3.2.36 \(\int \frac {(a^2+2 a b x+b^2 x^2)^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=37 \[ -\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 a x^4} \]

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Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {646, 37} \begin {gather*} -\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 a x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^5,x]

[Out]

-((a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*a*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 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^5} \, dx &=\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}{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}}{4 a x^4}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 53, normalized size = 1.43 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^3+4 a^2 b x+6 a b^2 x^2+4 b^3 x^3\right )}{4 x^4 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^5,x]

[Out]

-1/4*(Sqrt[(a + b*x)^2]*(a^3 + 4*a^2*b*x + 6*a*b^2*x^2 + 4*b^3*x^3))/(x^4*(a + b*x))

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IntegrateAlgebraic [B]  time = 1.14, size = 296, normalized size = 8.00 \begin {gather*} \frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} \left (-a^6 b-7 a^5 b^2 x-21 a^4 b^3 x^2-35 a^3 b^4 x^3-34 a^2 b^5 x^4-18 a b^6 x^5-4 b^7 x^6\right )+2 \sqrt {b^2} b^3 \left (a^7+8 a^6 b x+28 a^5 b^2 x^2+56 a^4 b^3 x^3+69 a^3 b^4 x^4+52 a^2 b^5 x^5+22 a b^6 x^6+4 b^7 x^7\right )}{\sqrt {b^2} x^4 \sqrt {a^2+2 a b x+b^2 x^2} \left (-8 a^3 b^3-24 a^2 b^4 x-24 a b^5 x^2-8 b^6 x^3\right )+x^4 \left (8 a^4 b^4+32 a^3 b^5 x+48 a^2 b^6 x^2+32 a b^7 x^3+8 b^8 x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^5,x]

[Out]

(2*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-(a^6*b) - 7*a^5*b^2*x - 21*a^4*b^3*x^2 - 35*a^3*b^4*x^3 - 34*a^2*b^5*x^
4 - 18*a*b^6*x^5 - 4*b^7*x^6) + 2*b^3*Sqrt[b^2]*(a^7 + 8*a^6*b*x + 28*a^5*b^2*x^2 + 56*a^4*b^3*x^3 + 69*a^3*b^
4*x^4 + 52*a^2*b^5*x^5 + 22*a*b^6*x^6 + 4*b^7*x^7))/(Sqrt[b^2]*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-8*a^3*b^3 -
24*a^2*b^4*x - 24*a*b^5*x^2 - 8*b^6*x^3) + x^4*(8*a^4*b^4 + 32*a^3*b^5*x + 48*a^2*b^6*x^2 + 32*a*b^7*x^3 + 8*
b^8*x^4))

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fricas [A]  time = 0.38, size = 33, normalized size = 0.89 \begin {gather*} -\frac {4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \end {gather*}

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^5,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.24, size = 73, normalized size = 1.97 \begin {gather*} -\frac {b^{4} \mathrm {sgn}\left (b x + a\right )}{4 \, a} - \frac {4 \, b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{2} b x \mathrm {sgn}\left (b x + a\right ) + a^{3} \mathrm {sgn}\left (b x + a\right )}{4 \, x^{4}} \end {gather*}

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^5,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.05, size = 50, normalized size = 1.35 \begin {gather*} -\frac {\left (4 b^{3} x^{3}+6 a \,b^{2} x^{2}+4 a^{2} b x +a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{4 \left (b x +a \right )^{3} x^{4}} \end {gather*}

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^5,x)

[Out]

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

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maxima [B]  time = 1.56, size = 138, normalized size = 3.73 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4}}{4 \, a^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{3}}{4 \, a^{3} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{2}}{4 \, a^{4} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b}{4 \, a^{3} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}{4 \, a^{2} x^{4}} \end {gather*}

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^5,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.19, size = 135, normalized size = 3.65 \begin {gather*} -\frac {a^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,x^4\,\left (a+b\,x\right )}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x\,\left (a+b\,x\right )}-\frac {3\,a\,b^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^2\,\left (a+b\,x\right )}-\frac {a^2\,b\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^3\,\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)^(3/2)/x^5,x)

[Out]

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

[Out]

Integral(((a + b*x)**2)**(3/2)/x**5, x)

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