3.296 \(\int \frac{a+b \log (c x^n)}{x^6 (d+e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=236 \[ -\frac{16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt{d+e x^2}}-\frac{8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt{d+e x^2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt{d+e x^2}}-\frac{a+b \log \left (c x^n\right )}{5 d x^5 \sqrt{d+e x^2}}-\frac{148 b e^2 n \sqrt{d+e x^2}}{75 d^4 x}+\frac{16 b e^{5/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{5 d^4}+\frac{14 b e n \sqrt{d+e x^2}}{75 d^3 x^3}-\frac{b n \sqrt{d+e x^2}}{25 d^2 x^5} \]

[Out]

-(b*n*Sqrt[d + e*x^2])/(25*d^2*x^5) + (14*b*e*n*Sqrt[d + e*x^2])/(75*d^3*x^3) - (148*b*e^2*n*Sqrt[d + e*x^2])/
(75*d^4*x) + (16*b*e^(5/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(5*d^4) - (a + b*Log[c*x^n])/(5*d*x^5*Sqrt[
d + e*x^2]) + (2*e*(a + b*Log[c*x^n]))/(5*d^2*x^3*Sqrt[d + e*x^2]) - (8*e^2*(a + b*Log[c*x^n]))/(5*d^3*x*Sqrt[
d + e*x^2]) - (16*e^3*x*(a + b*Log[c*x^n]))/(5*d^4*Sqrt[d + e*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.270337, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {271, 191, 2350, 12, 1807, 1585, 1265, 451, 217, 206} \[ -\frac{16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt{d+e x^2}}-\frac{8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt{d+e x^2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt{d+e x^2}}-\frac{a+b \log \left (c x^n\right )}{5 d x^5 \sqrt{d+e x^2}}-\frac{148 b e^2 n \sqrt{d+e x^2}}{75 d^4 x}+\frac{16 b e^{5/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{5 d^4}+\frac{14 b e n \sqrt{d+e x^2}}{75 d^3 x^3}-\frac{b n \sqrt{d+e x^2}}{25 d^2 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(x^6*(d + e*x^2)^(3/2)),x]

[Out]

-(b*n*Sqrt[d + e*x^2])/(25*d^2*x^5) + (14*b*e*n*Sqrt[d + e*x^2])/(75*d^3*x^3) - (148*b*e^2*n*Sqrt[d + e*x^2])/
(75*d^4*x) + (16*b*e^(5/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(5*d^4) - (a + b*Log[c*x^n])/(5*d*x^5*Sqrt[
d + e*x^2]) + (2*e*(a + b*Log[c*x^n]))/(5*d^2*x^3*Sqrt[d + e*x^2]) - (8*e^2*(a + b*Log[c*x^n]))/(5*d^3*x*Sqrt[
d + e*x^2]) - (16*e^3*x*(a + b*Log[c*x^n]))/(5*d^4*Sqrt[d + e*x^2])

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 191

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

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 1807

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[(R*(c*x)^(m + 1)*(a + b*x^2)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rule 1585

