3.3.42 \(\int \frac {1}{x^{10} (a+b x^3+c x^6)^{3/2}} \, dx\) [242]

Optimal. Leaf size=256 \[ \frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{36 a^3 \left (b^2-4 a c\right ) x^6}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a+b x^3+c x^6}}{72 a^4 \left (b^2-4 a c\right ) x^3}+\frac {5 b \left (7 b^2-12 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{48 a^{9/2}} \]

[Out]

5/48*b*(-12*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)+2/3*(b*c*x^3-2*a*c+b^2)/
a/(-4*a*c+b^2)/x^9/(c*x^6+b*x^3+a)^(1/2)-1/9*(-16*a*c+7*b^2)*(c*x^6+b*x^3+a)^(1/2)/a^2/(-4*a*c+b^2)/x^9+1/36*b
*(-116*a*c+35*b^2)*(c*x^6+b*x^3+a)^(1/2)/a^3/(-4*a*c+b^2)/x^6-1/72*(256*a^2*c^2-460*a*b^2*c+105*b^4)*(c*x^6+b*
x^3+a)^(1/2)/a^4/(-4*a*c+b^2)/x^3

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Rubi [A]
time = 0.20, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1371, 754, 848, 820, 738, 212} \begin {gather*} \frac {5 b \left (7 b^2-12 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{48 a^{9/2}}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{36 a^3 x^6 \left (b^2-4 a c\right )}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{9 a^2 x^9 \left (b^2-4 a c\right )}-\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a+b x^3+c x^6}}{72 a^4 x^3 \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x^3\right )}{3 a x^9 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^10*(a + b*x^3 + c*x^6)^(3/2)),x]

[Out]

(2*(b^2 - 2*a*c + b*c*x^3))/(3*a*(b^2 - 4*a*c)*x^9*Sqrt[a + b*x^3 + c*x^6]) - ((7*b^2 - 16*a*c)*Sqrt[a + b*x^3
+ c*x^6])/(9*a^2*(b^2 - 4*a*c)*x^9) + (b*(35*b^2 - 116*a*c)*Sqrt[a + b*x^3 + c*x^6])/(36*a^3*(b^2 - 4*a*c)*x^
6) - ((105*b^4 - 460*a*b^2*c + 256*a^2*c^2)*Sqrt[a + b*x^3 + c*x^6])/(72*a^4*(b^2 - 4*a*c)*x^3) + (5*b*(7*b^2
- 12*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(48*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 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 754

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

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 {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x^4 \left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^3\right )\\ &=\frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}-\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} \left (-7 b^2+16 a c\right )-3 b c x}{x^4 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{3 a \left (b^2-4 a c\right )}\\ &=\frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac {2 \text {Subst}\left (\int \frac {-\frac {1}{4} b \left (35 b^2-116 a c\right )-c \left (7 b^2-16 a c\right ) x}{x^3 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{9 a^2 \left (b^2-4 a c\right )}\\ &=\frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{36 a^3 \left (b^2-4 a c\right ) x^6}-\frac {\text {Subst}\left (\int \frac {\frac {1}{8} \left (-105 b^4+460 a b^2 c-256 a^2 c^2\right )-\frac {1}{4} b c \left (35 b^2-116 a c\right ) x}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{9 a^3 \left (b^2-4 a c\right )}\\ &=\frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{36 a^3 \left (b^2-4 a c\right ) x^6}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a+b x^3+c x^6}}{72 a^4 \left (b^2-4 a c\right ) x^3}-\frac {\left (5 b \left (7 b^2-12 a c\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{48 a^4}\\ &=\frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{36 a^3 \left (b^2-4 a c\right ) x^6}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a+b x^3+c x^6}}{72 a^4 \left (b^2-4 a c\right ) x^3}+\frac {\left (5 b \left (7 b^2-12 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 )}{24 a^4}\\ &=\frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{36 a^3 \left (b^2-4 a c\right ) x^6}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a+b x^3+c x^6}}{72 a^4 \left (b^2-4 a c\right ) x^3}+\frac {5 b \left (7 b^2-12 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{48 a^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.95, size = 210, normalized size = 0.82 \begin {gather*} \frac {-32 a^4 c+105 b^4 x^9 \left (b+c x^3\right )+5 a b^2 x^6 \left (7 b^2-106 b c x^3-92 c^2 x^6\right )+8 a^3 \left (b^2+7 b c x^3+16 c^2 x^6\right )+2 a^2 x^3 \left (-7 b^3-86 b^2 c x^3+244 b c^2 x^6+128 c^3 x^9\right )}{72 a^4 \left (-b^2+4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}+\frac {5 b \left (-7 b^2+12 a c\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )}{24 a^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^10*(a + b*x^3 + c*x^6)^(3/2)),x]

[Out]

(-32*a^4*c + 105*b^4*x^9*(b + c*x^3) + 5*a*b^2*x^6*(7*b^2 - 106*b*c*x^3 - 92*c^2*x^6) + 8*a^3*(b^2 + 7*b*c*x^3
+ 16*c^2*x^6) + 2*a^2*x^3*(-7*b^3 - 86*b^2*c*x^3 + 244*b*c^2*x^6 + 128*c^3*x^9))/(72*a^4*(-b^2 + 4*a*c)*x^9*S
qrt[a + b*x^3 + c*x^6]) + (5*b*(-7*b^2 + 12*a*c)*ArcTanh[(Sqrt[c]*x^3 - Sqrt[a + b*x^3 + c*x^6])/Sqrt[a]])/(24
*a^(9/2))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{10} \left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^10/(c*x^6+b*x^3+a)^(3/2),x)

