3.82 \(\int \frac{(a^2+2 a b x^3+b^2 x^6)^{5/2}}{x^{19}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{\left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{18 a x^{18}} \]

[Out]

-((a + b*x^3)^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(18*a*x^18)

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Rubi [A]  time = 0.0184203, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1355, 264} \[ -\frac{\left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{18 a x^{18}} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^19,x]

[Out]

-((a + b*x^3)^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(18*a*x^18)

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{19}} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \frac{\left (a b+b^2 x^3\right )^5}{x^{19}} \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=-\frac{\left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{18 a x^{18}}\\ \end{align*}

Mathematica [A]  time = 0.0164692, size = 81, normalized size = 1.98 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (20 a^2 b^3 x^9+15 a^3 b^2 x^6+6 a^4 b x^3+a^5+15 a b^4 x^{12}+6 b^5 x^{15}\right )}{18 x^{18} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^19,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(a^5 + 6*a^4*b*x^3 + 15*a^3*b^2*x^6 + 20*a^2*b^3*x^9 + 15*a*b^4*x^12 + 6*b^5*x^15))/(18*
x^18*(a + b*x^3))

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Maple [B]  time = 0.005, size = 78, normalized size = 1.9 \begin{align*} -{\frac{6\,{b}^{5}{x}^{15}+15\,a{b}^{4}{x}^{12}+20\,{a}^{2}{b}^{3}{x}^{9}+15\,{a}^{3}{b}^{2}{x}^{6}+6\,{a}^{4}b{x}^{3}+{a}^{5}}{18\,{x}^{18} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^19,x)

[Out]

-1/18*(6*b^5*x^15+15*a*b^4*x^12+20*a^2*b^3*x^9+15*a^3*b^2*x^6+6*a^4*b*x^3+a^5)*((b*x^3+a)^2)^(5/2)/x^18/(b*x^3
+a)^5

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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((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^19,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.80635, size = 128, normalized size = 3.12 \begin{align*} -\frac{6 \, b^{5} x^{15} + 15 \, a b^{4} x^{12} + 20 \, a^{2} b^{3} x^{9} + 15 \, a^{3} b^{2} x^{6} + 6 \, a^{4} b x^{3} + a^{5}}{18 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^19,x, algorithm="fricas")

[Out]

-1/18*(6*b^5*x^15 + 15*a*b^4*x^12 + 20*a^2*b^3*x^9 + 15*a^3*b^2*x^6 + 6*a^4*b*x^3 + a^5)/x^18

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x^{19}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**6+2*a*b*x**3+a**2)**(5/2)/x**19,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**19, x)

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Giac [B]  time = 1.12067, size = 143, normalized size = 3.49 \begin{align*} -\frac{6 \, b^{5} x^{15} \mathrm{sgn}\left (b x^{3} + a\right ) + 15 \, a b^{4} x^{12} \mathrm{sgn}\left (b x^{3} + a\right ) + 20 \, a^{2} b^{3} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) + 15 \, a^{3} b^{2} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + 6 \, a^{4} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + a^{5} \mathrm{sgn}\left (b x^{3} + a\right )}{18 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^19,x, algorithm="giac")

[Out]

-1/18*(6*b^5*x^15*sgn(b*x^3 + a) + 15*a*b^4*x^12*sgn(b*x^3 + a) + 20*a^2*b^3*x^9*sgn(b*x^3 + a) + 15*a^3*b^2*x
^6*sgn(b*x^3 + a) + 6*a^4*b*x^3*sgn(b*x^3 + a) + a^5*sgn(b*x^3 + a))/x^18