3.83 \(\int \frac{(a^2+2 a b x^3+b^2 x^6)^{5/2}}{x^{20}} \, dx\)

Optimal. Leaf size=253 \[ -\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 x^{19} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 x^{16} \left (a+b x^3\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac{a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^{10} \left (a+b x^3\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(19*x^19*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(16*x
^16*(a + b*x^3)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(13*x^13*(a + b*x^3)) - (a^2*b^3*Sqrt[a^2 + 2*
a*b*x^3 + b^2*x^6])/(x^10*(a + b*x^3)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*x^7*(a + b*x^3)) - (b^5*
Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*x^4*(a + b*x^3))

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Rubi [A]  time = 0.0614571, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1355, 270} \[ -\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 x^{19} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 x^{16} \left (a+b x^3\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac{a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^{10} \left (a+b x^3\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^20,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(19*x^19*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(16*x
^16*(a + b*x^3)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(13*x^13*(a + b*x^3)) - (a^2*b^3*Sqrt[a^2 + 2*
a*b*x^3 + b^2*x^6])/(x^10*(a + b*x^3)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*x^7*(a + b*x^3)) - (b^5*
Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*x^4*(a + b*x^3))

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 270

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

Rubi steps

\begin{align*} \int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{20}} \, 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^{20}} \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \left (\frac{a^5 b^5}{x^{20}}+\frac{5 a^4 b^6}{x^{17}}+\frac{10 a^3 b^7}{x^{14}}+\frac{10 a^2 b^8}{x^{11}}+\frac{5 a b^9}{x^8}+\frac{b^{10}}{x^5}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 x^{19} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 x^{16} \left (a+b x^3\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac{a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^{10} \left (a+b x^3\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}\\ \end{align*}

Mathematica [A]  time = 0.0191013, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (27664 a^2 b^3 x^9+21280 a^3 b^2 x^6+8645 a^4 b x^3+1456 a^5+19760 a b^4 x^{12}+6916 b^5 x^{15}\right )}{27664 x^{19} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^20,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(1456*a^5 + 8645*a^4*b*x^3 + 21280*a^3*b^2*x^6 + 27664*a^2*b^3*x^9 + 19760*a*b^4*x^12 +
6916*b^5*x^15))/(27664*x^19*(a + b*x^3))

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Maple [A]  time = 0.007, size = 80, normalized size = 0.3 \begin{align*} -{\frac{6916\,{b}^{5}{x}^{15}+19760\,a{b}^{4}{x}^{12}+27664\,{a}^{2}{b}^{3}{x}^{9}+21280\,{a}^{3}{b}^{2}{x}^{6}+8645\,{a}^{4}b{x}^{3}+1456\,{a}^{5}}{27664\,{x}^{19} \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^20,x)

[Out]

-1/27664*(6916*b^5*x^15+19760*a*b^4*x^12+27664*a^2*b^3*x^9+21280*a^3*b^2*x^6+8645*a^4*b*x^3+1456*a^5)*((b*x^3+
a)^2)^(5/2)/x^19/(b*x^3+a)^5

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Maxima [A]  time = 1.02547, size = 80, normalized size = 0.32 \begin{align*} -\frac{6916 \, b^{5} x^{15} + 19760 \, a b^{4} x^{12} + 27664 \, a^{2} b^{3} x^{9} + 21280 \, a^{3} b^{2} x^{6} + 8645 \, a^{4} b x^{3} + 1456 \, a^{5}}{27664 \, x^{19}} \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^20,x, algorithm="maxima")

[Out]

-1/27664*(6916*b^5*x^15 + 19760*a*b^4*x^12 + 27664*a^2*b^3*x^9 + 21280*a^3*b^2*x^6 + 8645*a^4*b*x^3 + 1456*a^5
)/x^19

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Fricas [A]  time = 1.78799, size = 159, normalized size = 0.63 \begin{align*} -\frac{6916 \, b^{5} x^{15} + 19760 \, a b^{4} x^{12} + 27664 \, a^{2} b^{3} x^{9} + 21280 \, a^{3} b^{2} x^{6} + 8645 \, a^{4} b x^{3} + 1456 \, a^{5}}{27664 \, x^{19}} \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^20,x, algorithm="fricas")

[Out]

-1/27664*(6916*b^5*x^15 + 19760*a*b^4*x^12 + 27664*a^2*b^3*x^9 + 21280*a^3*b^2*x^6 + 8645*a^4*b*x^3 + 1456*a^5
)/x^19

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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^{20}}\, 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**20,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**20, x)

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Giac [A]  time = 1.11302, size = 144, normalized size = 0.57 \begin{align*} -\frac{6916 \, b^{5} x^{15} \mathrm{sgn}\left (b x^{3} + a\right ) + 19760 \, a b^{4} x^{12} \mathrm{sgn}\left (b x^{3} + a\right ) + 27664 \, a^{2} b^{3} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) + 21280 \, a^{3} b^{2} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + 8645 \, a^{4} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 1456 \, a^{5} \mathrm{sgn}\left (b x^{3} + a\right )}{27664 \, x^{19}} \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^20,x, algorithm="giac")

[Out]

-1/27664*(6916*b^5*x^15*sgn(b*x^3 + a) + 19760*a*b^4*x^12*sgn(b*x^3 + a) + 27664*a^2*b^3*x^9*sgn(b*x^3 + a) +
21280*a^3*b^2*x^6*sgn(b*x^3 + a) + 8645*a^4*b*x^3*sgn(b*x^3 + a) + 1456*a^5*sgn(b*x^3 + a))/x^19