3.1.21 \(\int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^{10}} \, dx\)

Optimal. Leaf size=79 \[ -\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )} \]

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Rubi [A]  time = 0.02, antiderivative size = 79, 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, 14} \begin {gather*} -\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^10,x]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(9*x^9*(a + b*x^3)) - (b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(6*x^6*(a + b*x
^3))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^{10}} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {a b+b^2 x^3}{x^{10}} \, dx}{a b+b^2 x^3}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (\frac {a b}{x^{10}}+\frac {b^2}{x^7}\right ) \, dx}{a b+b^2 x^3}\\ &=-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 39, normalized size = 0.49 \begin {gather*} -\frac {\sqrt {\left (a+b x^3\right )^2} \left (2 a+3 b x^3\right )}{18 x^9 \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^10,x]

[Out]

-1/18*(Sqrt[(a + b*x^3)^2]*(2*a + 3*b*x^3))/(x^9*(a + b*x^3))

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IntegrateAlgebraic [B]  time = 3.56, size = 753, normalized size = 9.53 \begin {gather*} \frac {2 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (-2 a^{15} b^3-55 a^{14} b^4 x^3-704 a^{13} b^5 x^6-5563 a^{12} b^6 x^9-30344 a^{11} b^7 x^{12}-121000 a^{10} b^8 x^{15}-364320 a^9 b^9 x^{18}-843216 a^8 b^{10} x^{21}-1512192 a^7 b^{11} x^{24}-2100736 a^6 b^{12} x^{27}-2241536 a^5 b^{13} x^{30}-1803520 a^4 b^{14} x^{33}-1058816 a^3 b^{15} x^{36}-428032 a^2 b^{16} x^{39}-106496 a b^{17} x^{42}-12288 b^{18} x^{45}\right )+2 \sqrt {b^2} \left (2 a^{16} b^2+57 a^{15} b^3 x^3+759 a^{14} b^4 x^6+6267 a^{13} b^5 x^9+35907 a^{12} b^6 x^{12}+151344 a^{11} b^7 x^{15}+485320 a^{10} b^8 x^{18}+1207536 a^9 b^9 x^{21}+2355408 a^8 b^{10} x^{24}+3612928 a^7 b^{11} x^{27}+4342272 a^6 b^{12} x^{30}+4045056 a^5 b^{13} x^{33}+2862336 a^4 b^{14} x^{36}+1486848 a^3 b^{15} x^{39}+534528 a^2 b^{16} x^{42}+118784 a b^{17} x^{45}+12288 b^{18} x^{48}\right )}{9 \sqrt {b^2} x^9 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (-4 a^{14} b^2-104 a^{13} b^3 x^3-1252 a^{12} b^4 x^6-9248 a^{11} b^5 x^9-46816 a^{10} b^6 x^{12}-171776 a^9 b^7 x^{15}-470976 a^8 b^8 x^{18}-979968 a^7 b^9 x^{21}-1554432 a^6 b^{10} x^{24}-1869824 a^5 b^{11} x^{27}-1678336 a^4 b^{12} x^{30}-1089536 a^3 b^{13} x^{33}-483328 a^2 b^{14} x^{36}-131072 a b^{15} x^{39}-16384 b^{16} x^{42}\right )+9 x^9 \left (4 a^{15} b^3+108 a^{14} b^4 x^3+1356 a^{13} b^5 x^6+10500 a^{12} b^6 x^9+56064 a^{11} b^7 x^{12}+218592 a^{10} b^8 x^{15}+642752 a^9 b^9 x^{18}+1450944 a^8 b^{10} x^{21}+2534400 a^7 b^{11} x^{24}+3424256 a^6 b^{12} x^{27}+3548160 a^5 b^{13} x^{30}+2767872 a^4 b^{14} x^{33}+1572864 a^3 b^{15} x^{36}+614400 a^2 b^{16} x^{39}+147456 a b^{17} x^{42}+16384 b^{18} x^{45}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^10,x]

[Out]

