3.159 \(\int \frac{(a^2+2 a b x+b^2 x^2)^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=76 \[ \frac{b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{20 a^2 x^4}-\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{5 a x^5} \]

[Out]

-((a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*a*x^5) + (b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(20*a^2
*x^4)

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Rubi [A]  time = 0.0234828, antiderivative size = 76, normalized size of antiderivative = 1., 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} \[ \frac{b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{20 a^2 x^4}-\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^6,x]

[Out]

-((a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*a*x^5) + (b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(20*a^2
*x^4)

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 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 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]

Rubi steps

\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^6} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^3}{x^6} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{5 a x^5}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^3}{x^5} \, dx}{5 a b \left (a b+b^2 x\right )}\\ &=-\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{5 a x^5}+\frac{b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{20 a^2 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0154006, size = 55, normalized size = 0.72 \[ -\frac{\sqrt{(a+b x)^2} \left (15 a^2 b x+4 a^3+20 a b^2 x^2+10 b^3 x^3\right )}{20 x^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^6,x]

[Out]

-(Sqrt[(a + b*x)^2]*(4*a^3 + 15*a^2*b*x + 20*a*b^2*x^2 + 10*b^3*x^3))/(20*x^5*(a + b*x))

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Maple [A]  time = 0.184, size = 52, normalized size = 0.7 \begin{align*} -{\frac{10\,{b}^{3}{x}^{3}+20\,a{b}^{2}{x}^{2}+15\,b{a}^{2}x+4\,{a}^{3}}{20\,{x}^{5} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{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)^(3/2)/x^6,x)

[Out]

-1/20*(10*b^3*x^3+20*a*b^2*x^2+15*a^2*b*x+4*a^3)*((b*x+a)^2)^(3/2)/x^5/(b*x+a)^3

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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)^(3/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.01163, size = 81, normalized size = 1.07 \begin{align*} -\frac{10 \, b^{3} x^{3} + 20 \, a b^{2} x^{2} + 15 \, a^{2} b x + 4 \, a^{3}}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(3/2)/x^6,x, algorithm="fricas")

[Out]

-1/20*(10*b^3*x^3 + 20*a*b^2*x^2 + 15*a^2*b*x + 4*a^3)/x^5

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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{3}{2}}}{x^{6}}\, 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)**(3/2)/x**6,x)

[Out]

Integral(((a + b*x)**2)**(3/2)/x**6, x)

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Giac [A]  time = 1.22869, size = 100, normalized size = 1.32 \begin{align*} \frac{b^{5} \mathrm{sgn}\left (b x + a\right )}{20 \, a^{2}} - \frac{10 \, b^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + 20 \, a b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{2} b x \mathrm{sgn}\left (b x + a\right ) + 4 \, a^{3} \mathrm{sgn}\left (b x + a\right )}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(3/2)/x^6,x, algorithm="giac")

[Out]

1/20*b^5*sgn(b*x + a)/a^2 - 1/20*(10*b^3*x^3*sgn(b*x + a) + 20*a*b^2*x^2*sgn(b*x + a) + 15*a^2*b*x*sgn(b*x + a
) + 4*a^3*sgn(b*x + a))/x^5