3.2.54 \(\int \frac {(a^2+2 a b x+b^2 x^2)^{5/2}}{x^8} \, dx\)

Optimal. Leaf size=76 \[ \frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{42 a^2 x^6}-\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{7 a x^7} \]

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Rubi [A]  time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, 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 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{42 a^2 x^6}-\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{7 a x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^8,x]

[Out]

-((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*a*x^7) + (b*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(42*a^2
*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^8} \, dx &=\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}{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}}{7 a 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}{7 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}}{7 a x^7}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{42 a^2 x^6}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 77, normalized size = 1.01 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (6 a^5+35 a^4 b x+84 a^3 b^2 x^2+105 a^2 b^3 x^3+70 a b^4 x^4+21 b^5 x^5\right )}{42 x^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^8,x]

[Out]

-1/42*(Sqrt[(a + b*x)^2]*(6*a^5 + 35*a^4*b*x + 84*a^3*b^2*x^2 + 105*a^2*b^3*x^3 + 70*a*b^4*x^4 + 21*b^5*x^5))/
(x^7*(a + b*x))

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IntegrateAlgebraic [B]  time = 1.39, size = 476, normalized size = 6.26 \begin {gather*} \frac {32 b^6 \sqrt {a^2+2 a b x+b^2 x^2} \left (-6 a^{11} b-71 a^{10} b^2 x-384 a^9 b^3 x^2-1254 a^8 b^4 x^3-2750 a^7 b^5 x^4-4257 a^6 b^6 x^5-4752 a^5 b^7 x^6-3829 a^4 b^8 x^7-2184 a^3 b^9 x^8-840 a^2 b^{10} x^9-196 a b^{11} x^{10}-21 b^{12} x^{11}\right )+32 \sqrt {b^2} b^6 \left (6 a^{12}+77 a^{11} b x+455 a^{10} b^2 x^2+1638 a^9 b^3 x^3+4004 a^8 b^4 x^4+7007 a^7 b^5 x^5+9009 a^6 b^6 x^6+8581 a^5 b^7 x^7+6013 a^4 b^8 x^8+3024 a^3 b^9 x^9+1036 a^2 b^{10} x^{10}+217 a b^{11} x^{11}+21 b^{12} x^{12}\right )}{21 \sqrt {b^2} x^7 \sqrt {a^2+2 a b x+b^2 x^2} \left (-64 a^6 b^6-384 a^5 b^7 x-960 a^4 b^8 x^2-1280 a^3 b^9 x^3-960 a^2 b^{10} x^4-384 a b^{11} x^5-64 b^{12} x^6\right )+21 x^7 \left (64 a^7 b^7+448 a^6 b^8 x+1344 a^5 b^9 x^2+2240 a^4 b^{10} x^3+2240 a^3 b^{11} x^4+1344 a^2 b^{12} x^5+448 a b^{13} x^6+64 b^{14} x^7\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^8,x]

[Out]

(32*b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-6*a^11*b - 71*a^10*b^2*x - 384*a^9*b^3*x^2 - 1254*a^8*b^4*x^3 - 2750*a
^7*b^5*x^4 - 4257*a^6*b^6*x^5 - 4752*a^5*b^7*x^6 - 3829*a^4*b^8*x^7 - 2184*a^3*b^9*x^8 - 840*a^2*b^10*x^9 - 19
6*a*b^11*x^10 - 21*b^12*x^11) + 32*b^6*Sqrt[b^2]*(6*a^12 + 77*a^11*b*x + 455*a^10*b^2*x^2 + 1638*a^9*b^3*x^3 +
4004*a^8*b^4*x^4 + 7007*a^7*b^5*x^5 + 9009*a^6*b^6*x^6 + 8581*a^5*b^7*x^7 + 6013*a^4*b^8*x^8 + 3024*a^3*b^9*x
^9 + 1036*a^2*b^10*x^10 + 217*a*b^11*x^11 + 21*b^12*x^12))/(21*Sqrt[b^2]*x^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-6
4*a^6*b^6 - 384*a^5*b^7*x - 960*a^4*b^8*x^2 - 1280*a^3*b^9*x^3 - 960*a^2*b^10*x^4 - 384*a*b^11*x^5 - 64*b^12*x
^6) + 21*x^7*(64*a^7*b^7 + 448*a^6*b^8*x + 1344*a^5*b^9*x^2 + 2240*a^4*b^10*x^3 + 2240*a^3*b^11*x^4 + 1344*a^2
*b^12*x^5 + 448*a*b^13*x^6 + 64*b^14*x^7))

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fricas [A]  time = 0.38, size = 57, normalized size = 0.75 \begin {gather*} -\frac {21 \, b^{5} x^{5} + 70 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} + 84 \, a^{3} b^{2} x^{2} + 35 \, a^{4} b x + 6 \, a^{5}}{42 \, x^{7}} \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^8,x, algorithm="fricas")

[Out]

-1/42*(21*b^5*x^5 + 70*a*b^4*x^4 + 105*a^2*b^3*x^3 + 84*a^3*b^2*x^2 + 35*a^4*b*x + 6*a^5)/x^7

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giac [B]  time = 0.55, size = 108, normalized size = 1.42 \begin {gather*} \frac {b^{7} \mathrm {sgn}\left (b x + a\right )}{42 \, a^{2}} - \frac {21 \, b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 70 \, a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 105 \, a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 84 \, a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{5} \mathrm {sgn}\left (b x + a\right )}{42 \, x^{7}} \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^8,x, algorithm="giac")

[Out]

1/42*b^7*sgn(b*x + a)/a^2 - 1/42*(21*b^5*x^5*sgn(b*x + a) + 70*a*b^4*x^4*sgn(b*x + a) + 105*a^2*b^3*x^3*sgn(b*
x + a) + 84*a^3*b^2*x^2*sgn(b*x + a) + 35*a^4*b*x*sgn(b*x + a) + 6*a^5*sgn(b*x + a))/x^7

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maple [A]  time = 0.04, size = 74, normalized size = 0.97 \begin {gather*} -\frac {\left (21 b^{5} x^{5}+70 a \,b^{4} x^{4}+105 a^{2} b^{3} x^{3}+84 a^{3} b^{2} x^{2}+35 a^{4} b x +6 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{42 \left (b x +a \right )^{5} x^{7}} \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^8,x)

[Out]

-1/42*(21*b^5*x^5+70*a*b^4*x^4+105*a^2*b^3*x^3+84*a^3*b^2*x^2+35*a^4*b*x+6*a^5)*((b*x+a)^2)^(5/2)/x^7/(b*x+a)^
5

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maxima [B]  time = 1.62, size = 225, normalized size = 2.96 \begin {gather*} -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{7}}{6 \, a^{7}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{6}}{6 \, a^{6} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{5}}{6 \, a^{7} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{4}}{6 \, a^{6} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{3}}{6 \, a^{5} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{2}}{6 \, a^{4} x^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b}{6 \, a^{3} x^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{7 \, a^{2} x^{7}} \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^8,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.19, size = 207, normalized size = 2.72 \begin {gather*} -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^2\,\left (a+b\,x\right )}-\frac {5\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^4\,\left (a+b\,x\right )}-\frac {2\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^5\,\left (a+b\,x\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^3\,\left (a+b\,x\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,x^6\,\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^8,x)

[Out]

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

[Out]

Integral(((a + b*x)**2)**(5/2)/x**8, x)

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