3.5.31 \(\int \frac {(a^2+2 a b x^2+b^2 x^4)^{5/2}}{x^{23}} \, dx\)

Optimal. Leaf size=255 \[ -\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{12 x^{12} \left (a+b x^2\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^{16} \left (a+b x^2\right )}-\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{22 x^{22} \left (a+b x^2\right )}-\frac {a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^{20} \left (a+b x^2\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 x^{18} \left (a+b x^2\right )} \]

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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}}{22 x^{22} \left (a+b x^2\right )}-\frac {a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^{20} \left (a+b x^2\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 x^{18} \left (a+b x^2\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^{16} \left (a+b x^2\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{12 x^{12} \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^23,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(22*x^22*(a + b*x^2)) - (a^4*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(4*x^20
*(a + b*x^2)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(9*x^18*(a + b*x^2)) - (5*a^2*b^3*Sqrt[a^2 + 2*a*b
*x^2 + b^2*x^4])/(8*x^16*(a + b*x^2)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(14*x^14*(a + b*x^2)) - (b^5
*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(12*x^12*(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^{23}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{12}} \, 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^{12}} \, 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^{12}}+\frac {5 a^4 b^6}{x^{11}}+\frac {10 a^3 b^7}{x^{10}}+\frac {10 a^2 b^8}{x^9}+\frac {5 a b^9}{x^8}+\frac {b^{10}}{x^7}\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}}{22 x^{22} \left (a+b x^2\right )}-\frac {a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^{20} \left (a+b x^2\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 x^{18} \left (a+b x^2\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^{16} \left (a+b x^2\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{12 x^{12} \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 (252 a^5+1386 a^4 b x^2+3080 a^3 b^2 x^4+3465 a^2 b^3 x^6+1980 a b^4 x^8+462 b^5 x^{10}\right )}{5544 x^{22} \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^23,x]

[Out]

-1/5544*(Sqrt[(a + b*x^2)^2]*(252*a^5 + 1386*a^4*b*x^2 + 3080*a^3*b^2*x^4 + 3465*a^2*b^3*x^6 + 1980*a*b^4*x^8
+ 462*b^5*x^10))/(x^22*(a + b*x^2))

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IntegrateAlgebraic [B]  time = 1.78, size = 664, normalized size = 2.60 \begin {gather*} \frac {128 b^{10} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (-252 a^{15} b-3906 a^{14} b^2 x^2-28280 a^{13} b^3 x^4-126875 a^{12} b^4 x^6-394470 a^{11} b^5 x^8-900351 a^{10} b^6 x^{10}-1558512 a^9 b^7 x^{12}-2083500 a^8 b^8 x^{14}-2168880 a^7 b^9 x^{16}-1758120 a^6 b^{10} x^{18}-1100736 a^5 b^{11} x^{20}-522731 a^4 b^{12} x^{22}-182270 a^3 b^{13} x^{24}-44055 a^2 b^{14} x^{26}-6600 a b^{15} x^{28}-462 b^{16} x^{30}\right )+128 \sqrt {b^2} b^{10} \left (252 a^{16}+4158 a^{15} b x^2+32186 a^{14} b^2 x^4+155155 a^{13} b^3 x^6+521345 a^{12} b^4 x^8+1294821 a^{11} b^5 x^{10}+2458863 a^{10} b^6 x^{12}+3642012 a^9 b^7 x^{14}+4252380 a^8 b^8 x^{16}+3927000 a^7 b^9 x^{18}+2858856 a^6 b^{10} x^{20}+1623467 a^5 b^{11} x^{22}+705001 a^4 b^{12} x^{24}+226325 a^3 b^{13} x^{26}+50655 a^2 b^{14} x^{28}+7062 a b^{15} x^{30}+462 b^{16} x^{32}\right )}{693 \sqrt {b^2} x^{22} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (-1024 a^{10} b^{10}-10240 a^9 b^{11} x^2-46080 a^8 b^{12} x^4-122880 a^7 b^{13} x^6-215040 a^6 b^{14} x^8-258048 a^5 b^{15} x^{10}-215040 a^4 b^{16} x^{12}-122880 a^3 b^{17} x^{14}-46080 a^2 b^{18} x^{16}-10240 a b^{19} x^{18}-1024 b^{20} x^{20}\right )+693 x^{22} \left (1024 a^{11} b^{11}+11264 a^{10} b^{12} x^2+56320 a^9 b^{13} x^4+168960 a^8 b^{14} x^6+337920 a^7 b^{15} x^8+473088 a^6 b^{16} x^{10}+473088 a^5 b^{17} x^{12}+337920 a^4 b^{18} x^{14}+168960 a^3 b^{19} x^{16}+56320 a^2 b^{20} x^{18}+11264 a b^{21} x^{20}+1024 b^{22} x^{22}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^23,x]

[Out]