(2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*(-2*a^15*b^3 - 55*a^14*b^4*x^3 - 704*a^13*b^5*x^6 - 5563*a^12*b^6*x^9 - 303
44*a^11*b^7*x^12 - 121000*a^10*b^8*x^15 - 364320*a^9*b^9*x^18 - 843216*a^8*b^10*x^21 - 1512192*a^7*b^11*x^24 -
2100736*a^6*b^12*x^27 - 2241536*a^5*b^13*x^30 - 1803520*a^4*b^14*x^33 - 1058816*a^3*b^15*x^36 - 428032*a^2*b^
16*x^39 - 106496*a*b^17*x^42 - 12288*b^18*x^45) + 2*Sqrt[b^2]*(2*a^16*b^2 + 57*a^15*b^3*x^3 + 759*a^14*b^4*x^6
+ 6267*a^13*b^5*x^9 + 35907*a^12*b^6*x^12 + 151344*a^11*b^7*x^15 + 485320*a^10*b^8*x^18 + 1207536*a^9*b^9*x^2
1 + 2355408*a^8*b^10*x^24 + 3612928*a^7*b^11*x^27 + 4342272*a^6*b^12*x^30 + 4045056*a^5*b^13*x^33 + 2862336*a^
4*b^14*x^36 + 1486848*a^3*b^15*x^39 + 534528*a^2*b^16*x^42 + 118784*a*b^17*x^45 + 12288*b^18*x^48))/(9*Sqrt[b^
2]*x^9*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*(-4*a^14*b^2 - 104*a^13*b^3*x^3 - 1252*a^12*b^4*x^6 - 9248*a^11*b^5*x^9
- 46816*a^10*b^6*x^12 - 171776*a^9*b^7*x^15 - 470976*a^8*b^8*x^18 - 979968*a^7*b^9*x^21 - 1554432*a^6*b^10*x^
24 - 1869824*a^5*b^11*x^27 - 1678336*a^4*b^12*x^30 - 1089536*a^3*b^13*x^33 - 483328*a^2*b^14*x^36 - 131072*a*b
^15*x^39 - 16384*b^16*x^42) + 9*x^9*(4*a^15*b^3 + 108*a^14*b^4*x^3 + 1356*a^13*b^5*x^6 + 10500*a^12*b^6*x^9 +
56064*a^11*b^7*x^12 + 218592*a^10*b^8*x^15 + 642752*a^9*b^9*x^18 + 1450944*a^8*b^10*x^21 + 2534400*a^7*b^11*x^
24 + 3424256*a^6*b^12*x^27 + 3548160*a^5*b^13*x^30 + 2767872*a^4*b^14*x^33 + 1572864*a^3*b^15*x^36 + 614400*a^
2*b^16*x^39 + 147456*a*b^17*x^42 + 16384*b^18*x^45))

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fricas [A]  time = 1.37, size = 15, normalized size = 0.19 \begin {gather*} -\frac {3 \, b x^{3} + 2 \, a}{18 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^3+a)^2)^(1/2)/x^10,x, algorithm="fricas")

[Out]

-1/18*(3*b*x^3 + 2*a)/x^9

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giac [A]  time = 0.36, size = 31, normalized size = 0.39 \begin {gather*} -\frac {3 \, b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 2 \, a \mathrm {sgn}\left (b x^{3} + a\right )}{18 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^3+a)^2)^(1/2)/x^10,x, algorithm="giac")

[Out]

-1/18*(3*b*x^3*sgn(b*x^3 + a) + 2*a*sgn(b*x^3 + a))/x^9

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maple [A]  time = 0.00, size = 36, normalized size = 0.46 \begin {gather*} -\frac {\left (3 b \,x^{3}+2 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{18 \left (b \,x^{3}+a \right ) x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x^3+a)^2)^(1/2)/x^10,x)

[Out]

-1/18*(3*b*x^3+2*a)*((b*x^3+a)^2)^(1/2)/x^9/(b*x^3+a)

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maxima [B]  time = 0.47, size = 117, normalized size = 1.48 \begin {gather*} -\frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} b^{3}}{6 \, a^{3}} - \frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} b^{2}}{6 \, a^{2} x^{3}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} b}{6 \, a^{3} x^{6}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}}}{9 \, a^{2} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^3+a)^2)^(1/2)/x^10,x, algorithm="maxima")

[Out]

-1/6*sqrt(b^2*x^6 + 2*a*b*x^3 + a^2)*b^3/a^3 - 1/6*sqrt(b^2*x^6 + 2*a*b*x^3 + a^2)*b^2/(a^2*x^3) + 1/6*(b^2*x^
6 + 2*a*b*x^3 + a^2)^(3/2)*b/(a^3*x^6) - 1/9*(b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)/(a^2*x^9)

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mupad [B]  time = 1.15, size = 35, normalized size = 0.44 \begin {gather*} -\frac {\left (3\,b\,x^3+2\,a\right )\,\sqrt {{\left (b\,x^3+a\right )}^2}}{18\,x^9\,\left (b\,x^3+a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^3)^2)^(1/2)/x^10,x)

[Out]

-((2*a + 3*b*x^3)*((a + b*x^3)^2)^(1/2))/(18*x^9*(a + b*x^3))

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sympy [A]  time = 0.21, size = 15, normalized size = 0.19 \begin {gather*} \frac {- 2 a - 3 b x^{3}}{18 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x**3+a)**2)**(1/2)/x**10,x)

[Out]

(-2*a - 3*b*x**3)/(18*x**9)

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