3.3.13 \(\int \frac {(a+b x^3+c x^6)^{3/2}}{x^{22}} \, dx\) [213]

Optimal. Leaf size=255 \[ -\frac {b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{1024 a^5 x^6}+\frac {b \left (3 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 a^4 x^{12}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{21 a x^{21}}+\frac {b \left (a+b x^3+c x^6\right )^{5/2}}{28 a^2 x^{18}}-\frac {\left (21 b^2-16 a c\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 a^3 x^{15}}+\frac {b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{2048 a^{11/2}} \]

[Out]

1/384*b*(-4*a*c+3*b^2)*(b*x^3+2*a)*(c*x^6+b*x^3+a)^(3/2)/a^4/x^12-1/21*(c*x^6+b*x^3+a)^(5/2)/a/x^21+1/28*b*(c*
x^6+b*x^3+a)^(5/2)/a^2/x^18-1/840*(-16*a*c+21*b^2)*(c*x^6+b*x^3+a)^(5/2)/a^3/x^15+1/2048*b*(-4*a*c+b^2)^2*(-4*
a*c+3*b^2)*arctanh(1/2*(b*x^3+2*a)/a^(1/2)/(c*x^6+b*x^3+a)^(1/2))/a^(11/2)-1/1024*b*(-4*a*c+b^2)*(-4*a*c+3*b^2
)*(b*x^3+2*a)*(c*x^6+b*x^3+a)^(1/2)/a^5/x^6

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Rubi [A]
time = 0.21, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1371, 758, 848, 820, 734, 738, 212} \begin {gather*} \frac {b \left (b^2-4 a c\right )^2 \left (3 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 )}{2048 a^{11/2}}-\frac {b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{1024 a^5 x^6}+\frac {b \left (3 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 a^4 x^{12}}-\frac {\left (21 b^2-16 a c\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 a^3 x^{15}}+\frac {b \left (a+b x^3+c x^6\right )^{5/2}}{28 a^2 x^{18}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{21 a x^{21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3 + c*x^6)^(3/2)/x^22,x]

[Out]

-1/1024*(b*(b^2 - 4*a*c)*(3*b^2 - 4*a*c)*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(a^5*x^6) + (b*(3*b^2 - 4*a*c)
*(2*a + b*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(384*a^4*x^12) - (a + b*x^3 + c*x^6)^(5/2)/(21*a*x^21) + (b*(a + b*x
^3 + c*x^6)^(5/2))/(28*a^2*x^18) - ((21*b^2 - 16*a*c)*(a + b*x^3 + c*x^6)^(5/2))/(840*a^3*x^15) + (b*(b^2 - 4*
a*c)^2*(3*b^2 - 4*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(2048*a^(11/2))

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 848

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)/((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)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

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*} \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{22}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^8} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{21 a x^{21}}-\frac {\text {Subst}\left (\int \frac {\left (\frac {9 b}{2}+2 c x\right ) \left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx,x,x^3\right )}{21 a}\\ &=-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{21 a x^{21}}+\frac {b \left (a+b x^3+c x^6\right )^{5/2}}{28 a^2 x^{18}}+\frac {\text {Subst}\left (\int \frac {\left (\frac {3}{4} \left (21 b^2-16 a c\right )+\frac {9 b c x}{2}\right ) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx,x,x^3\right )}{126 a^2}\\ &=-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{21 a x^{21}}+\frac {b \left (a+b x^3+c x^6\right )^{5/2}}{28 a^2 x^{18}}-\frac {\left (21 b^2-16 a c\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 a^3 x^{15}}-\frac {\left (b \left (3 b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^3\right )}{48 a^3}\\ &=\frac {b \left (3 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 a^4 x^{12}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{21 a x^{21}}+\frac {b \left (a+b x^3+c x^6\right )^{5/2}}{28 a^2 x^{18}}-\frac {\left (21 b^2-16 a c\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 a^3 x^{15}}+\frac {\left (b \left (b^2-4 a c\right ) \left (3 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 )}{256 a^4}\\ &=-\frac {b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{1024 a^5 x^6}+\frac {b \left (3 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 a^4 x^{12}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{21 a x^{21}}+\frac {b \left (a+b x^3+c x^6\right )^{5/2}}{28 a^2 x^{18}}-\frac {\left (21 b^2-16 a c\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 a^3 x^{15}}-\frac {\left (b \left (b^2-4 a c\right )^2 \left (3 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 )}{2048 a^5}\\ &=-\frac {b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{1024 a^5 x^6}+\frac {b \left (3 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 a^4 x^{12}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{21 a x^{21}}+\frac {b \left (a+b x^3+c x^6\right )^{5/2}}{28 a^2 x^{18}}-\frac {\left (21 b^2-16 a c\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 a^3 x^{15}}+\frac {\left (b \left (b^2-4 a c\right )^2 \left (3 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 )}{1024 a^5}\\ &=-\frac {b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{1024 a^5 x^6}+\frac {b \left (3 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 a^4 x^{12}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{21 a x^{21}}+\frac {b \left (a+b x^3+c x^6\right )^{5/2}}{28 a^2 x^{18}}-\frac {\left (21 b^2-16 a c\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 a^3 x^{15}}+\frac {b \left (b^2-4 a c\right )^2 \left (3 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 )}{2048 a^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 1.54, size = 244, normalized size = 0.96 \begin {gather*} \frac {-\frac {\sqrt {a} \sqrt {a+b x^3+c x^6} \left (5120 a^6+315 b^6 x^{18}-210 a b^4 x^{15} \left (b+12 c x^3\right )+256 a^5 \left (25 b x^3+32 c x^6\right )+64 a^4 x^6 \left (2 b^2+11 b c x^3+16 c^2 x^6\right )+56 a^2 b^2 x^{12} \left (3 b^2+26 b c x^3+98 c^2 x^6\right )-16 a^3 x^9 \left (9 b^3+62 b^2 c x^3+146 b c^2 x^6+128 c^3 x^9\right )\right )}{x^{21}}-105 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )}{107520 a^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3 + c*x^6)^(3/2)/x^22,x]

