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

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

[Out]

-(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])/(1024*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))

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Rubi [A]  time = 0.311733, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {1357, 744, 834, 806, 720, 724, 206} \[ -\frac{\left (21 b^2-16 a c\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 a^3 x^{15}}+\frac{b \left (3 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 a^4 x^{12}}-\frac{b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{1024 a^5 x^6}+\frac{b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{2048 a^{11/2}}+\frac{b \left (a+b x^3+c x^6\right )^{5/2}}{28 a^2 x^{18}}-\frac{\left (a+b x^3+c x^6\right )^{5/2}}{21 a x^{21}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3 + c*x^6)^(3/2)/x^22,x]

[Out]

-(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])/(1024*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 1357

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]]

Rule 744

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 834

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 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((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[Sim
plify[m + 2*p + 3], 0]

Rule 720

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] /; FreeQ[
{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 724

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3+c x^6\right )^{3/2}}{x^{22}} \, dx &=\frac{1}{3} \operatorname{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{\operatorname{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{\operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [A]  time = 0.506127, size = 243, normalized size = 0.95 \[ -\frac{\frac{\left (21 b^2-16 a c\right ) \left (a+b x^3+c x^6\right )^{5/2}}{40 a^2 x^{15}}+\frac{\left (21 a b c-\frac{63 b^3}{4}\right ) \left (16 a^{3/2} \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}-3 x^6 \left (b^2-4 a c\right ) \left (2 \sqrt{a} \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}-x^6 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )\right )\right )}{1536 a^{9/2} x^{12}}-\frac{3 b \left (a+b x^3+c x^6\right )^{5/2}}{4 a x^{18}}+\frac{\left (a+b x^3+c x^6\right )^{5/2}}{x^{21}}}{21 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*x^3 + c*x^6)^(5/2)/x^21 - (3*b*(a + b*x^3 + c*x^6)^(5/2))/(4*a*x^18) + ((21*b^2 - 16*a*c)*(a + b*x^3
+ c*x^6)^(5/2))/(40*a^2*x^15) + (((-63*b^3)/4 + 21*a*b*c)*(16*a^(3/2)*(2*a + b*x^3)*(a + b*x^3 + c*x^6)^(3/2)
- 3*(b^2 - 4*a*c)*x^6*(2*Sqrt[a]*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6] - (b^2 - 4*a*c)*x^6*ArcTanh[(2*a + b*x^
3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])))/(1536*a^(9/2)*x^12))/(21*a)

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Maple [F]  time = 0.084, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{22}} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [A]  time = 4.65438, size = 1318, normalized size = 5.17 \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{x^{22}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}}}{x^{22}}\,{d x} \end{align*}

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)