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

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

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Rubi [A]  time = 0.06, antiderivative size = 253, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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 {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.02, size = 83, normalized size = 0.33 \begin {gather*} -\frac {\sqrt {\left (a+b x^3\right )^2} \left (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}\right )}{27664 x^{19} \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/27664*(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))/(x^19*(a + b*x^3))

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IntegrateAlgebraic [A]  time = 21.32, size = 83, normalized size = 0.33 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (-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}\right )}{27664 x^{19} \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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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fricas [A]  time = 1.10, size = 59, normalized size = 0.23 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.36, size = 107, normalized size = 0.42 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.01, size = 80, normalized size = 0.32 \begin {gather*} -\frac {\left (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}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{27664 \left (b \,x^{3}+a \right )^{5} x^{19}} \end {gather*}

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 = 1.04, size = 59, normalized size = 0.23 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 1.31, size = 231, normalized size = 0.91 \begin {gather*} -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{19\,x^{19}\,\left (b\,x^3+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{4\,x^4\,\left (b\,x^3+a\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{7\,x^7\,\left (b\,x^3+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{16\,x^{16}\,\left (b\,x^3+a\right )}-\frac {a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{x^{10}\,\left (b\,x^3+a\right )}-\frac {10\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{13\,x^{13}\,\left (b\,x^3+a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2)/x^20,x)

[Out]

- (a^5*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(19*x^19*(a + b*x^3)) - (b^5*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(4*x
^4*(a + b*x^3)) - (5*a*b^4*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(7*x^7*(a + b*x^3)) - (5*a^4*b*(a^2 + b^2*x^6 +
2*a*b*x^3)^(1/2))/(16*x^16*(a + b*x^3)) - (a^2*b^3*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(x^10*(a + b*x^3)) - (10
*a^3*b^2*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(13*x^13*(a + b*x^3))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x^{20}}\, dx \end {gather*}

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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