3.15 \(\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^4} \, dx\)

Optimal. Leaf size=75 \[ \frac{b \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.0209114, antiderivative size = 75, 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} \[ \frac{b \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^4,x]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*x^3*(a + b*x^3)) + (b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*Log[x])/(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^4} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \frac{a b+b^2 x^3}{x^4} \, 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^4}+\frac{b^2}{x}\right ) \, dx}{a b+b^2 x^3}\\ &=-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3}\\ \end{align*}

Mathematica [A]  time = 0.0112664, size = 39, normalized size = 0.52 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (a-3 b x^3 \log (x)\right )}{3 x^3 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^4,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(a - 3*b*x^3*Log[x]))/(3*x^3*(a + b*x^3))

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Maple [A]  time = 0.008, size = 38, normalized size = 0.5 \begin{align*}{\frac{3\,b\ln \left ( x \right ){x}^{3}-a}{ \left ( 3\,b{x}^{3}+3\,a \right ){x}^{3}}\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^4,x)

[Out]

1/3*((b*x^3+a)^2)^(1/2)*(3*b*ln(x)*x^3-a)/(b*x^3+a)/x^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*x^3+a)^2)^(1/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.72024, size = 41, normalized size = 0.55 \begin{align*} \frac{3 \, b x^{3} \log \left (x\right ) - a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^3+a)^2)^(1/2)/x^4,x, algorithm="fricas")

[Out]

1/3*(3*b*x^3*log(x) - a)/x^3

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Sympy [A]  time = 0.296766, size = 10, normalized size = 0.13 \begin{align*} - \frac{a}{3 x^{3}} + b \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x**3+a)**2)**(1/2)/x**4,x)

[Out]

-a/(3*x**3) + b*log(x)

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Giac [A]  time = 1.10787, size = 58, normalized size = 0.77 \begin{align*} b \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x^{3} + a\right ) - \frac{b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + a \mathrm{sgn}\left (b x^{3} + a\right )}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^3+a)^2)^(1/2)/x^4,x, algorithm="giac")

[Out]

b*log(abs(x))*sgn(b*x^3 + a) - 1/3*(b*x^3*sgn(b*x^3 + a) + a*sgn(b*x^3 + a))/x^3