3.2.58 \(\int \frac {(a^2+2 a b x+b^2 x^2)^{5/2}}{x^{12}} \, dx\)

Optimal. Leaf size=231 \[ -\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {5 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^{10} (a+b x)}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {646, 43} \begin {gather*} -\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^{10} (a+b x)}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac {5 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^12,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*x^11*(a + b*x)) - (a^4*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*x^10*(a +
b*x)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*x^9*(a + b*x)) - (5*a^2*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(4*x^8*(a + b*x)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*x^7*(a + b*x)) - (b^5*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(6*x^6*(a + b*x))

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]

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{12}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{x^{12}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \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}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^{10} (a+b x)}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac {5 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 77, normalized size = 0.33 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (252 a^5+1386 a^4 b x+3080 a^3 b^2 x^2+3465 a^2 b^3 x^3+1980 a b^4 x^4+462 b^5 x^5\right )}{2772 x^{11} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^12,x]

[Out]

-1/2772*(Sqrt[(a + b*x)^2]*(252*a^5 + 1386*a^4*b*x + 3080*a^3*b^2*x^2 + 3465*a^2*b^3*x^3 + 1980*a*b^4*x^4 + 46
2*b^5*x^5))/(x^11*(a + b*x))

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 1.80, size = 652, normalized size = 2.82 \begin {gather*} \frac {256 b^{10} \sqrt {a^2+2 a b x+b^2 x^2} \left (-252 a^{15} b-3906 a^{14} b^2 x-28280 a^{13} b^3 x^2-126875 a^{12} b^4 x^3-394470 a^{11} b^5 x^4-900351 a^{10} b^6 x^5-1558512 a^9 b^7 x^6-2083500 a^8 b^8 x^7-2168880 a^7 b^9 x^8-1758120 a^6 b^{10} x^9-1100736 a^5 b^{11} x^{10}-522731 a^4 b^{12} x^{11}-182270 a^3 b^{13} x^{12}-44055 a^2 b^{14} x^{13}-6600 a b^{15} x^{14}-462 b^{16} x^{15}\right )+256 \sqrt {b^2} b^{10} \left (252 a^{16}+4158 a^{15} b x+32186 a^{14} b^2 x^2+155155 a^{13} b^3 x^3+521345 a^{12} b^4 x^4+1294821 a^{11} b^5 x^5+2458863 a^{10} b^6 x^6+3642012 a^9 b^7 x^7+4252380 a^8 b^8 x^8+3927000 a^7 b^9 x^9+2858856 a^6 b^{10} x^{10}+1623467 a^5 b^{11} x^{11}+705001 a^4 b^{12} x^{12}+226325 a^3 b^{13} x^{13}+50655 a^2 b^{14} x^{14}+7062 a b^{15} x^{15}+462 b^{16} x^{16}\right )}{693 \sqrt {b^2} x^{11} \sqrt {a^2+2 a b x+b^2 x^2} \left (-1024 a^{10} b^{10}-10240 a^9 b^{11} x-46080 a^8 b^{12} x^2-122880 a^7 b^{13} x^3-215040 a^6 b^{14} x^4-258048 a^5 b^{15} x^5-215040 a^4 b^{16} x^6-122880 a^3 b^{17} x^7-46080 a^2 b^{18} x^8-10240 a b^{19} x^9-1024 b^{20} x^{10}\right )+693 x^{11} \left (1024 a^{11} b^{11}+11264 a^{10} b^{12} x+56320 a^9 b^{13} x^2+168960 a^8 b^{14} x^3+337920 a^7 b^{15} x^4+473088 a^6 b^{16} x^5+473088 a^5 b^{17} x^6+337920 a^4 b^{18} x^7+168960 a^3 b^{19} x^8+56320 a^2 b^{20} x^9+11264 a b^{21} x^{10}+1024 b^{22} x^{11}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^12,x]

[Out]

(256*b^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-252*a^15*b - 3906*a^14*b^2*x - 28280*a^13*b^3*x^2 - 126875*a^12*b^4*
x^3 - 394470*a^11*b^5*x^4 - 900351*a^10*b^6*x^5 - 1558512*a^9*b^7*x^6 - 2083500*a^8*b^8*x^7 - 2168880*a^7*b^9*
x^8 - 1758120*a^6*b^10*x^9 - 1100736*a^5*b^11*x^10 - 522731*a^4*b^12*x^11 - 182270*a^3*b^13*x^12 - 44055*a^2*b
^14*x^13 - 6600*a*b^15*x^14 - 462*b^16*x^15) + 256*b^10*Sqrt[b^2]*(252*a^16 + 4158*a^15*b*x + 32186*a^14*b^2*x
^2 + 155155*a^13*b^3*x^3 + 521345*a^12*b^4*x^4 + 1294821*a^11*b^5*x^5 + 2458863*a^10*b^6*x^6 + 3642012*a^9*b^7
*x^7 + 4252380*a^8*b^8*x^8 + 3927000*a^7*b^9*x^9 + 2858856*a^6*b^10*x^10 + 1623467*a^5*b^11*x^11 + 705001*a^4*
b^12*x^12 + 226325*a^3*b^13*x^13 + 50655*a^2*b^14*x^14 + 7062*a*b^15*x^15 + 462*b^16*x^16))/(693*Sqrt[b^2]*x^1
1*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1024*a^10*b^10 - 10240*a^9*b^11*x - 46080*a^8*b^12*x^2 - 122880*a^7*b^13*x^3
- 215040*a^6*b^14*x^4 - 258048*a^5*b^15*x^5 - 215040*a^4*b^16*x^6 - 122880*a^3*b^17*x^7 - 46080*a^2*b^18*x^8
- 10240*a*b^19*x^9 - 1024*b^20*x^10) + 693*x^11*(1024*a^11*b^11 + 11264*a^10*b^12*x + 56320*a^9*b^13*x^2 + 168
960*a^8*b^14*x^3 + 337920*a^7*b^15*x^4 + 473088*a^6*b^16*x^5 + 473088*a^5*b^17*x^6 + 337920*a^4*b^18*x^7 + 168
960*a^3*b^19*x^8 + 56320*a^2*b^20*x^9 + 11264*a*b^21*x^10 + 1024*b^22*x^11))

