3.5.30 \(\int \frac {(a^2+2 a b x^2+b^2 x^4)^{5/2}}{x^{21}} \, dx\)

Optimal. Leaf size=255 \[ -\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{12 x^{12} \left (a+b x^2\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 x^{14} \left (a+b x^2\right )}-\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{20 x^{20} \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{18 x^{18} \left (a+b x^2\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^{16} \left (a+b x^2\right )} \]

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Rubi [A]  time = 0.15, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1111, 646, 43} \begin {gather*} -\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{20 x^{20} \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{18 x^{18} \left (a+b x^2\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^{16} \left (a+b x^2\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 x^{14} \left (a+b x^2\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{12 x^{12} \left (a+b x^2\right )}-\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^21,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(20*x^20*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(18*x
^18*(a + b*x^2)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*x^16*(a + b*x^2)) - (5*a^2*b^3*Sqrt[a^2 + 2*
a*b*x^2 + b^2*x^4])/(7*x^14*(a + b*x^2)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(12*x^12*(a + b*x^2)) - (
b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(10*x^10*(a + b*x^2))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rule 1111

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && Integ
erQ[(m - 1)/2] && (GtQ[m, 0] || LtQ[0, 4*p, -m - 1])

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{21}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{11}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^5}{x^{11}} \, dx,x,x^2\right )}{2 b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int \left (\frac {a^5 b^5}{x^{11}}+\frac {5 a^4 b^6}{x^{10}}+\frac {10 a^3 b^7}{x^9}+\frac {10 a^2 b^8}{x^8}+\frac {5 a b^9}{x^7}+\frac {b^{10}}{x^6}\right ) \, dx,x,x^2\right )}{2 b^4 \left (a b+b^2 x^2\right )}\\ &=-\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{20 x^{20} \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{18 x^{18} \left (a+b x^2\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^{16} \left (a+b x^2\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 x^{14} \left (a+b x^2\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{12 x^{12} \left (a+b x^2\right )}-\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\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^2\right )^2} \left (126 a^5+700 a^4 b x^2+1575 a^3 b^2 x^4+1800 a^2 b^3 x^6+1050 a b^4 x^8+252 b^5 x^{10}\right )}{2520 x^{20} \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^21,x]

[Out]

-1/2520*(Sqrt[(a + b*x^2)^2]*(126*a^5 + 700*a^4*b*x^2 + 1575*a^3*b^2*x^4 + 1800*a^2*b^3*x^6 + 1050*a*b^4*x^8 +
252*b^5*x^10))/(x^20*(a + b*x^2))

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IntegrateAlgebraic [B]  time = 1.67, size = 620, normalized size = 2.43 \begin {gather*} \frac {64 b^9 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (-126 a^{14} b-1834 a^{13} b^2 x^2-12411 a^{12} b^3 x^4-51759 a^{11} b^4 x^6-148626 a^{10} b^5 x^8-310878 a^9 b^6 x^{10}-488502 a^8 b^7 x^{12}-585858 a^7 b^8 x^{14}-538902 a^6 b^9 x^{16}-378378 a^5 b^{10} x^{18}-199627 a^4 b^{11} x^{20}-76743 a^3 b^{12} x^{22}-20322 a^2 b^{13} x^{24}-3318 a b^{14} x^{26}-252 b^{15} x^{28}\right )+64 \sqrt {b^2} b^9 \left (126 a^{15}+1960 a^{14} b x^2+14245 a^{13} b^2 x^4+64170 a^{12} b^3 x^6+200385 a^{11} b^4 x^8+459504 a^{10} b^5 x^{10}+799380 a^9 b^6 x^{12}+1074360 a^8 b^7 x^{14}+1124760 a^7 b^8 x^{16}+917280 a^6 b^9 x^{18}+578005 a^5 b^{10} x^{20}+276370 a^4 b^{11} x^{22}+97065 a^3 b^{12} x^{24}+23640 a^2 b^{13} x^{26}+3570 a b^{14} x^{28}+252 b^{15} x^{30}\right )}{315 \sqrt {b^2} x^{20} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (-512 a^9 b^9-4608 a^8 b^{10} x^2-18432 a^7 b^{11} x^4-43008 a^6 b^{12} x^6-64512 a^5 b^{13} x^8-64512 a^4 b^{14} x^{10}-43008 a^3 b^{15} x^{12}-18432 a^2 b^{16} x^{14}-4608 a b^{17} x^{16}-512 b^{18} x^{18}\right )+315 x^{20} \left (512 a^{10} b^{10}+5120 a^9 b^{11} x^2+23040 a^8 b^{12} x^4+61440 a^7 b^{13} x^6+107520 a^6 b^{14} x^8+129024 a^5 b^{15} x^{10}+107520 a^4 b^{16} x^{12}+61440 a^3 b^{17} x^{14}+23040 a^2 b^{18} x^{16}+5120 a b^{19} x^{18}+512 b^{20} x^{20}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^21,x]

[Out]

