3.260 \(\int \frac{\sqrt{d+e x^2} (a+b \log (c x^n))}{x^4} \, dx\)

Optimal. Leaf size=112 \[ -\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}+\frac{b e^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{3 d}-\frac{b e n \sqrt{d+e x^2}}{3 d x}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 d x^3} \]

[Out]

-(b*e*n*Sqrt[d + e*x^2])/(3*d*x) - (b*n*(d + e*x^2)^(3/2))/(9*d*x^3) + (b*e^(3/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d
+ e*x^2]])/(3*d) - ((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(3*d*x^3)

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Rubi [A]  time = 0.104005, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2335, 277, 217, 206} \[ -\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}+\frac{b e^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{3 d}-\frac{b e n \sqrt{d+e x^2}}{3 d x}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 d x^3} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/x^4,x]

[Out]

-(b*e*n*Sqrt[d + e*x^2])/(3*d*x) - (b*n*(d + e*x^2)^(3/2))/(9*d*x^3) + (b*e^(3/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d
+ e*x^2]])/(3*d) - ((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(3*d*x^3)

Rule 2335

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp
[((f*x)^(m + 1)*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n]))/(d*f*(m + 1)), x] - Dist[(b*n)/(d*(m + 1)), Int[(f*x)^
m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m + r*(q + 1) + 1, 0] && NeQ[
m, -1]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx &=-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}+\frac{(b n) \int \frac{\left (d+e x^2\right )^{3/2}}{x^4} \, dx}{3 d}\\ &=-\frac{b n \left (d+e x^2\right )^{3/2}}{9 d x^3}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}+\frac{(b e n) \int \frac{\sqrt{d+e x^2}}{x^2} \, dx}{3 d}\\ &=-\frac{b e n \sqrt{d+e x^2}}{3 d x}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 d x^3}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}+\frac{\left (b e^2 n\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx}{3 d}\\ &=-\frac{b e n \sqrt{d+e x^2}}{3 d x}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 d x^3}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}+\frac{\left (b e^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{3 d}\\ &=-\frac{b e n \sqrt{d+e x^2}}{3 d x}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 d x^3}+\frac{b e^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{3 d}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}\\ \end{align*}

Mathematica [A]  time = 0.145032, size = 99, normalized size = 0.88 \[ -\frac{\sqrt{d+e x^2} \left (3 a \left (d+e x^2\right )+b n \left (d+4 e x^2\right )\right )+3 b \left (d+e x^2\right )^{3/2} \log \left (c x^n\right )-3 b e^{3/2} n x^3 \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )}{9 d x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/x^4,x]

[Out]

-(Sqrt[d + e*x^2]*(3*a*(d + e*x^2) + b*n*(d + 4*e*x^2)) + 3*b*(d + e*x^2)^(3/2)*Log[c*x^n] - 3*b*e^(3/2)*n*x^3
*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(9*d*x^3)

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Maple [F]  time = 0.475, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{4}}\sqrt{e{x}^{2}+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))*(e*x^2+d)^(1/2)/x^4,x)

[Out]

int((a+b*ln(c*x^n))*(e*x^2+d)^(1/2)/x^4,x)

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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((a+b*log(c*x^n))*(e*x^2+d)^(1/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.48604, size = 529, normalized size = 4.72 \begin{align*} \left [\frac{3 \, b e^{\frac{3}{2}} n x^{3} \log \left (-2 \, e x^{2} - 2 \, \sqrt{e x^{2} + d} \sqrt{e} x - d\right ) - 2 \,{\left (b d n +{\left (4 \, b e n + 3 \, a e\right )} x^{2} + 3 \, a d + 3 \,{\left (b e x^{2} + b d\right )} \log \left (c\right ) + 3 \,{\left (b e n x^{2} + b d n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{18 \, d x^{3}}, -\frac{3 \, b \sqrt{-e} e n x^{3} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) +{\left (b d n +{\left (4 \, b e n + 3 \, a e\right )} x^{2} + 3 \, a d + 3 \,{\left (b e x^{2} + b d\right )} \log \left (c\right ) + 3 \,{\left (b e n x^{2} + b d n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{9 \, d x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*(e*x^2+d)^(1/2)/x^4,x, algorithm="fricas")

[Out]

[1/18*(3*b*e^(3/2)*n*x^3*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) - 2*(b*d*n + (4*b*e*n + 3*a*e)*x^2 +
3*a*d + 3*(b*e*x^2 + b*d)*log(c) + 3*(b*e*n*x^2 + b*d*n)*log(x))*sqrt(e*x^2 + d))/(d*x^3), -1/9*(3*b*sqrt(-e)*
e*n*x^3*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + (b*d*n + (4*b*e*n + 3*a*e)*x^2 + 3*a*d + 3*(b*e*x^2 + b*d)*log(c)
+ 3*(b*e*n*x^2 + b*d*n)*log(x))*sqrt(e*x^2 + d))/(d*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c x^{n} \right )}\right ) \sqrt{d + e x^{2}}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))*(e*x**2+d)**(1/2)/x**4,x)

[Out]

Integral((a + b*log(c*x**n))*sqrt(d + e*x**2)/x**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x^{2} + d}{\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*(e*x^2+d)^(1/2)/x^4,x, algorithm="giac")

[Out]

integrate(sqrt(e*x^2 + d)*(b*log(c*x^n) + a)/x^4, x)