3.2.56 \(\int \frac {(a^2+2 a b x+b^2 x^2)^{5/2}}{x^{10}} \, dx\)

Optimal. Leaf size=229 \[ -\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{x^5 (a+b x)}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^6 (a+b x)}-\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)} \]

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Rubi [A]  time = 0.05, antiderivative size = 229, 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}}{9 x^9 (a+b x)}-\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {10 a^3 b^2 \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}}{3 x^6 (a+b x)}-\frac {a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{x^5 (a+b x)}-\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^10,x]

[Out]

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

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Mathematica [A]  time = 0.02, size = 77, normalized size = 0.34 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (56 a^5+315 a^4 b x+720 a^3 b^2 x^2+840 a^2 b^3 x^3+504 a b^4 x^4+126 b^5 x^5\right )}{504 x^9 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^10,x]

[Out]

-1/504*(Sqrt[(a + b*x)^2]*(56*a^5 + 315*a^4*b*x + 720*a^3*b^2*x^2 + 840*a^2*b^3*x^3 + 504*a*b^4*x^4 + 126*b^5*
x^5))/(x^9*(a + b*x))

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IntegrateAlgebraic [B]  time = 1.56, size = 564, normalized size = 2.46 \begin {gather*} \frac {32 b^8 \sqrt {a^2+2 a b x+b^2 x^2} \left (-56 a^{13} b-763 a^{12} b^2 x-4808 a^{11} b^3 x^2-18556 a^{10} b^4 x^3-48944 a^9 b^5 x^4-93184 a^8 b^6 x^5-131768 a^7 b^7 x^6-140140 a^6 b^8 x^7-112112 a^5 b^9 x^8-66639 a^4 b^{10} x^9-28608 a^3 b^{11} x^{10}-8400 a^2 b^{12} x^{11}-1512 a b^{13} x^{12}-126 b^{14} x^{13}\right )+32 \sqrt {b^2} b^8 \left (56 a^{14}+819 a^{13} b x+5571 a^{12} b^2 x^2+23364 a^{11} b^3 x^3+67500 a^{10} b^4 x^4+142128 a^9 b^5 x^5+224952 a^8 b^6 x^6+271908 a^7 b^7 x^7+252252 a^6 b^8 x^8+178751 a^5 b^9 x^9+95247 a^4 b^{10} x^{10}+37008 a^3 b^{11} x^{11}+9912 a^2 b^{12} x^{12}+1638 a b^{13} x^{13}+126 b^{14} x^{14}\right )}{63 \sqrt {b^2} x^9 \sqrt {a^2+2 a b x+b^2 x^2} \left (-256 a^8 b^8-2048 a^7 b^9 x-7168 a^6 b^{10} x^2-14336 a^5 b^{11} x^3-17920 a^4 b^{12} x^4-14336 a^3 b^{13} x^5-7168 a^2 b^{14} x^6-2048 a b^{15} x^7-256 b^{16} x^8\right )+63 x^9 \left (256 a^9 b^9+2304 a^8 b^{10} x+9216 a^7 b^{11} x^2+21504 a^6 b^{12} x^3+32256 a^5 b^{13} x^4+32256 a^4 b^{14} x^5+21504 a^3 b^{15} x^6+9216 a^2 b^{16} x^7+2304 a b^{17} x^8+256 b^{18} x^9\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^10,x]

[Out]

(32*b^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-56*a^13*b - 763*a^12*b^2*x - 4808*a^11*b^3*x^2 - 18556*a^10*b^4*x^3 -
48944*a^9*b^5*x^4 - 93184*a^8*b^6*x^5 - 131768*a^7*b^7*x^6 - 140140*a^6*b^8*x^7 - 112112*a^5*b^9*x^8 - 66639*a
^4*b^10*x^9 - 28608*a^3*b^11*x^10 - 8400*a^2*b^12*x^11 - 1512*a*b^13*x^12 - 126*b^14*x^13) + 32*b^8*Sqrt[b^2]*
(56*a^14 + 819*a^13*b*x + 5571*a^12*b^2*x^2 + 23364*a^11*b^3*x^3 + 67500*a^10*b^4*x^4 + 142128*a^9*b^5*x^5 + 2
24952*a^8*b^6*x^6 + 271908*a^7*b^7*x^7 + 252252*a^6*b^8*x^8 + 178751*a^5*b^9*x^9 + 95247*a^4*b^10*x^10 + 37008
*a^3*b^11*x^11 + 9912*a^2*b^12*x^12 + 1638*a*b^13*x^13 + 126*b^14*x^14))/(63*Sqrt[b^2]*x^9*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]*(-256*a^8*b^8 - 2048*a^7*b^9*x - 7168*a^6*b^10*x^2 - 14336*a^5*b^11*x^3 - 17920*a^4*b^12*x^4 - 1433
6*a^3*b^13*x^5 - 7168*a^2*b^14*x^6 - 2048*a*b^15*x^7 - 256*b^16*x^8) + 63*x^9*(256*a^9*b^9 + 2304*a^8*b^10*x +
9216*a^7*b^11*x^2 + 21504*a^6*b^12*x^3 + 32256*a^5*b^13*x^4 + 32256*a^4*b^14*x^5 + 21504*a^3*b^15*x^6 + 9216*
a^2*b^16*x^7 + 2304*a*b^17*x^8 + 256*b^18*x^9))

