3.2.55 \(\int \frac {(a^2+2 a b x+b^2 x^2)^{5/2}}{x^9} \, dx\)

Optimal. Leaf size=116 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 a x^8}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{28 a^2 x^7}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{168 a^3 x^6} \]

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Rubi [A]  time = 0.03, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {646, 45, 37} \begin {gather*} -\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{168 a^3 x^6}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{28 a^2 x^7}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 a x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^9,x]

[Out]

-((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*a*x^8) + (b*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(28*a^2
*x^7) - (b^2*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(168*a^3*x^6)

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 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^9} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{x^9} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{8 a x^8}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{x^8} \, dx}{4 a b^3 \left (a b+b^2 x\right )}\\ &=-\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{8 a x^8}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{28 a^2 x^7}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{x^7} \, dx}{28 a^2 b^2 \left (a b+b^2 x\right )}\\ &=-\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{8 a x^8}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{28 a^2 x^7}-\frac {b^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{168 a^3 x^6}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 77, normalized size = 0.66 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (21 a^5+120 a^4 b x+280 a^3 b^2 x^2+336 a^2 b^3 x^3+210 a b^4 x^4+56 b^5 x^5\right )}{168 x^8 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^9,x]

[Out]

-1/168*(Sqrt[(a + b*x)^2]*(21*a^5 + 120*a^4*b*x + 280*a^3*b^2*x^2 + 336*a^2*b^3*x^3 + 210*a*b^4*x^4 + 56*b^5*x
^5))/(x^8*(a + b*x))

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IntegrateAlgebraic [B]  time = 1.41, size = 520, normalized size = 4.48 \begin {gather*} \frac {16 b^7 \sqrt {a^2+2 a b x+b^2 x^2} \left (-21 a^{12} b-267 a^{11} b^2 x-1561 a^{10} b^3 x^2-5551 a^9 b^4 x^3-13377 a^8 b^5 x^4-23023 a^7 b^6 x^5-29029 a^6 b^7 x^6-27027 a^5 b^8 x^7-18446 a^4 b^9 x^8-9002 a^3 b^{10} x^9-2982 a^2 b^{11} x^{10}-602 a b^{12} x^{11}-56 b^{13} x^{12}\right )+16 \sqrt {b^2} b^7 \left (21 a^{13}+288 a^{12} b x+1828 a^{11} b^2 x^2+7112 a^{10} b^3 x^3+18928 a^9 b^4 x^4+36400 a^8 b^5 x^5+52052 a^7 b^6 x^6+56056 a^6 b^7 x^7+45473 a^5 b^8 x^8+27448 a^4 b^9 x^9+11984 a^3 b^{10} x^{10}+3584 a^2 b^{11} x^{11}+658 a b^{12} x^{12}+56 b^{13} x^{13}\right )}{21 \sqrt {b^2} x^8 \sqrt {a^2+2 a b x+b^2 x^2} \left (-128 a^7 b^7-896 a^6 b^8 x-2688 a^5 b^9 x^2-4480 a^4 b^{10} x^3-4480 a^3 b^{11} x^4-2688 a^2 b^{12} x^5-896 a b^{13} x^6-128 b^{14} x^7\right )+21 x^8 \left (128 a^8 b^8+1024 a^7 b^9 x+3584 a^6 b^{10} x^2+7168 a^5 b^{11} x^3+8960 a^4 b^{12} x^4+7168 a^3 b^{13} x^5+3584 a^2 b^{14} x^6+1024 a b^{15} x^7+128 b^{16} x^8\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^9,x]

[Out]

(16*b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-21*a^12*b - 267*a^11*b^2*x - 1561*a^10*b^3*x^2 - 5551*a^9*b^4*x^3 - 13
377*a^8*b^5*x^4 - 23023*a^7*b^6*x^5 - 29029*a^6*b^7*x^6 - 27027*a^5*b^8*x^7 - 18446*a^4*b^9*x^8 - 9002*a^3*b^1
0*x^9 - 2982*a^2*b^11*x^10 - 602*a*b^12*x^11 - 56*b^13*x^12) + 16*b^7*Sqrt[b^2]*(21*a^13 + 288*a^12*b*x + 1828
*a^11*b^2*x^2 + 7112*a^10*b^3*x^3 + 18928*a^9*b^4*x^4 + 36400*a^8*b^5*x^5 + 52052*a^7*b^6*x^6 + 56056*a^6*b^7*
x^7 + 45473*a^5*b^8*x^8 + 27448*a^4*b^9*x^9 + 11984*a^3*b^10*x^10 + 3584*a^2*b^11*x^11 + 658*a*b^12*x^12 + 56*
b^13*x^13))/(21*Sqrt[b^2]*x^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-128*a^7*b^7 - 896*a^6*b^8*x - 2688*a^5*b^9*x^2 -
4480*a^4*b^10*x^3 - 4480*a^3*b^11*x^4 - 2688*a^2*b^12*x^5 - 896*a*b^13*x^6 - 128*b^14*x^7) + 21*x^8*(128*a^8*
b^8 + 1024*a^7*b^9*x + 3584*a^6*b^10*x^2 + 7168*a^5*b^11*x^3 + 8960*a^4*b^12*x^4 + 7168*a^3*b^13*x^5 + 3584*a^
2*b^14*x^6 + 1024*a*b^15*x^7 + 128*b^16*x^8))