[Out]

(-((Sqrt[a]*Sqrt[a + b*x^3 + c*x^6]*(5120*a^6 + 315*b^6*x^18 - 210*a*b^4*x^15*(b + 12*c*x^3) + 256*a^5*(25*b*x
^3 + 32*c*x^6) + 64*a^4*x^6*(2*b^2 + 11*b*c*x^3 + 16*c^2*x^6) + 56*a^2*b^2*x^12*(3*b^2 + 26*b*c*x^3 + 98*c^2*x
^6) - 16*a^3*x^9*(9*b^3 + 62*b^2*c*x^3 + 146*b*c^2*x^6 + 128*c^3*x^9)))/x^21) - 105*b*(b^2 - 4*a*c)^2*(3*b^2 -
4*a*c)*ArcTanh[(Sqrt[c]*x^3 - Sqrt[a + b*x^3 + c*x^6])/Sqrt[a]])/(107520*a^(11/2))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}{x^{22}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^6+b*x^3+a)^(3/2)/x^22,x)

[Out]

int((c*x^6+b*x^3+a)^(3/2)/x^22,x)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^6+b*x^3+a)^(3/2)/x^22,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

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Fricas [A]
time = 0.65, size = 557, normalized size = 2.18 \begin {gather*} \left [-\frac {105 \, {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} \sqrt {a} x^{21} \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 (315 \, a b^{6} - 2520 \, a^{2} b^{4} c + 5488 \, a^{3} b^{2} c^{2} - 2048 \, a^{4} c^{3}\right )} x^{18} - 2 \, {\left (105 \, a^{2} b^{5} - 728 \, a^{3} b^{3} c + 1168 \, a^{4} b c^{2}\right )} x^{15} + 8 \, {\left (21 \, a^{3} b^{4} - 124 \, a^{4} b^{2} c + 128 \, a^{5} c^{2}\right )} x^{12} + 6400 \, a^{6} b x^{3} - 16 \, {\left (9 \, a^{4} b^{3} - 44 \, a^{5} b c\right )} x^{9} + 5120 \, a^{7} + 128 \, {\left (a^{5} b^{2} + 64 \, a^{6} c\right )} x^{6}\right )} \sqrt {c x^{6} + b x^{3} + a}}{430080 \, a^{6} x^{21}}, -\frac {105 \, {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} \sqrt {-a} x^{21} \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 (315 \, a b^{6} - 2520 \, a^{2} b^{4} c + 5488 \, a^{3} b^{2} c^{2} - 2048 \, a^{4} c^{3}\right )} x^{18} - 2 \, {\left (105 \, a^{2} b^{5} - 728 \, a^{3} b^{3} c + 1168 \, a^{4} b c^{2}\right )} x^{15} + 8 \, {\left (21 \, a^{3} b^{4} - 124 \, a^{4} b^{2} c + 128 \, a^{5} c^{2}\right )} x^{12} + 6400 \, a^{6} b x^{3} - 16 \, {\left (9 \, a^{4} b^{3} - 44 \, a^{5} b c\right )} x^{9} + 5120 \, a^{7} + 128 \, {\left (a^{5} b^{2} + 64 \, a^{6} c\right )} x^{6}\right )} \sqrt {c x^{6} + b x^{3} + a}}{215040 \, a^{6} x^{21}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^6+b*x^3+a)^(3/2)/x^22,x, algorithm="fricas")

[Out]

[-1/430080*(105*(3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*c^3)*sqrt(a)*x^21*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*((315*a*b^6 - 2520*a^2*b^4*c + 548
8*a^3*b^2*c^2 - 2048*a^4*c^3)*x^18 - 2*(105*a^2*b^5 - 728*a^3*b^3*c + 1168*a^4*b*c^2)*x^15 + 8*(21*a^3*b^4 - 1
24*a^4*b^2*c + 128*a^5*c^2)*x^12 + 6400*a^6*b*x^3 - 16*(9*a^4*b^3 - 44*a^5*b*c)*x^9 + 5120*a^7 + 128*(a^5*b^2
+ 64*a^6*c)*x^6)*sqrt(c*x^6 + b*x^3 + a))/(a^6*x^21), -1/215040*(105*(3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64
*a^3*b*c^3)*sqrt(-a)*x^21*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*((315*a*b^6 - 2520*a^2*b^4*c + 5488*a^3*b^2*c^2 - 2048*a^4*c^3)*x^18 - 2*(105*a^2*b^5 - 728*a^3*b^3*c + 1
168*a^4*b*c^2)*x^15 + 8*(21*a^3*b^4 - 124*a^4*b^2*c + 128*a^5*c^2)*x^12 + 6400*a^6*b*x^3 - 16*(9*a^4*b^3 - 44*
a^5*b*c)*x^9 + 5120*a^7 + 128*(a^5*b^2 + 64*a^6*c)*x^6)*sqrt(c*x^6 + b*x^3 + a))/(a^6*x^21)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}{x^{22}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**6+b*x**3+a)**(3/2)/x**22,x)

[Out]

Integral((a + b*x**3 + c*x**6)**(3/2)/x**22, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^6+b*x^3+a)^(3/2)/x^22,x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)^(3/2)/x^22, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{x^{22}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3 + c*x^6)^(3/2)/x^22,x)

[Out]

int((a + b*x^3 + c*x^6)^(3/2)/x^22, x)

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