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fricas [A]  time = 0.39, size = 57, normalized size = 0.25 \begin {gather*} -\frac {126 \, b^{5} x^{5} + 504 \, a b^{4} x^{4} + 840 \, a^{2} b^{3} x^{3} + 720 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b x + 56 \, a^{5}}{504 \, x^{9}} \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^10,x, algorithm="fricas")

[Out]

-1/504*(126*b^5*x^5 + 504*a*b^4*x^4 + 840*a^2*b^3*x^3 + 720*a^3*b^2*x^2 + 315*a^4*b*x + 56*a^5)/x^9

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giac [A]  time = 0.16, size = 108, normalized size = 0.47 \begin {gather*} \frac {b^{9} \mathrm {sgn}\left (b x + a\right )}{504 \, a^{4}} - \frac {126 \, b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 504 \, a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 840 \, a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 720 \, a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 315 \, a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 56 \, a^{5} \mathrm {sgn}\left (b x + a\right )}{504 \, x^{9}} \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^10,x, algorithm="giac")

[Out]

1/504*b^9*sgn(b*x + a)/a^4 - 1/504*(126*b^5*x^5*sgn(b*x + a) + 504*a*b^4*x^4*sgn(b*x + a) + 840*a^2*b^3*x^3*sg
n(b*x + a) + 720*a^3*b^2*x^2*sgn(b*x + a) + 315*a^4*b*x*sgn(b*x + a) + 56*a^5*sgn(b*x + a))/x^9

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maple [A]  time = 0.05, size = 74, normalized size = 0.32 \begin {gather*} -\frac {\left (126 b^{5} x^{5}+504 a \,b^{4} x^{4}+840 a^{2} b^{3} x^{3}+720 a^{3} b^{2} x^{2}+315 a^{4} b x +56 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 \left (b x +a \right )^{5} x^{9}} \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^10,x)

[Out]

-1/504*(126*b^5*x^5+504*a*b^4*x^4+840*a^2*b^3*x^3+720*a^3*b^2*x^2+315*a^4*b*x+56*a^5)*((b*x+a)^2)^(5/2)/x^9/(b
*x+a)^5

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maxima [A]  time = 1.59, size = 283, normalized size = 1.24 \begin {gather*} -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{9}}{6 \, a^{9}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{8}}{6 \, a^{8} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{7}}{6 \, a^{9} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{6}}{6 \, a^{8} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{5}}{6 \, a^{7} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{4}}{6 \, a^{6} x^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{3}}{6 \, a^{5} x^{6}} - \frac {83 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{2}}{504 \, a^{4} x^{7}} + \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b}{72 \, a^{3} x^{8}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{9 \, a^{2} x^{9}} \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^10,x, algorithm="maxima")

[Out]

-1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*b^9/a^9 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*b^8/(a^8*x) + 1/6*(b^2*x^2
+ 2*a*b*x + a^2)^(7/2)*b^7/(a^9*x^2) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b^6/(a^8*x^3) + 1/6*(b^2*x^2 + 2*a*
b*x + a^2)^(7/2)*b^5/(a^7*x^4) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b^4/(a^6*x^5) + 1/6*(b^2*x^2 + 2*a*b*x +
a^2)^(7/2)*b^3/(a^5*x^6) - 83/504*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b^2/(a^4*x^7) + 11/72*(b^2*x^2 + 2*a*b*x + a
^2)^(7/2)*b/(a^3*x^8) - 1/9*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)/(a^2*x^9)

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mupad [B]  time = 0.19, size = 207, normalized size = 0.90 \begin {gather*} -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{9\,x^9\,\left (a+b\,x\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,x^4\,\left (a+b\,x\right )}-\frac {5\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^6\,\left (a+b\,x\right )}-\frac {10\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )}-\frac {a\,b^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^5\,\left (a+b\,x\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{8\,x^8\,\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^10,x)

[Out]

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

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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^{10}}\, 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**10,x)

[Out]

Integral(((a + b*x)**2)**(5/2)/x**10, x)

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