(64*b^9*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-126*a^14*b - 1834*a^13*b^2*x^2 - 12411*a^12*b^3*x^4 - 51759*a^11*b^4
*x^6 - 148626*a^10*b^5*x^8 - 310878*a^9*b^6*x^10 - 488502*a^8*b^7*x^12 - 585858*a^7*b^8*x^14 - 538902*a^6*b^9*
x^16 - 378378*a^5*b^10*x^18 - 199627*a^4*b^11*x^20 - 76743*a^3*b^12*x^22 - 20322*a^2*b^13*x^24 - 3318*a*b^14*x
^26 - 252*b^15*x^28) + 64*b^9*Sqrt[b^2]*(126*a^15 + 1960*a^14*b*x^2 + 14245*a^13*b^2*x^4 + 64170*a^12*b^3*x^6
+ 200385*a^11*b^4*x^8 + 459504*a^10*b^5*x^10 + 799380*a^9*b^6*x^12 + 1074360*a^8*b^7*x^14 + 1124760*a^7*b^8*x^
16 + 917280*a^6*b^9*x^18 + 578005*a^5*b^10*x^20 + 276370*a^4*b^11*x^22 + 97065*a^3*b^12*x^24 + 23640*a^2*b^13*
x^26 + 3570*a*b^14*x^28 + 252*b^15*x^30))/(315*Sqrt[b^2]*x^20*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-512*a^9*b^9 -
4608*a^8*b^10*x^2 - 18432*a^7*b^11*x^4 - 43008*a^6*b^12*x^6 - 64512*a^5*b^13*x^8 - 64512*a^4*b^14*x^10 - 43008
*a^3*b^15*x^12 - 18432*a^2*b^16*x^14 - 4608*a*b^17*x^16 - 512*b^18*x^18) + 315*x^20*(512*a^10*b^10 + 5120*a^9*
b^11*x^2 + 23040*a^8*b^12*x^4 + 61440*a^7*b^13*x^6 + 107520*a^6*b^14*x^8 + 129024*a^5*b^15*x^10 + 107520*a^4*b
^16*x^12 + 61440*a^3*b^17*x^14 + 23040*a^2*b^18*x^16 + 5120*a*b^19*x^18 + 512*b^20*x^20))

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fricas [A]  time = 0.73, size = 59, normalized size = 0.23 \begin {gather*} -\frac {252 \, b^{5} x^{10} + 1050 \, a b^{4} x^{8} + 1800 \, a^{2} b^{3} x^{6} + 1575 \, a^{3} b^{2} x^{4} + 700 \, a^{4} b x^{2} + 126 \, a^{5}}{2520 \, x^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^21,x, algorithm="fricas")

[Out]

-1/2520*(252*b^5*x^10 + 1050*a*b^4*x^8 + 1800*a^2*b^3*x^6 + 1575*a^3*b^2*x^4 + 700*a^4*b*x^2 + 126*a^5)/x^20

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giac [A]  time = 0.16, size = 107, normalized size = 0.42 \begin {gather*} -\frac {252 \, b^{5} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + 1050 \, a b^{4} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + 1800 \, a^{2} b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 1575 \, a^{3} b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 700 \, a^{4} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 126 \, a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{2520 \, x^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^21,x, algorithm="giac")

[Out]

-1/2520*(252*b^5*x^10*sgn(b*x^2 + a) + 1050*a*b^4*x^8*sgn(b*x^2 + a) + 1800*a^2*b^3*x^6*sgn(b*x^2 + a) + 1575*
a^3*b^2*x^4*sgn(b*x^2 + a) + 700*a^4*b*x^2*sgn(b*x^2 + a) + 126*a^5*sgn(b*x^2 + a))/x^20

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maple [A]  time = 0.01, size = 80, normalized size = 0.31 \begin {gather*} -\frac {\left (252 b^{5} x^{10}+1050 a \,b^{4} x^{8}+1800 a^{2} b^{3} x^{6}+1575 a^{3} b^{2} x^{4}+700 a^{4} b \,x^{2}+126 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}}}{2520 \left (b \,x^{2}+a \right )^{5} x^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^21,x)

[Out]

-1/2520*(252*b^5*x^10+1050*a*b^4*x^8+1800*a^2*b^3*x^6+1575*a^3*b^2*x^4+700*a^4*b*x^2+126*a^5)*((b*x^2+a)^2)^(5
/2)/x^20/(b*x^2+a)^5

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maxima [A]  time = 1.34, size = 57, normalized size = 0.22 \begin {gather*} -\frac {b^{5}}{10 \, x^{10}} - \frac {5 \, a b^{4}}{12 \, x^{12}} - \frac {5 \, a^{2} b^{3}}{7 \, x^{14}} - \frac {5 \, a^{3} b^{2}}{8 \, x^{16}} - \frac {5 \, a^{4} b}{18 \, x^{18}} - \frac {a^{5}}{20 \, x^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^21,x, algorithm="maxima")

[Out]

-1/10*b^5/x^10 - 5/12*a*b^4/x^12 - 5/7*a^2*b^3/x^14 - 5/8*a^3*b^2/x^16 - 5/18*a^4*b/x^18 - 1/20*a^5/x^20

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mupad [B]  time = 4.22, size = 231, normalized size = 0.91 \begin {gather*} -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{20\,x^{20}\,\left (b\,x^2+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{10\,x^{10}\,\left (b\,x^2+a\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{12\,x^{12}\,\left (b\,x^2+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{18\,x^{18}\,\left (b\,x^2+a\right )}-\frac {5\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{7\,x^{14}\,\left (b\,x^2+a\right )}-\frac {5\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{8\,x^{16}\,\left (b\,x^2+a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)/x^21,x)

[Out]

- (a^5*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(20*x^20*(a + b*x^2)) - (b^5*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(10*
x^10*(a + b*x^2)) - (5*a*b^4*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(12*x^12*(a + b*x^2)) - (5*a^4*b*(a^2 + b^2*x^
4 + 2*a*b*x^2)^(1/2))/(18*x^18*(a + b*x^2)) - (5*a^2*b^3*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(7*x^14*(a + b*x^2
)) - (5*a^3*b^2*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(8*x^16*(a + b*x^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}{x^{21}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/x**21,x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/x**21, x)

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