3.285 \(\int \frac{a+b \log (c x^n)}{x^6 \sqrt{d+e x^2}} \, dx\)

Optimal. Leaf size=204 \[ -\frac{8 e^2 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^3 x}+\frac{4 e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^2 x^3}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}-\frac{8 b e^2 n \sqrt{d+e x^2}}{15 d^3 x}+\frac{8 b e^{5/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{15 d^3}+\frac{26 b e n \left (d+e x^2\right )^{3/2}}{225 d^3 x^3}-\frac{b n \left (d+e x^2\right )^{3/2}}{25 d^2 x^5} \]

[Out]

(-8*b*e^2*n*Sqrt[d + e*x^2])/(15*d^3*x) - (b*n*(d + e*x^2)^(3/2))/(25*d^2*x^5) + (26*b*e*n*(d + e*x^2)^(3/2))/
(225*d^3*x^3) + (8*b*e^(5/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(15*d^3) - (Sqrt[d + e*x^2]*(a + b*Log[c*
x^n]))/(5*d*x^5) + (4*e*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(15*d^2*x^3) - (8*e^2*Sqrt[d + e*x^2]*(a + b*Log[c
*x^n]))/(15*d^3*x)

________________________________________________________________________________________

Rubi [A]  time = 0.177343, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {271, 264, 2350, 12, 1265, 451, 277, 217, 206} \[ -\frac{8 e^2 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^3 x}+\frac{4 e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^2 x^3}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}-\frac{8 b e^2 n \sqrt{d+e x^2}}{15 d^3 x}+\frac{8 b e^{5/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{15 d^3}+\frac{26 b e n \left (d+e x^2\right )^{3/2}}{225 d^3 x^3}-\frac{b n \left (d+e x^2\right )^{3/2}}{25 d^2 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(x^6*Sqrt[d + e*x^2]),x]

[Out]

(-8*b*e^2*n*Sqrt[d + e*x^2])/(15*d^3*x) - (b*n*(d + e*x^2)^(3/2))/(25*d^2*x^5) + (26*b*e*n*(d + e*x^2)^(3/2))/
(225*d^3*x^3) + (8*b*e^(5/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(15*d^3) - (Sqrt[d + e*x^2]*(a + b*Log[c*
x^n]))/(5*d*x^5) + (4*e*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(15*d^2*x^3) - (8*e^2*Sqrt[d + e*x^2]*(a + b*Log[c
*x^n]))/(15*d^3*x)

Rule 271

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

Rule 264

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

Rule 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x,
x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1265

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Wit
h[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x,
x]}, Simp[(R*(f*x)^(m + 1)*(d + e*x^2)^(q + 1))/(d*f*(m + 1)), x] + Dist[1/(d*f^2*(m + 1)), Int[(f*x)^(m + 2)
*(d + e*x^2)^q*ExpandToSum[(d*f*(m + 1)*Qx)/x - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q},
x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]

Rule 451

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[d/e^n, Int[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a,
b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && (IntegerQ[n] || GtQ[e, 0]) && (
(GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -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{a+b \log \left (c x^n\right )}{x^6 \sqrt{d+e x^2}} \, dx &=-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}+\frac{4 e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^2 x^3}-\frac{8 e^2 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^3 x}-(b n) \int \frac{\sqrt{d+e x^2} \left (-3 d^2+4 d e x^2-8 e^2 x^4\right )}{15 d^3 x^6} \, dx\\ &=-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}+\frac{4 e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^2 x^3}-\frac{8 e^2 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^3 x}-\frac{(b n) \int \frac{\sqrt{d+e x^2} \left (-3 d^2+4 d e x^2-8 e^2 x^4\right )}{x^6} \, dx}{15 d^3}\\ &=-\frac{b n \left (d+e x^2\right )^{3/2}}{25 d^2 x^5}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}+\frac{4 e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^2 x^3}-\frac{8 e^2 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^3 x}+\frac{(b n) \int \frac{\sqrt{d+e x^2} \left (-26 d^2 e+40 d e^2 x^2\right )}{x^4} \, dx}{75 d^4}\\ &=-\frac{b n \left (d+e x^2\right )^{3/2}}{25 d^2 x^5}+\frac{26 b e n \left (d+e x^2\right )^{3/2}}{225 d^3 x^3}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}+\frac{4 e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^2 x^3}-\frac{8 e^2 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^3 x}+\frac{\left (8 b e^2 n\right ) \int \frac{\sqrt{d+e x^2}}{x^2} \, dx}{15 d^3}\\ &=-\frac{8 b e^2 n \sqrt{d+e x^2}}{15 d^3 x}-\frac{b n \left (d+e x^2\right )^{3/2}}{25 d^2 x^5}+\frac{26 b e n \left (d+e x^2\right )^{3/2}}{225 d^3 x^3}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}+\frac{4 e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^2 x^3}-\frac{8 e^2 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^3 x}+\frac{\left (8 b e^3 n\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx}{15 d^3}\\ &=-\frac{8 b e^2 n \sqrt{d+e x^2}}{15 d^3 x}-\frac{b n \left (d+e x^2\right )^{3/2}}{25 d^2 x^5}+\frac{26 b e n \left (d+e x^2\right )^{3/2}}{225 d^3 x^3}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}+\frac{4 e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^2 x^3}-\frac{8 e^2 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^3 x}+\frac{\left (8 b e^3 n\right ) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{15 d^3}\\ &=-\frac{8 b e^2 n \sqrt{d+e x^2}}{15 d^3 x}-\frac{b n \left (d+e x^2\right )^{3/2}}{25 d^2 x^5}+\frac{26 b e n \left (d+e x^2\right )^{3/2}}{225 d^3 x^3}+\frac{8 b e^{5/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{15 d^3}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}+\frac{4 e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^2 x^3}-\frac{8 e^2 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{15 d^3 x}\\ \end{align*}

