\(\int \frac {(a+b x^3+c x^6)^{3/2}}{x^{19}} \, dx\) [212]

Optimal result

Integrand size = 20, antiderivative size = 216 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{19}} \, dx=\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 a^4 x^6}-\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 a^3 x^{12}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}}+\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 a^2 x^{15}}-\frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{3072 a^{9/2}} \]

[Out]

-1/576*(-4*a*c+7*b^2)*(b*x^3+2*a)*(c*x^6+b*x^3+a)^(3/2)/a^3/x^12-1/18*(c*x^6+b*x^3+a)^(5/2)/a/x^18+7/180*b*(c*
x^6+b*x^3+a)^(5/2)/a^2/x^15-1/3072*(-4*a*c+b^2)^2*(-4*a*c+7*b^2)*arctanh(1/2*(b*x^3+2*a)/a^(1/2)/(c*x^6+b*x^3+
a)^(1/2))/a^(9/2)+1/1536*(-4*a*c+b^2)*(-4*a*c+7*b^2)*(b*x^3+2*a)*(c*x^6+b*x^3+a)^(1/2)/a^4/x^6

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1371, 758, 820, 734, 738, 212} \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{19}} \, dx=-\frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{3072 a^{9/2}}+\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 a^4 x^6}-\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 a^3 x^{12}}+\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 a^2 x^{15}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}} \]

[In]

Int[(a + b*x^3 + c*x^6)^(3/2)/x^19,x]

[Out]

((b^2 - 4*a*c)*(7*b^2 - 4*a*c)*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(1536*a^4*x^6) - ((7*b^2 - 4*a*c)*(2*a +
b*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(576*a^3*x^12) - (a + b*x^3 + c*x^6)^(5/2)/(18*a*x^18) + (7*b*(a + b*x^3 +
c*x^6)^(5/2))/(180*a^2*x^15) - ((b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^
3 + c*x^6])])/(3072*a^(9/2))

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))
*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[p*((b^2
- 4*a*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m
+ 2*p + 2, 0] && GtQ[p, 0]

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-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] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 758

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*
((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e
, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[S
implify[m + 2*p + 3], 0]

Rule 1371

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(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] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx,x,x^3\right ) \\ & = -\frac {\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}}-\frac {\text {Subst}\left (\int \frac {\left (\frac {7 b}{2}+c x\right ) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx,x,x^3\right )}{18 a} \\ & = -\frac {\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}}+\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 a^2 x^{15}}+\frac {\left (7 b^2-4 a c\right ) \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^3\right )}{72 a^2} \\ & = -\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 a^3 x^{12}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}}+\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 a^2 x^{15}}-\frac {\left (\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx,x,x^3\right )}{384 a^3} \\ & = \frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 a^4 x^6}-\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 a^3 x^{12}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}}+\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 a^2 x^{15}}+\frac {\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{3072 a^4} \\ & = \frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 a^4 x^6}-\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 a^3 x^{12}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}}+\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 a^2 x^{15}}-\frac {\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^3}{\sqrt {a+b x^3+c x^6}}\right )}{1536 a^4} \\ & = \frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 a^4 x^6}-\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 a^3 x^{12}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}}+\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 a^2 x^{15}}-\frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{3072 a^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.33 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{19}} \, dx=\frac {-\frac {\sqrt {a} \sqrt {a+b x^3+c x^6} \left (1280 a^5-105 b^5 x^{15}+10 a b^3 x^{12} \left (7 b+76 c x^3\right )+64 a^4 \left (26 b x^3+35 c x^6\right )+48 a^3 x^6 \left (b^2+6 b c x^3+10 c^2 x^6\right )-8 a^2 b x^9 \left (7 b^2+54 b c x^3+162 c^2 x^6\right )\right )}{x^{18}}+15 \left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )}{23040 a^{9/2}} \]

[In]

Integrate[(a + b*x^3 + c*x^6)^(3/2)/x^19,x]

[Out]

(-((Sqrt[a]*Sqrt[a + b*x^3 + c*x^6]*(1280*a^5 - 105*b^5*x^15 + 10*a*b^3*x^12*(7*b + 76*c*x^3) + 64*a^4*(26*b*x
^3 + 35*c*x^6) + 48*a^3*x^6*(b^2 + 6*b*c*x^3 + 10*c^2*x^6) - 8*a^2*b*x^9*(7*b^2 + 54*b*c*x^3 + 162*c^2*x^6)))/
x^18) + 15*(b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*ArcTanh[(Sqrt[c]*x^3 - Sqrt[a + b*x^3 + c*x^6])/Sqrt[a]])/(23040*a^
(9/2))

