3.1.88 \(\int \frac {(a^2+2 a b x^3+b^2 x^6)^{5/2}}{x^{25}} \, dx\)

Optimal. Leaf size=128 \[ -\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{24 a x^{24}}+\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{84 a^2 x^{21}}-\frac {b^2 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{504 a^3 x^{18}} \]

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Rubi [A]  time = 0.06, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1355, 266, 45, 37} \begin {gather*} -\frac {b^2 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{504 a^3 x^{18}}+\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{84 a^2 x^{21}}-\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{24 a x^{24}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^25,x]

[Out]

-((a + b*x^3)^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(24*a*x^24) + (b*(a + b*x^3)^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6
])/(84*a^2*x^21) - (b^2*(a + b*x^3)^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(504*a^3*x^18)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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^{25}} \, 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^{25}} \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^5}{x^9} \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=-\frac {\left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{24 a x^{24}}-\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^5}{x^8} \, dx,x,x^3\right )}{12 a b^3 \left (a b+b^2 x^3\right )}\\ &=-\frac {\left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{24 a x^{24}}+\frac {b \left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{84 a^2 x^{21}}+\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^5}{x^7} \, dx,x,x^3\right )}{84 a^2 b^2 \left (a b+b^2 x^3\right )}\\ &=-\frac {\left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{24 a x^{24}}+\frac {b \left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{84 a^2 x^{21}}-\frac {b^2 \left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{504 a^3 x^{18}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 83, normalized size = 0.65 \begin {gather*} -\frac {\sqrt {\left (a+b x^3\right )^2} \left (21 a^5+120 a^4 b x^3+280 a^3 b^2 x^6+336 a^2 b^3 x^9+210 a b^4 x^{12}+56 b^5 x^{15}\right )}{504 x^{24} \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^25,x]

[Out]

-1/504*(Sqrt[(a + b*x^3)^2]*(21*a^5 + 120*a^4*b*x^3 + 280*a^3*b^2*x^6 + 336*a^2*b^3*x^9 + 210*a*b^4*x^12 + 56*
b^5*x^15))/(x^24*(a + b*x^3))

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IntegrateAlgebraic [B]  time = 1.54, size = 532, normalized size = 4.16 \begin {gather*} \frac {16 b^7 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (-21 a^{12} b-267 a^{11} b^2 x^3-1561 a^{10} b^3 x^6-5551 a^9 b^4 x^9-13377 a^8 b^5 x^{12}-23023 a^7 b^6 x^{15}-29029 a^6 b^7 x^{18}-27027 a^5 b^8 x^{21}-18446 a^4 b^9 x^{24}-9002 a^3 b^{10} x^{27}-2982 a^2 b^{11} x^{30}-602 a b^{12} x^{33}-56 b^{13} x^{36}\right )+16 \sqrt {b^2} b^7 \left (21 a^{13}+288 a^{12} b x^3+1828 a^{11} b^2 x^6+7112 a^{10} b^3 x^9+18928 a^9 b^4 x^{12}+36400 a^8 b^5 x^{15}+52052 a^7 b^6 x^{18}+56056 a^6 b^7 x^{21}+45473 a^5 b^8 x^{24}+27448 a^4 b^9 x^{27}+11984 a^3 b^{10} x^{30}+3584 a^2 b^{11} x^{33}+658 a b^{12} x^{36}+56 b^{13} x^{39}\right )}{63 \sqrt {b^2} x^{24} \sqrt {a^2+2 a b x^3+b^2 x^6} \left (-128 a^7 b^7-896 a^6 b^8 x^3-2688 a^5 b^9 x^6-4480 a^4 b^{10} x^9-4480 a^3 b^{11} x^{12}-2688 a^2 b^{12} x^{15}-896 a b^{13} x^{18}-128 b^{14} x^{21}\right )+63 x^{24} \left (128 a^8 b^8+1024 a^7 b^9 x^3+3584 a^6 b^{10} x^6+7168 a^5 b^{11} x^9+8960 a^4 b^{12} x^{12}+7168 a^3 b^{13} x^{15}+3584 a^2 b^{14} x^{18}+1024 a b^{15} x^{21}+128 b^{16} x^{24}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^25,x]

[Out]

(16*b^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*(-21*a^12*b - 267*a^11*b^2*x^3 - 1561*a^10*b^3*x^6 - 5551*a^9*b^4*x^9
- 13377*a^8*b^5*x^12 - 23023*a^7*b^6*x^15 - 29029*a^6*b^7*x^18 - 27027*a^5*b^8*x^21 - 18446*a^4*b^9*x^24 - 900
2*a^3*b^10*x^27 - 2982*a^2*b^11*x^30 - 602*a*b^12*x^33 - 56*b^13*x^36) + 16*b^7*Sqrt[b^2]*(21*a^13 + 288*a^12*
b*x^3 + 1828*a^11*b^2*x^6 + 7112*a^10*b^3*x^9 + 18928*a^9*b^4*x^12 + 36400*a^8*b^5*x^15 + 52052*a^7*b^6*x^18 +
56056*a^6*b^7*x^21 + 45473*a^5*b^8*x^24 + 27448*a^4*b^9*x^27 + 11984*a^3*b^10*x^30 + 3584*a^2*b^11*x^33 + 658
*a*b^12*x^36 + 56*b^13*x^39))/(63*Sqrt[b^2]*x^24*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*(-128*a^7*b^7 - 896*a^6*b^8*x
^3 - 2688*a^5*b^9*x^6 - 4480*a^4*b^10*x^9 - 4480*a^3*b^11*x^12 - 2688*a^2*b^12*x^15 - 896*a*b^13*x^18 - 128*b^
14*x^21) + 63*x^24*(128*a^8*b^8 + 1024*a^7*b^9*x^3 + 3584*a^6*b^10*x^6 + 7168*a^5*b^11*x^9 + 8960*a^4*b^12*x^1
2 + 7168*a^3*b^13*x^15 + 3584*a^2*b^14*x^18 + 1024*a*b^15*x^21 + 128*b^16*x^24))

