3.19 \(\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^8} \, dx\)

Optimal. Leaf size=79 \[ -\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )} \]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*x^7*(a + b*x^3)) - (b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*x^4*(a + b*x
^3))

________________________________________________________________________________________

Rubi [A]  time = 0.0223182, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1355, 14} \[ -\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^8,x]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*x^7*(a + b*x^3)) - (b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*x^4*(a + b*x
^3))

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^8} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \frac{a b+b^2 x^3}{x^8} \, dx}{a b+b^2 x^3}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \left (\frac{a b}{x^8}+\frac{b^2}{x^5}\right ) \, dx}{a b+b^2 x^3}\\ &=-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}\\ \end{align*}

Mathematica [A]  time = 0.0081179, size = 39, normalized size = 0.49 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (4 a+7 b x^3\right )}{28 x^7 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^8,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(4*a + 7*b*x^3))/(28*x^7*(a + b*x^3))

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 36, normalized size = 0.5 \begin{align*} -{\frac{7\,b{x}^{3}+4\,a}{28\,{x}^{7} \left ( b{x}^{3}+a \right ) }\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x^3+a)^2)^(1/2)/x^8,x)

[Out]

-1/28*(7*b*x^3+4*a)*((b*x^3+a)^2)^(1/2)/x^7/(b*x^3+a)

________________________________________________________________________________________

Maxima [A]  time = 1.01311, size = 20, normalized size = 0.25 \begin{align*} -\frac{7 \, b x^{3} + 4 \, a}{28 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^3+a)^2)^(1/2)/x^8,x, algorithm="maxima")

[Out]

-1/28*(7*b*x^3 + 4*a)/x^7

________________________________________________________________________________________

Fricas [A]  time = 1.72289, size = 36, normalized size = 0.46 \begin{align*} -\frac{7 \, b x^{3} + 4 \, a}{28 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^3+a)^2)^(1/2)/x^8,x, algorithm="fricas")

[Out]

-1/28*(7*b*x^3 + 4*a)/x^7

________________________________________________________________________________________

Sympy [A]  time = 0.328466, size = 15, normalized size = 0.19 \begin{align*} - \frac{4 a + 7 b x^{3}}{28 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x**3+a)**2)**(1/2)/x**8,x)

[Out]

-(4*a + 7*b*x**3)/(28*x**7)

________________________________________________________________________________________

Giac [A]  time = 1.11539, size = 42, normalized size = 0.53 \begin{align*} -\frac{7 \, b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 4 \, a \mathrm{sgn}\left (b x^{3} + a\right )}{28 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^3+a)^2)^(1/2)/x^8,x, algorithm="giac")

[Out]

-1/28*(7*b*x^3*sgn(b*x^3 + a) + 4*a*sgn(b*x^3 + a))/x^7