### 3.180 $$\int \frac{(a^2+2 a b x+b^2 x^2)^{5/2}}{x^{12}} \, dx$$

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

[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))

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Rubi [A]  time = 0.0560486, antiderivative size = 231, 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} $-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac{a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^{10} (a+b x)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac{5 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}$

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 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^{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 = 0.0163567, size = 77, normalized size = 0.33 $-\frac{\sqrt{(a+b x)^2} \left (3080 a^3 b^2 x^2+3465 a^2 b^3 x^3+1386 a^4 b x+252 a^5+1980 a b^4 x^4+462 b^5 x^5\right )}{2772 x^{11} (a+b x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(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 + 462*b^5*x
^5))/(2772*x^11*(a + b*x))

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Maple [A]  time = 0.237, size = 74, normalized size = 0.3 \begin{align*} -{\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}bx+252\,{a}^{5}}{2772\,{x}^{11} \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^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

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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^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.74668, size = 146, normalized size = 0.63 \begin{align*} -\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{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^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

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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^{12}}\, 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**12,x)

[Out]

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

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Giac [A]  time = 1.26166, size = 146, normalized size = 0.63 \begin{align*} \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{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^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