[Out]

int(1/x^10/(c*x^6+b*x^3+a)^(3/2),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(1/x^10/(c*x^6+b*x^3+a)^(3/2),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 = 0.55, size = 705, normalized size = 2.75 \begin {gather*} \left [-\frac {15 \, {\left ({\left (7 \, b^{5} c - 40 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x^{15} + {\left (7 \, b^{6} - 40 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2}\right )} x^{12} + {\left (7 \, a b^{5} - 40 \, a^{2} b^{3} c + 48 \, a^{3} b c^{2}\right )} x^{9}\right )} \sqrt {a} \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^{4} c - 460 \, a^{2} b^{2} c^{2} + 256 \, a^{3} c^{3}\right )} x^{12} + {\left (105 \, a b^{5} - 530 \, a^{2} b^{3} c + 488 \, a^{3} b c^{2}\right )} x^{9} + {\left (35 \, a^{2} b^{4} - 172 \, a^{3} b^{2} c + 128 \, a^{4} c^{2}\right )} x^{6} + 8 \, a^{4} b^{2} - 32 \, a^{5} c - 14 \, {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{288 \, {\left ({\left (a^{5} b^{2} c - 4 \, a^{6} c^{2}\right )} x^{15} + {\left (a^{5} b^{3} - 4 \, a^{6} b c\right )} x^{12} + {\left (a^{6} b^{2} - 4 \, a^{7} c\right )} x^{9}\right )}}, -\frac {15 \, {\left ({\left (7 \, b^{5} c - 40 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x^{15} + {\left (7 \, b^{6} - 40 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2}\right )} x^{12} + {\left (7 \, a b^{5} - 40 \, a^{2} b^{3} c + 48 \, a^{3} b c^{2}\right )} x^{9}\right )} \sqrt {-a} \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^{4} c - 460 \, a^{2} b^{2} c^{2} + 256 \, a^{3} c^{3}\right )} x^{12} + {\left (105 \, a b^{5} - 530 \, a^{2} b^{3} c + 488 \, a^{3} b c^{2}\right )} x^{9} + {\left (35 \, a^{2} b^{4} - 172 \, a^{3} b^{2} c + 128 \, a^{4} c^{2}\right )} x^{6} + 8 \, a^{4} b^{2} - 32 \, a^{5} c - 14 \, {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{144 \, {\left ({\left (a^{5} b^{2} c - 4 \, a^{6} c^{2}\right )} x^{15} + {\left (a^{5} b^{3} - 4 \, a^{6} b c\right )} x^{12} + {\left (a^{6} b^{2} - 4 \, a^{7} c\right )} x^{9}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(c*x^6+b*x^3+a)^(3/2),x, algorithm="fricas")

[Out]

[-1/288*(15*((7*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*x^15 + (7*b^6 - 40*a*b^4*c + 48*a^2*b^2*c^2)*x^12 + (7*a*
b^5 - 40*a^2*b^3*c + 48*a^3*b*c^2)*x^9)*sqrt(a)*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^4*c - 460*a^2*b^2*c^2 + 256*a^3*c^3)*x^12 + (105*a*b^5 - 5
30*a^2*b^3*c + 488*a^3*b*c^2)*x^9 + (35*a^2*b^4 - 172*a^3*b^2*c + 128*a^4*c^2)*x^6 + 8*a^4*b^2 - 32*a^5*c - 14
*(a^3*b^3 - 4*a^4*b*c)*x^3)*sqrt(c*x^6 + b*x^3 + a))/((a^5*b^2*c - 4*a^6*c^2)*x^15 + (a^5*b^3 - 4*a^6*b*c)*x^1
2 + (a^6*b^2 - 4*a^7*c)*x^9), -1/144*(15*((7*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*x^15 + (7*b^6 - 40*a*b^4*c +
48*a^2*b^2*c^2)*x^12 + (7*a*b^5 - 40*a^2*b^3*c + 48*a^3*b*c^2)*x^9)*sqrt(-a)*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^4*c - 460*a^2*b^2*c^2 + 256*a^3*c^3)*x^12 +
(105*a*b^5 - 530*a^2*b^3*c + 488*a^3*b*c^2)*x^9 + (35*a^2*b^4 - 172*a^3*b^2*c + 128*a^4*c^2)*x^6 + 8*a^4*b^2
- 32*a^5*c - 14*(a^3*b^3 - 4*a^4*b*c)*x^3)*sqrt(c*x^6 + b*x^3 + a))/((a^5*b^2*c - 4*a^6*c^2)*x^15 + (a^5*b^3 -
4*a^6*b*c)*x^12 + (a^6*b^2 - 4*a^7*c)*x^9)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{10} \left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**10/(c*x**6+b*x**3+a)**(3/2),x)

[Out]

Integral(1/(x**10*(a + b*x**3 + c*x**6)**(3/2)), 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(1/x^10/(c*x^6+b*x^3+a)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((c*x^6 + b*x^3 + a)^(3/2)*x^10), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^{10}\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^10*(a + b*x^3 + c*x^6)^(3/2)),x)

[Out]

int(1/(x^10*(a + b*x^3 + c*x^6)^(3/2)), x)

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