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

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 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 \left (d+e x^2\right )^{3/2}} \, dx &=-\frac{a+b \log \left (c x^n\right )}{5 d x^5 \sqrt{d+e x^2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt{d+e x^2}}-\frac{8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt{d+e x^2}}-\frac{16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt{d+e x^2}}-(b n) \int \frac{-d^3+2 d^2 e x^2-8 d e^2 x^4-16 e^3 x^6}{5 d^4 x^6 \sqrt{d+e x^2}} \, dx\\ &=-\frac{a+b \log \left (c x^n\right )}{5 d x^5 \sqrt{d+e x^2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt{d+e x^2}}-\frac{8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt{d+e x^2}}-\frac{16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt{d+e x^2}}-\frac{(b n) \int \frac{-d^3+2 d^2 e x^2-8 d e^2 x^4-16 e^3 x^6}{x^6 \sqrt{d+e x^2}} \, dx}{5 d^4}\\ &=-\frac{b n \sqrt{d+e x^2}}{25 d^2 x^5}-\frac{a+b \log \left (c x^n\right )}{5 d x^5 \sqrt{d+e x^2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt{d+e x^2}}-\frac{8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt{d+e x^2}}-\frac{16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt{d+e x^2}}+\frac{(b n) \int \frac{-14 d^3 e x+40 d^2 e^2 x^3+80 d e^3 x^5}{x^5 \sqrt{d+e x^2}} \, dx}{25 d^5}\\ &=-\frac{b n \sqrt{d+e x^2}}{25 d^2 x^5}-\frac{a+b \log \left (c x^n\right )}{5 d x^5 \sqrt{d+e x^2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt{d+e x^2}}-\frac{8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt{d+e x^2}}-\frac{16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt{d+e x^2}}+\frac{(b n) \int \frac{-14 d^3 e+40 d^2 e^2 x^2+80 d e^3 x^4}{x^4 \sqrt{d+e x^2}} \, dx}{25 d^5}\\ &=-\frac{b n \sqrt{d+e x^2}}{25 d^2 x^5}+\frac{14 b e n \sqrt{d+e x^2}}{75 d^3 x^3}-\frac{a+b \log \left (c x^n\right )}{5 d x^5 \sqrt{d+e x^2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt{d+e x^2}}-\frac{8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt{d+e x^2}}-\frac{16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt{d+e x^2}}-\frac{(b n) \int \frac{-148 d^3 e^2-240 d^2 e^3 x^2}{x^2 \sqrt{d+e x^2}} \, dx}{75 d^6}\\ &=-\frac{b n \sqrt{d+e x^2}}{25 d^2 x^5}+\frac{14 b e n \sqrt{d+e x^2}}{75 d^3 x^3}-\frac{148 b e^2 n \sqrt{d+e x^2}}{75 d^4 x}-\frac{a+b \log \left (c x^n\right )}{5 d x^5 \sqrt{d+e x^2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt{d+e x^2}}-\frac{8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt{d+e x^2}}-\frac{16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt{d+e x^2}}+\frac{\left (16 b e^3 n\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx}{5 d^4}\\ &=-\frac{b n \sqrt{d+e x^2}}{25 d^2 x^5}+\frac{14 b e n \sqrt{d+e x^2}}{75 d^3 x^3}-\frac{148 b e^2 n \sqrt{d+e x^2}}{75 d^4 x}-\frac{a+b \log \left (c x^n\right )}{5 d x^5 \sqrt{d+e x^2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt{d+e x^2}}-\frac{8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt{d+e x^2}}-\frac{16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt{d+e x^2}}+\frac{\left (16 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 )}{5 d^4}\\ &=-\frac{b n \sqrt{d+e x^2}}{25 d^2 x^5}+\frac{14 b e n \sqrt{d+e x^2}}{75 d^3 x^3}-\frac{148 b e^2 n \sqrt{d+e x^2}}{75 d^4 x}+\frac{16 b e^{5/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{5 d^4}-\frac{a+b \log \left (c x^n\right )}{5 d x^5 \sqrt{d+e x^2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^2 x^3 \sqrt{d+e x^2}}-\frac{8 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^3 x \sqrt{d+e x^2}}-\frac{16 e^3 x \left (a+b \log \left (c x^n\right )\right )}{5 d^4 \sqrt{d+e x^2}}\\ \end{align*}

Mathematica [A]  time = 0.189937, size = 180, normalized size = 0.76 \[ \frac{30 a d^2 e x^2-15 a d^3-120 a d e^2 x^4-240 a e^3 x^6-15 b \left (-2 d^2 e x^2+d^3+8 d e^2 x^4+16 e^3 x^6\right ) \log \left (c x^n\right )+11 b d^2 e n x^2-3 b d^3 n-134 b d e^2 n x^4+240 b e^{5/2} n x^5 \sqrt{d+e x^2} \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )-148 b e^3 n x^6}{75 d^4 x^5 \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/(x^6*(d + e*x^2)^(3/2)),x]

[Out]

(-15*a*d^3 - 3*b*d^3*n + 30*a*d^2*e*x^2 + 11*b*d^2*e*n*x^2 - 120*a*d*e^2*x^4 - 134*b*d*e^2*n*x^4 - 240*a*e^3*x
^6 - 148*b*e^3*n*x^6 - 15*b*(d^3 - 2*d^2*e*x^2 + 8*d*e^2*x^4 + 16*e^3*x^6)*Log[c*x^n] + 240*b*e^(5/2)*n*x^5*Sq
rt[d + e*x^2]*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(75*d^4*x^5*Sqrt[d + e*x^2])

