3.87 \(\int \frac {(a^2+2 a b x^3+b^2 x^6)^{5/2}}{x^{24}} \, dx\)

Optimal. Leaf size=255 \[ -\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^{14} \left (a+b x^3\right )}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{23 x^{23} \left (a+b x^3\right )}-\frac {a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^{20} \left (a+b x^3\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 x^{17} \left (a+b x^3\right )} \]

[Out]

-1/23*a^5*((b*x^3+a)^2)^(1/2)/x^23/(b*x^3+a)-1/4*a^4*b*((b*x^3+a)^2)^(1/2)/x^20/(b*x^3+a)-10/17*a^3*b^2*((b*x^
3+a)^2)^(1/2)/x^17/(b*x^3+a)-5/7*a^2*b^3*((b*x^3+a)^2)^(1/2)/x^14/(b*x^3+a)-5/11*a*b^4*((b*x^3+a)^2)^(1/2)/x^1
1/(b*x^3+a)-1/8*b^5*((b*x^3+a)^2)^(1/2)/x^8/(b*x^3+a)

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Rubi [A]  time = 0.06, antiderivative size = 255, 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} \[ -\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{23 x^{23} \left (a+b x^3\right )}-\frac {a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^{20} \left (a+b x^3\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 x^{17} \left (a+b x^3\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^{14} \left (a+b x^3\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^24,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(23*x^23*(a + b*x^3)) - (a^4*b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*x^20
*(a + b*x^3)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(17*x^17*(a + b*x^3)) - (5*a^2*b^3*Sqrt[a^2 + 2*a
*b*x^3 + b^2*x^6])/(7*x^14*(a + b*x^3)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(11*x^11*(a + b*x^3)) - (b
^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(8*x^8*(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^{24}} \, 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^{24}} \, 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^{24}}+\frac {5 a^4 b^6}{x^{21}}+\frac {10 a^3 b^7}{x^{18}}+\frac {10 a^2 b^8}{x^{15}}+\frac {5 a b^9}{x^{12}}+\frac {b^{10}}{x^9}\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}}{23 x^{23} \left (a+b x^3\right )}-\frac {a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^{20} \left (a+b x^3\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 x^{17} \left (a+b x^3\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^{14} \left (a+b x^3\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 83, normalized size = 0.33 \[ -\frac {\sqrt {\left (a+b x^3\right )^2} \left (10472 a^5+60214 a^4 b x^3+141680 a^3 b^2 x^6+172040 a^2 b^3 x^9+109480 a b^4 x^{12}+30107 b^5 x^{15}\right )}{240856 x^{23} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^24,x]

[Out]

-1/240856*(Sqrt[(a + b*x^3)^2]*(10472*a^5 + 60214*a^4*b*x^3 + 141680*a^3*b^2*x^6 + 172040*a^2*b^3*x^9 + 109480
*a*b^4*x^12 + 30107*b^5*x^15))/(x^23*(a + b*x^3))

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fricas [A]  time = 0.92, size = 59, normalized size = 0.23 \[ -\frac {30107 \, b^{5} x^{15} + 109480 \, a b^{4} x^{12} + 172040 \, a^{2} b^{3} x^{9} + 141680 \, a^{3} b^{2} x^{6} + 60214 \, a^{4} b x^{3} + 10472 \, a^{5}}{240856 \, x^{23}} \]

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^24,x, algorithm="fricas")

[Out]

-1/240856*(30107*b^5*x^15 + 109480*a*b^4*x^12 + 172040*a^2*b^3*x^9 + 141680*a^3*b^2*x^6 + 60214*a^4*b*x^3 + 10
472*a^5)/x^23

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giac [A]  time = 0.34, size = 107, normalized size = 0.42 \[ -\frac {30107 \, b^{5} x^{15} \mathrm {sgn}\left (b x^{3} + a\right ) + 109480 \, a b^{4} x^{12} \mathrm {sgn}\left (b x^{3} + a\right ) + 172040 \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 141680 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 60214 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 10472 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{240856 \, x^{23}} \]

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^24,x, algorithm="giac")

[Out]

-1/240856*(30107*b^5*x^15*sgn(b*x^3 + a) + 109480*a*b^4*x^12*sgn(b*x^3 + a) + 172040*a^2*b^3*x^9*sgn(b*x^3 + a
) + 141680*a^3*b^2*x^6*sgn(b*x^3 + a) + 60214*a^4*b*x^3*sgn(b*x^3 + a) + 10472*a^5*sgn(b*x^3 + a))/x^23

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maple [A]  time = 0.01, size = 80, normalized size = 0.31 \[ -\frac {\left (30107 b^{5} x^{15}+109480 a \,b^{4} x^{12}+172040 a^{2} b^{3} x^{9}+141680 a^{3} b^{2} x^{6}+60214 a^{4} b \,x^{3}+10472 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{240856 \left (b \,x^{3}+a \right )^{5} x^{23}} \]

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^24,x)

[Out]

-1/240856*(30107*b^5*x^15+109480*a*b^4*x^12+172040*a^2*b^3*x^9+141680*a^3*b^2*x^6+60214*a^4*b*x^3+10472*a^5)*(
(b*x^3+a)^2)^(5/2)/x^23/(b*x^3+a)^5

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maxima [A]  time = 0.83, size = 59, normalized size = 0.23 \[ -\frac {30107 \, b^{5} x^{15} + 109480 \, a b^{4} x^{12} + 172040 \, a^{2} b^{3} x^{9} + 141680 \, a^{3} b^{2} x^{6} + 60214 \, a^{4} b x^{3} + 10472 \, a^{5}}{240856 \, x^{23}} \]

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^24,x, algorithm="maxima")

[Out]

-1/240856*(30107*b^5*x^15 + 109480*a*b^4*x^12 + 172040*a^2*b^3*x^9 + 141680*a^3*b^2*x^6 + 60214*a^4*b*x^3 + 10
472*a^5)/x^23

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mupad [B]  time = 1.23, size = 231, normalized size = 0.91 \[ -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{23\,x^{23}\,\left (b\,x^3+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{8\,x^8\,\left (b\,x^3+a\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{11\,x^{11}\,\left (b\,x^3+a\right )}-\frac {a^4\,b\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{4\,x^{20}\,\left (b\,x^3+a\right )}-\frac {5\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{7\,x^{14}\,\left (b\,x^3+a\right )}-\frac {10\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{17\,x^{17}\,\left (b\,x^3+a\right )} \]

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^24,x)

[Out]

- (a^5*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(23*x^23*(a + b*x^3)) - (b^5*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(8*x
^8*(a + b*x^3)) - (5*a*b^4*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(11*x^11*(a + b*x^3)) - (a^4*b*(a^2 + b^2*x^6 +
2*a*b*x^3)^(1/2))/(4*x^20*(a + b*x^3)) - (5*a^2*b^3*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(7*x^14*(a + b*x^3)) -
(10*a^3*b^2*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(17*x^17*(a + b*x^3))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x^{24}}\, dx \]

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**24,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**24, x)

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