Mathematica [A]  time = 0.224176, size = 147, normalized size = 0.72 \[ -\frac{\sqrt{d+e x^2} \left (15 a \left (3 d^2-4 d e x^2+8 e^2 x^4\right )+b n \left (9 d^2-17 d e x^2+94 e^2 x^4\right )\right )+15 b \sqrt{d+e x^2} \left (3 d^2-4 d e x^2+8 e^2 x^4\right ) \log \left (c x^n\right )-120 b e^{5/2} n x^5 \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )}{225 d^3 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/(x^6*Sqrt[d + e*x^2]),x]

[Out]

-(Sqrt[d + e*x^2]*(15*a*(3*d^2 - 4*d*e*x^2 + 8*e^2*x^4) + b*n*(9*d^2 - 17*d*e*x^2 + 94*e^2*x^4)) + 15*b*Sqrt[d
+ e*x^2]*(3*d^2 - 4*d*e*x^2 + 8*e^2*x^4)*Log[c*x^n] - 120*b*e^(5/2)*n*x^5*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])
/(225*d^3*x^5)

________________________________________________________________________________________

Maple [F]  time = 0.444, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{6}}{\frac{1}{\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))/x^6/(e*x^2+d)^(1/2),x)

[Out]

int((a+b*ln(c*x^n))/x^6/(e*x^2+d)^(1/2),x)

________________________________________________________________________________________

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))/x^6/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.62303, size = 794, normalized size = 3.89 \begin{align*} \left [\frac{60 \, b e^{\frac{5}{2}} n x^{5} \log \left (-2 \, e x^{2} - 2 \, \sqrt{e x^{2} + d} \sqrt{e} x - d\right ) -{\left (2 \,{\left (47 \, b e^{2} n + 60 \, a e^{2}\right )} x^{4} + 9 \, b d^{2} n + 45 \, a d^{2} -{\left (17 \, b d e n + 60 \, a d e\right )} x^{2} + 15 \,{\left (8 \, b e^{2} x^{4} - 4 \, b d e x^{2} + 3 \, b d^{2}\right )} \log \left (c\right ) + 15 \,{\left (8 \, b e^{2} n x^{4} - 4 \, b d e n x^{2} + 3 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{225 \, d^{3} x^{5}}, -\frac{120 \, b \sqrt{-e} e^{2} n x^{5} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) +{\left (2 \,{\left (47 \, b e^{2} n + 60 \, a e^{2}\right )} x^{4} + 9 \, b d^{2} n + 45 \, a d^{2} -{\left (17 \, b d e n + 60 \, a d e\right )} x^{2} + 15 \,{\left (8 \, b e^{2} x^{4} - 4 \, b d e x^{2} + 3 \, b d^{2}\right )} \log \left (c\right ) + 15 \,{\left (8 \, b e^{2} n x^{4} - 4 \, b d e n x^{2} + 3 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{225 \, d^{3} x^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^6/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

[1/225*(60*b*e^(5/2)*n*x^5*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) - (2*(47*b*e^2*n + 60*a*e^2)*x^4 +
9*b*d^2*n + 45*a*d^2 - (17*b*d*e*n + 60*a*d*e)*x^2 + 15*(8*b*e^2*x^4 - 4*b*d*e*x^2 + 3*b*d^2)*log(c) + 15*(8*b
*e^2*n*x^4 - 4*b*d*e*n*x^2 + 3*b*d^2*n)*log(x))*sqrt(e*x^2 + d))/(d^3*x^5), -1/225*(120*b*sqrt(-e)*e^2*n*x^5*a
rctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + (2*(47*b*e^2*n + 60*a*e^2)*x^4 + 9*b*d^2*n + 45*a*d^2 - (17*b*d*e*n + 60*a
*d*e)*x^2 + 15*(8*b*e^2*x^4 - 4*b*d*e*x^2 + 3*b*d^2)*log(c) + 15*(8*b*e^2*n*x^4 - 4*b*d*e*n*x^2 + 3*b*d^2*n)*l
og(x))*sqrt(e*x^2 + d))/(d^3*x^5)]

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/x**6/(e*x**2+d)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c x^{n}\right ) + a}{\sqrt{e x^{2} + d} x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^6/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)/(sqrt(e*x^2 + d)*x^6), x)