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fricas [A]  time = 0.40, size = 57, normalized size = 0.49 \begin {gather*} -\frac {56 \, b^{5} x^{5} + 210 \, a b^{4} x^{4} + 336 \, a^{2} b^{3} x^{3} + 280 \, a^{3} b^{2} x^{2} + 120 \, a^{4} b x + 21 \, a^{5}}{168 \, x^{8}} \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^9,x, algorithm="fricas")

[Out]

-1/168*(56*b^5*x^5 + 210*a*b^4*x^4 + 336*a^2*b^3*x^3 + 280*a^3*b^2*x^2 + 120*a^4*b*x + 21*a^5)/x^8

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giac [A]  time = 0.16, size = 108, normalized size = 0.93 \begin {gather*} -\frac {b^{8} \mathrm {sgn}\left (b x + a\right )}{168 \, a^{3}} - \frac {56 \, b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 210 \, a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 336 \, a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 280 \, a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 120 \, a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{5} \mathrm {sgn}\left (b x + a\right )}{168 \, x^{8}} \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^9,x, algorithm="giac")

[Out]

-1/168*b^8*sgn(b*x + a)/a^3 - 1/168*(56*b^5*x^5*sgn(b*x + a) + 210*a*b^4*x^4*sgn(b*x + a) + 336*a^2*b^3*x^3*sg
n(b*x + a) + 280*a^3*b^2*x^2*sgn(b*x + a) + 120*a^4*b*x*sgn(b*x + a) + 21*a^5*sgn(b*x + a))/x^8

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maple [A]  time = 0.05, size = 74, normalized size = 0.64 \begin {gather*} -\frac {\left (56 b^{5} x^{5}+210 a \,b^{4} x^{4}+336 a^{2} b^{3} x^{3}+280 a^{3} b^{2} x^{2}+120 a^{4} b x +21 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{168 \left (b x +a \right )^{5} x^{8}} \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^9,x)

[Out]

-1/168*(56*b^5*x^5+210*a*b^4*x^4+336*a^2*b^3*x^3+280*a^3*b^2*x^2+120*a^4*b*x+21*a^5)*((b*x+a)^2)^(5/2)/x^8/(b*
x+a)^5

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maxima [B]  time = 1.64, size = 254, normalized size = 2.19 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{8}}{6 \, a^{8}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{7}}{6 \, a^{7} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{6}}{6 \, a^{8} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{5}}{6 \, a^{7} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{4}}{6 \, a^{6} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{3}}{6 \, a^{5} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{2}}{6 \, a^{4} x^{6}} + \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b}{56 \, a^{3} x^{7}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{8 \, a^{2} x^{8}} \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^9,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.19, size = 207, normalized size = 1.78 \begin {gather*} -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{8\,x^8\,\left (a+b\,x\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^3\,\left (a+b\,x\right )}-\frac {2\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^5\,\left (a+b\,x\right )}-\frac {5\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^6\,\left (a+b\,x\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,x^4\,\left (a+b\,x\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\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^9,x)

[Out]

- (a^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(8*x^8*(a + b*x)) - (b^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(3*x^3*(a +
b*x)) - (2*a^2*b^3*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^5*(a + b*x)) - (5*a^3*b^2*(a^2 + b^2*x^2 + 2*a*b*x)^(1/
2))/(3*x^6*(a + b*x)) - (5*a*b^4*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(4*x^4*(a + b*x)) - (5*a^4*b*(a^2 + b^2*x^2
+ 2*a*b*x)^(1/2))/(7*x^7*(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^{9}}\, 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**9,x)

[Out]

Integral(((a + b*x)**2)**(5/2)/x**9, x)

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