3.1.81 \(\int \frac {(a^2+2 a b x^3+b^2 x^6)^{5/2}}{x^{18}} \, dx\)

Optimal. Leaf size=253 \[ -\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}-\frac {a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^8 \left (a+b x^3\right )}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 x^{17} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )} \]

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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}}{17 x^{17} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^8 \left (a+b x^3\right )}-\frac {a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \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^18,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(17*x^17*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(14*x
^14*(a + b*x^3)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(11*x^11*(a + b*x^3)) - (5*a^2*b^3*Sqrt[a^2 +
2*a*b*x^3 + b^2*x^6])/(4*x^8*(a + b*x^3)) - (a*b^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(x^5*(a + b*x^3)) - (b^5*S
qrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*x^2*(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^{18}} \, 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^{18}} \, 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^{18}}+\frac {5 a^4 b^6}{x^{15}}+\frac {10 a^3 b^7}{x^{12}}+\frac {10 a^2 b^8}{x^9}+\frac {5 a b^9}{x^6}+\frac {b^{10}}{x^3}\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}}{17 x^{17} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^8 \left (a+b x^3\right )}-\frac {a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \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 (308 a^5+1870 a^4 b x^3+4760 a^3 b^2 x^6+6545 a^2 b^3 x^9+5236 a b^4 x^{12}+2618 b^5 x^{15}\right )}{5236 x^{17} \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^18,x]

[Out]

-1/5236*(Sqrt[(a + b*x^3)^2]*(308*a^5 + 1870*a^4*b*x^3 + 4760*a^3*b^2*x^6 + 6545*a^2*b^3*x^9 + 5236*a*b^4*x^12
+ 2618*b^5*x^15))/(x^17*(a + b*x^3))

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IntegrateAlgebraic [A]  time = 20.55, size = 83, normalized size = 0.33 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (-308 a^5-1870 a^4 b x^3-4760 a^3 b^2 x^6-6545 a^2 b^3 x^9-5236 a b^4 x^{12}-2618 b^5 x^{15}\right )}{5236 x^{17} \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^18,x]

[Out]

(Sqrt[(a + b*x^3)^2]*(-308*a^5 - 1870*a^4*b*x^3 - 4760*a^3*b^2*x^6 - 6545*a^2*b^3*x^9 - 5236*a*b^4*x^12 - 2618
*b^5*x^15))/(5236*x^17*(a + b*x^3))

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fricas [A]  time = 1.21, size = 59, normalized size = 0.23 \begin {gather*} -\frac {2618 \, b^{5} x^{15} + 5236 \, a b^{4} x^{12} + 6545 \, a^{2} b^{3} x^{9} + 4760 \, a^{3} b^{2} x^{6} + 1870 \, a^{4} b x^{3} + 308 \, a^{5}}{5236 \, x^{17}} \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^18,x, algorithm="fricas")

[Out]

-1/5236*(2618*b^5*x^15 + 5236*a*b^4*x^12 + 6545*a^2*b^3*x^9 + 4760*a^3*b^2*x^6 + 1870*a^4*b*x^3 + 308*a^5)/x^1
7

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giac [A]  time = 0.45, size = 107, normalized size = 0.42 \begin {gather*} -\frac {2618 \, b^{5} x^{15} \mathrm {sgn}\left (b x^{3} + a\right ) + 5236 \, a b^{4} x^{12} \mathrm {sgn}\left (b x^{3} + a\right ) + 6545 \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 4760 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 1870 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 308 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{5236 \, x^{17}} \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^18,x, algorithm="giac")

[Out]

-1/5236*(2618*b^5*x^15*sgn(b*x^3 + a) + 5236*a*b^4*x^12*sgn(b*x^3 + a) + 6545*a^2*b^3*x^9*sgn(b*x^3 + a) + 476
0*a^3*b^2*x^6*sgn(b*x^3 + a) + 1870*a^4*b*x^3*sgn(b*x^3 + a) + 308*a^5*sgn(b*x^3 + a))/x^17

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maple [A]  time = 0.01, size = 80, normalized size = 0.32 \begin {gather*} -\frac {\left (2618 b^{5} x^{15}+5236 a \,b^{4} x^{12}+6545 a^{2} b^{3} x^{9}+4760 a^{3} b^{2} x^{6}+1870 a^{4} b \,x^{3}+308 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{5236 \left (b \,x^{3}+a \right )^{5} x^{17}} \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^18,x)

[Out]

-1/5236*(2618*b^5*x^15+5236*a*b^4*x^12+6545*a^2*b^3*x^9+4760*a^3*b^2*x^6+1870*a^4*b*x^3+308*a^5)*((b*x^3+a)^2)
^(5/2)/x^17/(b*x^3+a)^5

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maxima [A]  time = 1.16, size = 59, normalized size = 0.23 \begin {gather*} -\frac {2618 \, b^{5} x^{15} + 5236 \, a b^{4} x^{12} + 6545 \, a^{2} b^{3} x^{9} + 4760 \, a^{3} b^{2} x^{6} + 1870 \, a^{4} b x^{3} + 308 \, a^{5}}{5236 \, x^{17}} \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^18,x, algorithm="maxima")

[Out]

-1/5236*(2618*b^5*x^15 + 5236*a*b^4*x^12 + 6545*a^2*b^3*x^9 + 4760*a^3*b^2*x^6 + 1870*a^4*b*x^3 + 308*a^5)/x^1
7

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mupad [B]  time = 1.32, size = 231, normalized size = 0.91 \begin {gather*} -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{17\,x^{17}\,\left (b\,x^3+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{2\,x^2\,\left (b\,x^3+a\right )}-\frac {a\,b^4\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{x^5\,\left (b\,x^3+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{14\,x^{14}\,\left (b\,x^3+a\right )}-\frac {5\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{4\,x^8\,\left (b\,x^3+a\right )}-\frac {10\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{11\,x^{11}\,\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^18,x)

[Out]

- (a^5*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(17*x^17*(a + b*x^3)) - (b^5*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(2*x
^2*(a + b*x^3)) - (a*b^4*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(x^5*(a + b*x^3)) - (5*a^4*b*(a^2 + b^2*x^6 + 2*a*
b*x^3)^(1/2))/(14*x^14*(a + b*x^3)) - (5*a^2*b^3*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(4*x^8*(a + b*x^3)) - (10*
a^3*b^2*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(11*x^11*(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^{18}}\, 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**18,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**18, x)

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