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

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

[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))

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Rubi [A]  time = 0.0529566, antiderivative size = 229, normalized size of antiderivative = 1., 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} \[ -\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^6 (a+b x)}-\frac{a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{x^5 (a+b x)}-\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)} \]

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 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 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])

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*}

Mathematica [A]  time = 0.018933, size = 77, normalized size = 0.34 \[ -\frac{\sqrt{(a+b x)^2} \left (720 a^3 b^2 x^2+840 a^2 b^3 x^3+315 a^4 b x+56 a^5+504 a b^4 x^4+126 b^5 x^5\right )}{504 x^9 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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))/
(504*x^9*(a + b*x))

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Maple [A]  time = 0.176, size = 74, normalized size = 0.3 \begin{align*} -{\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}bx+56\,{a}^{5}}{504\,{x}^{9} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 1.66672, size = 136, normalized size = 0.59 \begin{align*} -\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{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{10}}\, dx \end{align*}

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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Giac [A]  time = 1.32478, size = 146, normalized size = 0.64 \begin{align*} \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{align*}

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