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]

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

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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} \[ \frac {b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{20 a^2 x^4}-\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 a x^5} \]

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 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 )^{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*}

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Mathematica [A]  time = 0.02, size = 55, normalized size = 0.72 \[ -\frac {\sqrt {(a+b x)^2} \left (4 a^3+15 a^2 b x+20 a b^2 x^2+10 b^3 x^3\right )}{20 x^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.55, size = 35, normalized size = 0.46 \[ -\frac {10 \, b^{3} x^{3} + 20 \, a b^{2} x^{2} + 15 \, a^{2} b x + 4 \, a^{3}}{20 \, x^{5}} \]

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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giac [A]  time = 0.24, size = 74, normalized size = 0.97 \[ \frac {b^{5} \mathrm {sgn}\left (b x + a\right )}{20 \, a^{2}} - \frac {10 \, b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{3} \mathrm {sgn}\left (b x + a\right )}{20 \, x^{5}} \]

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

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maple [A]  time = 0.05, size = 52, normalized size = 0.68 \[ -\frac {\left (10 b^{3} x^{3}+20 a \,b^{2} x^{2}+15 a^{2} b x +4 a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{20 \left (b x +a \right )^{3} x^{5}} \]

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 [B]  time = 1.57, size = 167, normalized size = 2.20 \[ -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{5}}{4 \, a^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4}}{4 \, a^{4} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{3}}{4 \, a^{5} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{2}}{4 \, a^{4} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b}{4 \, a^{3} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}{5 \, a^{2} x^{5}} \]

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]

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

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mupad [B]  time = 0.19, size = 135, normalized size = 1.78 \[ -\frac {a^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,x^5\,\left (a+b\,x\right )}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^2\,\left (a+b\,x\right )}-\frac {a\,b^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^3\,\left (a+b\,x\right )}-\frac {3\,a^2\,b\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,x^4\,\left (a+b\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{x^{6}}\, dx \]

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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