*a^2*sqrt(-c)*c*x^ 9*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)) + 3*(b^3 - 12*a*b*c)*sqrt(a)*x^9*log(-((b^2 + 4*a*c)*x^6 + 8* a*b*x^3 - 4*sqrt(c*x^6 + b*x^3 + a)*(b*x^3 + 2*a)*sqrt(a) + 8*a^2)/x^6) + 4*((3*a*b^2 + 32*a^2*c)*x^6 + 14*a^2*b*x^3 + 8*a^3)*sqrt(c*x^6 + b*x^3 + a ))/(a^2*x^9), 1/144*(24*a^2*c^(3/2)*x^9*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) - 3*(b^3 - 12*a* b*c)*sqrt(-a)*x^9*arctan(1/2*sqrt(c*x^6 + b*x^3 + a)*(b*x^3 + 2*a)*sqrt(-a )/(a*c*x^6 + a*b*x^3 + a^2)) - 2*((3*a*b^2 + 32*a^2*c)*x^6 + 14*a^2*b*x^3 + 8*a^3)*sqrt(c*x^6 + b*x^3 + a))/(a^2*x^9), -1/144*(48*a^2*sqrt(-c)*c*x^9 *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)) + 3*(b^3 - 12*a*b*c)*sqrt(-a)*x^9*arctan(1/2*sqrt(c*x^6 + b*x^ 3 + a)*(b*x^3 + 2*a)*sqrt(-a)/(a*c*x^6 + a*b*x^3 + a^2)) + 2*((3*a*b^2 + 3 2*a^2*c)*x^6 + 14*a^2*b*x^3 + 8*a^3)*sqrt(c*x^6 + b*x^3 + a))/(a^2*x^9)]
\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{10}} \, dx=\int \frac {\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}{x^{10}}\, dx \]
Exception generated. \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{10}} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{10}} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}}{x^{10}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{10}} \, dx=\int \frac {{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{x^{10}} \,d x \]