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 57, normalized size = 0.25 \begin {gather*} -\frac {462 \, b^{5} x^{5} + 1980 \, a b^{4} x^{4} + 3465 \, a^{2} b^{3} x^{3} + 3080 \, a^{3} b^{2} x^{2} + 1386 \, a^{4} b x + 252 \, a^{5}}{2772 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^12,x, algorithm="fricas")

[Out]

-1/2772*(462*b^5*x^5 + 1980*a*b^4*x^4 + 3465*a^2*b^3*x^3 + 3080*a^3*b^2*x^2 + 1386*a^4*b*x + 252*a^5)/x^11

________________________________________________________________________________________

giac [A]  time = 0.19, size = 108, normalized size = 0.47 \begin {gather*} \frac {b^{11} \mathrm {sgn}\left (b x + a\right )}{2772 \, a^{6}} - \frac {462 \, b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 1980 \, a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 3465 \, a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 3080 \, a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 1386 \, a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 252 \, a^{5} \mathrm {sgn}\left (b x + a\right )}{2772 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^12,x, algorithm="giac")

[Out]

1/2772*b^11*sgn(b*x + a)/a^6 - 1/2772*(462*b^5*x^5*sgn(b*x + a) + 1980*a*b^4*x^4*sgn(b*x + a) + 3465*a^2*b^3*x
^3*sgn(b*x + a) + 3080*a^3*b^2*x^2*sgn(b*x + a) + 1386*a^4*b*x*sgn(b*x + a) + 252*a^5*sgn(b*x + a))/x^11

________________________________________________________________________________________

maple [A]  time = 0.05, size = 74, normalized size = 0.32 \begin {gather*} -\frac {\left (462 b^{5} x^{5}+1980 a \,b^{4} x^{4}+3465 a^{2} b^{3} x^{3}+3080 a^{3} b^{2} x^{2}+1386 a^{4} b x +252 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2772 \left (b x +a \right )^{5} x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^12,x)

[Out]

-1/2772*(462*b^5*x^5+1980*a*b^4*x^4+3465*a^2*b^3*x^3+3080*a^3*b^2*x^2+1386*a^4*b*x+252*a^5)*((b*x+a)^2)^(5/2)/
x^11/(b*x+a)^5

________________________________________________________________________________________

maxima [B]  time = 1.74, size = 341, normalized size = 1.48 \begin {gather*} -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{11}}{6 \, a^{11}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{10}}{6 \, a^{10} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{9}}{6 \, a^{11} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{8}}{6 \, a^{10} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{7}}{6 \, a^{9} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{6}}{6 \, a^{8} x^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{5}}{6 \, a^{7} x^{6}} - \frac {461 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{4}}{2772 \, a^{6} x^{7}} + \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{3}}{396 \, a^{5} x^{8}} - \frac {31 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{2}}{198 \, a^{4} x^{9}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b}{22 \, a^{3} x^{10}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{11 \, a^{2} x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^12,x, algorithm="maxima")

[Out]

-1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*b^11/a^11 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*b^10/(a^10*x) + 1/6*(b^2*
x^2 + 2*a*b*x + a^2)^(7/2)*b^9/(a^11*x^2) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b^8/(a^10*x^3) + 1/6*(b^2*x^2
+ 2*a*b*x + a^2)^(7/2)*b^7/(a^9*x^4) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b^6/(a^8*x^5) + 1/6*(b^2*x^2 + 2*a*
b*x + a^2)^(7/2)*b^5/(a^7*x^6) - 461/2772*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b^4/(a^6*x^7) + 65/396*(b^2*x^2 + 2*
a*b*x + a^2)^(7/2)*b^3/(a^5*x^8) - 31/198*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b^2/(a^4*x^9) + 3/22*(b^2*x^2 + 2*a*
b*x + a^2)^(7/2)*b/(a^3*x^10) - 1/11*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)/(a^2*x^11)

________________________________________________________________________________________

mupad [B]  time = 0.20, size = 207, normalized size = 0.90 \begin {gather*} -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{11\,x^{11}\,\left (a+b\,x\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,x^6\,\left (a+b\,x\right )}-\frac {5\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,x^8\,\left (a+b\,x\right )}-\frac {10\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{9\,x^9\,\left (a+b\,x\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )}-\frac {a^4\,b\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^{10}\,\left (a+b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^(5/2)/x^12,x)

[Out]

- (a^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(11*x^11*(a + b*x)) - (b^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(6*x^6*(a
+ b*x)) - (5*a^2*b^3*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(4*x^8*(a + b*x)) - (10*a^3*b^2*(a^2 + b^2*x^2 + 2*a*b*x
)^(1/2))/(9*x^9*(a + b*x)) - (5*a*b^4*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(7*x^7*(a + b*x)) - (a^4*b*(a^2 + b^2*x
^2 + 2*a*b*x)^(1/2))/(2*x^10*(a + b*x))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{12}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/x**12,x)

[Out]

Integral(((a + b*x)**2)**(5/2)/x**12, x)

________________________________________________________________________________________