Maple [F]

\[\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}{x^{19}}d x\]

[In]

int((c*x^6+b*x^3+a)^(3/2)/x^19,x)

[Out]

int((c*x^6+b*x^3+a)^(3/2)/x^19,x)

Fricas [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.19 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{19}} \, dx=\left [-\frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {a} x^{18} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 4 \, {\left ({\left (105 \, a b^{5} - 760 \, a^{2} b^{3} c + 1296 \, a^{3} b c^{2}\right )} x^{15} - 2 \, {\left (35 \, a^{2} b^{4} - 216 \, a^{3} b^{2} c + 240 \, a^{4} c^{2}\right )} x^{12} + 8 \, {\left (7 \, a^{3} b^{3} - 36 \, a^{4} b c\right )} x^{9} - 1664 \, a^{5} b x^{3} - 16 \, {\left (3 \, a^{4} b^{2} + 140 \, a^{5} c\right )} x^{6} - 1280 \, a^{6}\right )} \sqrt {c x^{6} + b x^{3} + a}}{92160 \, a^{5} x^{18}}, \frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-a} x^{18} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 2 \, {\left ({\left (105 \, a b^{5} - 760 \, a^{2} b^{3} c + 1296 \, a^{3} b c^{2}\right )} x^{15} - 2 \, {\left (35 \, a^{2} b^{4} - 216 \, a^{3} b^{2} c + 240 \, a^{4} c^{2}\right )} x^{12} + 8 \, {\left (7 \, a^{3} b^{3} - 36 \, a^{4} b c\right )} x^{9} - 1664 \, a^{5} b x^{3} - 16 \, {\left (3 \, a^{4} b^{2} + 140 \, a^{5} c\right )} x^{6} - 1280 \, a^{6}\right )} \sqrt {c x^{6} + b x^{3} + a}}{46080 \, a^{5} x^{18}}\right ] \]

[In]

integrate((c*x^6+b*x^3+a)^(3/2)/x^19,x, algorithm="fricas")

[Out]

[-1/92160*(15*(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(a)*x^18*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*((105*a*b^5 - 760*a^2*b^3*c + 1296*a^
3*b*c^2)*x^15 - 2*(35*a^2*b^4 - 216*a^3*b^2*c + 240*a^4*c^2)*x^12 + 8*(7*a^3*b^3 - 36*a^4*b*c)*x^9 - 1664*a^5*
b*x^3 - 16*(3*a^4*b^2 + 140*a^5*c)*x^6 - 1280*a^6)*sqrt(c*x^6 + b*x^3 + a))/(a^5*x^18), 1/46080*(15*(7*b^6 - 6
0*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(-a)*x^18*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*((105*a*b^5 - 760*a^2*b^3*c + 1296*a^3*b*c^2)*x^15 - 2*(35*a^2*b^4 - 216*a^
3*b^2*c + 240*a^4*c^2)*x^12 + 8*(7*a^3*b^3 - 36*a^4*b*c)*x^9 - 1664*a^5*b*x^3 - 16*(3*a^4*b^2 + 140*a^5*c)*x^6
- 1280*a^6)*sqrt(c*x^6 + b*x^3 + a))/(a^5*x^18)]

Sympy [F]

\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{19}} \, dx=\int \frac {\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}{x^{19}}\, dx \]

[In]

integrate((c*x**6+b*x**3+a)**(3/2)/x**19,x)

[Out]

Integral((a + b*x**3 + c*x**6)**(3/2)/x**19, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{19}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^6+b*x^3+a)^(3/2)/x^19,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{19}} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}}{x^{19}} \,d x } \]

[In]

integrate((c*x^6+b*x^3+a)^(3/2)/x^19,x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)^(3/2)/x^19, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{19}} \, dx=\int \frac {{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{x^{19}} \,d x \]

[In]

int((a + b*x^3 + c*x^6)^(3/2)/x^19,x)

[Out]

int((a + b*x^3 + c*x^6)^(3/2)/x^19, x)