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fricas [A]  time = 1.27, size = 59, normalized size = 0.46 \begin {gather*} -\frac {56 \, b^{5} x^{15} + 210 \, a b^{4} x^{12} + 336 \, a^{2} b^{3} x^{9} + 280 \, a^{3} b^{2} x^{6} + 120 \, a^{4} b x^{3} + 21 \, a^{5}}{504 \, x^{24}} \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^25,x, algorithm="fricas")

[Out]

-1/504*(56*b^5*x^15 + 210*a*b^4*x^12 + 336*a^2*b^3*x^9 + 280*a^3*b^2*x^6 + 120*a^4*b*x^3 + 21*a^5)/x^24

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giac [A]  time = 0.39, size = 107, normalized size = 0.84 \begin {gather*} -\frac {56 \, b^{5} x^{15} \mathrm {sgn}\left (b x^{3} + a\right ) + 210 \, a b^{4} x^{12} \mathrm {sgn}\left (b x^{3} + a\right ) + 336 \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 280 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 120 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 21 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{504 \, x^{24}} \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^25,x, algorithm="giac")

[Out]

-1/504*(56*b^5*x^15*sgn(b*x^3 + a) + 210*a*b^4*x^12*sgn(b*x^3 + a) + 336*a^2*b^3*x^9*sgn(b*x^3 + a) + 280*a^3*
b^2*x^6*sgn(b*x^3 + a) + 120*a^4*b*x^3*sgn(b*x^3 + a) + 21*a^5*sgn(b*x^3 + a))/x^24

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maple [A]  time = 0.01, size = 80, normalized size = 0.62 \begin {gather*} -\frac {\left (56 b^{5} x^{15}+210 a \,b^{4} x^{12}+336 a^{2} b^{3} x^{9}+280 a^{3} b^{2} x^{6}+120 a^{4} b \,x^{3}+21 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{504 \left (b \,x^{3}+a \right )^{5} x^{24}} \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^25,x)

[Out]

-1/504*(56*b^5*x^15+210*a*b^4*x^12+336*a^2*b^3*x^9+280*a^3*b^2*x^6+120*a^4*b*x^3+21*a^5)*((b*x^3+a)^2)^(5/2)/x
^24/(b*x^3+a)^5

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maxima [B]  time = 1.02, size = 272, normalized size = 2.12 \begin {gather*} \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b^{8}}{18 \, a^{8}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b^{7}}{18 \, a^{7} x^{3}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{6}}{18 \, a^{8} x^{6}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{5}}{18 \, a^{7} x^{9}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{4}}{18 \, a^{6} x^{12}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{3}}{18 \, a^{5} x^{15}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{2}}{18 \, a^{4} x^{18}} + \frac {3 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b}{56 \, a^{3} x^{21}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}}}{24 \, a^{2} x^{24}} \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^25,x, algorithm="maxima")

[Out]

1/18*(b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)*b^8/a^8 + 1/18*(b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)*b^7/(a^7*x^3) - 1/18*(
b^2*x^6 + 2*a*b*x^3 + a^2)^(7/2)*b^6/(a^8*x^6) + 1/18*(b^2*x^6 + 2*a*b*x^3 + a^2)^(7/2)*b^5/(a^7*x^9) - 1/18*(
b^2*x^6 + 2*a*b*x^3 + a^2)^(7/2)*b^4/(a^6*x^12) + 1/18*(b^2*x^6 + 2*a*b*x^3 + a^2)^(7/2)*b^3/(a^5*x^15) - 1/18
*(b^2*x^6 + 2*a*b*x^3 + a^2)^(7/2)*b^2/(a^4*x^18) + 3/56*(b^2*x^6 + 2*a*b*x^3 + a^2)^(7/2)*b/(a^3*x^21) - 1/24
*(b^2*x^6 + 2*a*b*x^3 + a^2)^(7/2)/(a^2*x^24)

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mupad [B]  time = 1.22, size = 231, normalized size = 1.80 \begin {gather*} -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{24\,x^{24}\,\left (b\,x^3+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{9\,x^9\,\left (b\,x^3+a\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{12\,x^{12}\,\left (b\,x^3+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{21\,x^{21}\,\left (b\,x^3+a\right )}-\frac {2\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{3\,x^{15}\,\left (b\,x^3+a\right )}-\frac {5\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{9\,x^{18}\,\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^25,x)

[Out]

- (a^5*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(24*x^24*(a + b*x^3)) - (b^5*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(9*x
^9*(a + b*x^3)) - (5*a*b^4*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(12*x^12*(a + b*x^3)) - (5*a^4*b*(a^2 + b^2*x^6
+ 2*a*b*x^3)^(1/2))/(21*x^21*(a + b*x^3)) - (2*a^2*b^3*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(3*x^15*(a + b*x^3))
- (5*a^3*b^2*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(9*x^18*(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^{25}}\, 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**25,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**25, x)

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