(128*b^10*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-252*a^15*b - 3906*a^14*b^2*x^2 - 28280*a^13*b^3*x^4 - 126875*a^12*
b^4*x^6 - 394470*a^11*b^5*x^8 - 900351*a^10*b^6*x^10 - 1558512*a^9*b^7*x^12 - 2083500*a^8*b^8*x^14 - 2168880*a
^7*b^9*x^16 - 1758120*a^6*b^10*x^18 - 1100736*a^5*b^11*x^20 - 522731*a^4*b^12*x^22 - 182270*a^3*b^13*x^24 - 44
055*a^2*b^14*x^26 - 6600*a*b^15*x^28 - 462*b^16*x^30) + 128*b^10*Sqrt[b^2]*(252*a^16 + 4158*a^15*b*x^2 + 32186
*a^14*b^2*x^4 + 155155*a^13*b^3*x^6 + 521345*a^12*b^4*x^8 + 1294821*a^11*b^5*x^10 + 2458863*a^10*b^6*x^12 + 36
42012*a^9*b^7*x^14 + 4252380*a^8*b^8*x^16 + 3927000*a^7*b^9*x^18 + 2858856*a^6*b^10*x^20 + 1623467*a^5*b^11*x^
22 + 705001*a^4*b^12*x^24 + 226325*a^3*b^13*x^26 + 50655*a^2*b^14*x^28 + 7062*a*b^15*x^30 + 462*b^16*x^32))/(6
93*Sqrt[b^2]*x^22*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-1024*a^10*b^10 - 10240*a^9*b^11*x^2 - 46080*a^8*b^12*x^4 -
122880*a^7*b^13*x^6 - 215040*a^6*b^14*x^8 - 258048*a^5*b^15*x^10 - 215040*a^4*b^16*x^12 - 122880*a^3*b^17*x^1
4 - 46080*a^2*b^18*x^16 - 10240*a*b^19*x^18 - 1024*b^20*x^20) + 693*x^22*(1024*a^11*b^11 + 11264*a^10*b^12*x^2
+ 56320*a^9*b^13*x^4 + 168960*a^8*b^14*x^6 + 337920*a^7*b^15*x^8 + 473088*a^6*b^16*x^10 + 473088*a^5*b^17*x^1
2 + 337920*a^4*b^18*x^14 + 168960*a^3*b^19*x^16 + 56320*a^2*b^20*x^18 + 11264*a*b^21*x^20 + 1024*b^22*x^22))

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fricas [A]  time = 0.96, size = 59, normalized size = 0.23 \begin {gather*} -\frac {462 \, b^{5} x^{10} + 1980 \, a b^{4} x^{8} + 3465 \, a^{2} b^{3} x^{6} + 3080 \, a^{3} b^{2} x^{4} + 1386 \, a^{4} b x^{2} + 252 \, a^{5}}{5544 \, x^{22}} \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^23,x, algorithm="fricas")

[Out]

-1/5544*(462*b^5*x^10 + 1980*a*b^4*x^8 + 3465*a^2*b^3*x^6 + 3080*a^3*b^2*x^4 + 1386*a^4*b*x^2 + 252*a^5)/x^22

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giac [A]  time = 0.19, size = 107, normalized size = 0.42 \begin {gather*} -\frac {462 \, b^{5} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + 1980 \, a b^{4} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + 3465 \, a^{2} b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 3080 \, a^{3} b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 1386 \, a^{4} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 252 \, a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{5544 \, x^{22}} \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^23,x, algorithm="giac")

[Out]

-1/5544*(462*b^5*x^10*sgn(b*x^2 + a) + 1980*a*b^4*x^8*sgn(b*x^2 + a) + 3465*a^2*b^3*x^6*sgn(b*x^2 + a) + 3080*
a^3*b^2*x^4*sgn(b*x^2 + a) + 1386*a^4*b*x^2*sgn(b*x^2 + a) + 252*a^5*sgn(b*x^2 + a))/x^22

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maple [A]  time = 0.01, size = 80, normalized size = 0.31 \begin {gather*} -\frac {\left (462 b^{5} x^{10}+1980 a \,b^{4} x^{8}+3465 a^{2} b^{3} x^{6}+3080 a^{3} b^{2} x^{4}+1386 a^{4} b \,x^{2}+252 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}}}{5544 \left (b \,x^{2}+a \right )^{5} x^{22}} \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^23,x)

[Out]

-1/5544*(462*b^5*x^10+1980*a*b^4*x^8+3465*a^2*b^3*x^6+3080*a^3*b^2*x^4+1386*a^4*b*x^2+252*a^5)*((b*x^2+a)^2)^(
5/2)/x^22/(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}}{12 \, x^{12}} - \frac {5 \, a b^{4}}{14 \, x^{14}} - \frac {5 \, a^{2} b^{3}}{8 \, x^{16}} - \frac {5 \, a^{3} b^{2}}{9 \, x^{18}} - \frac {a^{4} b}{4 \, x^{20}} - \frac {a^{5}}{22 \, x^{22}} \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^23,x, algorithm="maxima")

[Out]

-1/12*b^5/x^12 - 5/14*a*b^4/x^14 - 5/8*a^2*b^3/x^16 - 5/9*a^3*b^2/x^18 - 1/4*a^4*b/x^20 - 1/22*a^5/x^22

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mupad [B]  time = 4.23, size = 231, normalized size = 0.91 \begin {gather*} -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{22\,x^{22}\,\left (b\,x^2+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{12\,x^{12}\,\left (b\,x^2+a\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{14\,x^{14}\,\left (b\,x^2+a\right )}-\frac {a^4\,b\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{4\,x^{20}\,\left (b\,x^2+a\right )}-\frac {5\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{8\,x^{16}\,\left (b\,x^2+a\right )}-\frac {5\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{9\,x^{18}\,\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^23,x)

[Out]

- (a^5*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(22*x^22*(a + b*x^2)) - (b^5*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(12*
x^12*(a + b*x^2)) - (5*a*b^4*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(14*x^14*(a + b*x^2)) - (a^4*b*(a^2 + b^2*x^4
+ 2*a*b*x^2)^(1/2))/(4*x^20*(a + b*x^2)) - (5*a^2*b^3*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(8*x^16*(a + b*x^2))
- (5*a^3*b^2*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(9*x^18*(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^{23}}\, 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**23,x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/x**23, x)

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