________________________________________________________________________________________

Maple [F]  time = 0.407, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{6}} \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, 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)^(3/2),x)

[Out]

int((a+b*ln(c*x^n))/x^6/(e*x^2+d)^(3/2),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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.84014, size = 1084, normalized size = 4.59 \begin{align*} \left [\frac{120 \,{\left (b e^{3} n x^{7} + b d e^{2} n x^{5}\right )} \sqrt{e} \log \left (-2 \, e x^{2} - 2 \, \sqrt{e x^{2} + d} \sqrt{e} x - d\right ) -{\left (4 \,{\left (37 \, b e^{3} n + 60 \, a e^{3}\right )} x^{6} + 3 \, b d^{3} n + 2 \,{\left (67 \, b d e^{2} n + 60 \, a d e^{2}\right )} x^{4} + 15 \, a d^{3} -{\left (11 \, b d^{2} e n + 30 \, a d^{2} e\right )} x^{2} + 15 \,{\left (16 \, b e^{3} x^{6} + 8 \, b d e^{2} x^{4} - 2 \, b d^{2} e x^{2} + b d^{3}\right )} \log \left (c\right ) + 15 \,{\left (16 \, b e^{3} n x^{6} + 8 \, b d e^{2} n x^{4} - 2 \, b d^{2} e n x^{2} + b d^{3} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{75 \,{\left (d^{4} e x^{7} + d^{5} x^{5}\right )}}, -\frac{240 \,{\left (b e^{3} n x^{7} + b d e^{2} n x^{5}\right )} \sqrt{-e} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) +{\left (4 \,{\left (37 \, b e^{3} n + 60 \, a e^{3}\right )} x^{6} + 3 \, b d^{3} n + 2 \,{\left (67 \, b d e^{2} n + 60 \, a d e^{2}\right )} x^{4} + 15 \, a d^{3} -{\left (11 \, b d^{2} e n + 30 \, a d^{2} e\right )} x^{2} + 15 \,{\left (16 \, b e^{3} x^{6} + 8 \, b d e^{2} x^{4} - 2 \, b d^{2} e x^{2} + b d^{3}\right )} \log \left (c\right ) + 15 \,{\left (16 \, b e^{3} n x^{6} + 8 \, b d e^{2} n x^{4} - 2 \, b d^{2} e n x^{2} + b d^{3} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{75 \,{\left (d^{4} e x^{7} + d^{5} x^{5}\right )}}\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)^(3/2),x, algorithm="fricas")

[Out]

[1/75*(120*(b*e^3*n*x^7 + b*d*e^2*n*x^5)*sqrt(e)*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) - (4*(37*b*e^
3*n + 60*a*e^3)*x^6 + 3*b*d^3*n + 2*(67*b*d*e^2*n + 60*a*d*e^2)*x^4 + 15*a*d^3 - (11*b*d^2*e*n + 30*a*d^2*e)*x
^2 + 15*(16*b*e^3*x^6 + 8*b*d*e^2*x^4 - 2*b*d^2*e*x^2 + b*d^3)*log(c) + 15*(16*b*e^3*n*x^6 + 8*b*d*e^2*n*x^4 -
2*b*d^2*e*n*x^2 + b*d^3*n)*log(x))*sqrt(e*x^2 + d))/(d^4*e*x^7 + d^5*x^5), -1/75*(240*(b*e^3*n*x^7 + b*d*e^2*
n*x^5)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + (4*(37*b*e^3*n + 60*a*e^3)*x^6 + 3*b*d^3*n + 2*(67*b*d*e^
2*n + 60*a*d*e^2)*x^4 + 15*a*d^3 - (11*b*d^2*e*n + 30*a*d^2*e)*x^2 + 15*(16*b*e^3*x^6 + 8*b*d*e^2*x^4 - 2*b*d^
2*e*x^2 + b*d^3)*log(c) + 15*(16*b*e^3*n*x^6 + 8*b*d*e^2*n*x^4 - 2*b*d^2*e*n*x^2 + b*d^3*n)*log(x))*sqrt(e*x^2
+ d))/(d^4*e*x^7 + d^5*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)**(3/2),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}{{\left (e x^{2} + d\right )}^{\frac{3}{2}} 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)^(3/2),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)/((e*x^2 + d)^(3/2)*x^6), x)