3.308 \(\int \frac{a+b \log (c x^n)}{x^4 (d+e x^2)^{5/2}} \, dx\)

Optimal. Leaf size=230 \[ \frac{16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}-\frac{a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}-\frac{b e^2 n x}{3 d^4 \sqrt{d+e x^2}}-\frac{16 b e^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{3 d^4}+\frac{23 b e n \sqrt{d+e x^2}}{9 d^4 x}-\frac{b n \sqrt{d+e x^2}}{9 d^3 x^3} \]

[Out]

-(b*e^2*n*x)/(3*d^4*Sqrt[d + e*x^2]) - (b*n*Sqrt[d + e*x^2])/(9*d^3*x^3) + (23*b*e*n*Sqrt[d + e*x^2])/(9*d^4*x
) - (16*b*e^(3/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(3*d^4) - (a + b*Log[c*x^n])/(3*d*x^3*(d + e*x^2)^(3
/2)) + (2*e*(a + b*Log[c*x^n]))/(d^2*x*(d + e*x^2)^(3/2)) + (8*e^2*x*(a + b*Log[c*x^n]))/(3*d^3*(d + e*x^2)^(3
/2)) + (16*e^2*x*(a + b*Log[c*x^n]))/(3*d^4*Sqrt[d + e*x^2])

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Rubi [A]  time = 0.25889, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {271, 192, 191, 2350, 12, 1805, 1265, 451, 217, 206} \[ \frac{16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}-\frac{a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}-\frac{b e^2 n x}{3 d^4 \sqrt{d+e x^2}}-\frac{16 b e^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{3 d^4}+\frac{23 b e n \sqrt{d+e x^2}}{9 d^4 x}-\frac{b n \sqrt{d+e x^2}}{9 d^3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(x^4*(d + e*x^2)^(5/2)),x]

[Out]

-(b*e^2*n*x)/(3*d^4*Sqrt[d + e*x^2]) - (b*n*Sqrt[d + e*x^2])/(9*d^3*x^3) + (23*b*e*n*Sqrt[d + e*x^2])/(9*d^4*x
) - (16*b*e^(3/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(3*d^4) - (a + b*Log[c*x^n])/(3*d*x^3*(d + e*x^2)^(3
/2)) + (2*e*(a + b*Log[c*x^n]))/(d^2*x*(d + e*x^2)^(3/2)) + (8*e^2*x*(a + b*Log[c*x^n]))/(3*d^3*(d + e*x^2)^(3
/2)) + (16*e^2*x*(a + b*Log[c*x^n]))/(3*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 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -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 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

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^4 \left (d+e x^2\right )^{5/2}} \, dx &=-\frac{a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac{8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt{d+e x^2}}-(b n) \int \frac{-d^3+6 d^2 e x^2+24 d e^2 x^4+16 e^3 x^6}{3 d^4 x^4 \left (d+e x^2\right )^{3/2}} \, dx\\ &=-\frac{a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac{8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt{d+e x^2}}-\frac{(b n) \int \frac{-d^3+6 d^2 e x^2+24 d e^2 x^4+16 e^3 x^6}{x^4 \left (d+e x^2\right )^{3/2}} \, dx}{3 d^4}\\ &=-\frac{b e^2 n x}{3 d^4 \sqrt{d+e x^2}}-\frac{a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac{8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt{d+e x^2}}+\frac{(b n) \int \frac{d^3-7 d^2 e x^2-16 d e^2 x^4}{x^4 \sqrt{d+e x^2}} \, dx}{3 d^5}\\ &=-\frac{b e^2 n x}{3 d^4 \sqrt{d+e x^2}}-\frac{b n \sqrt{d+e x^2}}{9 d^3 x^3}-\frac{a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac{8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt{d+e x^2}}-\frac{(b n) \int \frac{23 d^3 e+48 d^2 e^2 x^2}{x^2 \sqrt{d+e x^2}} \, dx}{9 d^6}\\ &=-\frac{b e^2 n x}{3 d^4 \sqrt{d+e x^2}}-\frac{b n \sqrt{d+e x^2}}{9 d^3 x^3}+\frac{23 b e n \sqrt{d+e x^2}}{9 d^4 x}-\frac{a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac{8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt{d+e x^2}}-\frac{\left (16 b e^2 n\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx}{3 d^4}\\ &=-\frac{b e^2 n x}{3 d^4 \sqrt{d+e x^2}}-\frac{b n \sqrt{d+e x^2}}{9 d^3 x^3}+\frac{23 b e n \sqrt{d+e x^2}}{9 d^4 x}-\frac{a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac{8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt{d+e x^2}}-\frac{\left (16 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^4}\\ &=-\frac{b e^2 n x}{3 d^4 \sqrt{d+e x^2}}-\frac{b n \sqrt{d+e x^2}}{9 d^3 x^3}+\frac{23 b e n \sqrt{d+e x^2}}{9 d^4 x}-\frac{16 b e^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{3 d^4}-\frac{a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac{8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt{d+e x^2}}\\ \end{align*}

Mathematica [A]  time = 0.216622, size = 182, normalized size = 0.79 \[ \frac{18 a d^2 e x^2-3 a d^3+72 a d e^2 x^4+48 a e^3 x^6+3 b \left (6 d^2 e x^2-d^3+24 d e^2 x^4+16 e^3 x^6\right ) \log \left (c x^n\right )+21 b d^2 e n x^2-b d^3 n+42 b d e^2 n x^4-48 b e^{3/2} n x^3 \left (d+e x^2\right )^{3/2} \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )+20 b e^3 n x^6}{9 d^4 x^3 \left (d+e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/(x^4*(d + e*x^2)^(5/2)),x]

[Out]

(-3*a*d^3 - b*d^3*n + 18*a*d^2*e*x^2 + 21*b*d^2*e*n*x^2 + 72*a*d*e^2*x^4 + 42*b*d*e^2*n*x^4 + 48*a*e^3*x^6 + 2
0*b*e^3*n*x^6 + 3*b*(-d^3 + 6*d^2*e*x^2 + 24*d*e^2*x^4 + 16*e^3*x^6)*Log[c*x^n] - 48*b*e^(3/2)*n*x^3*(d + e*x^
2)^(3/2)*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(9*d^4*x^3*(d + e*x^2)^(3/2))

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Maple [F]  time = 0.416, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{4}} \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/x^4/(e*x^2+d)^(5/2),x)

[Out]

int((a+b*ln(c*x^n))/x^4/(e*x^2+d)^(5/2),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))/x^4/(e*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.00739, size = 1156, normalized size = 5.03 \begin{align*} \left [\frac{24 \,{\left (b e^{3} n x^{7} + 2 \, b d e^{2} n x^{5} + b d^{2} e n x^{3}\right )} \sqrt{e} \log \left (-2 \, e x^{2} + 2 \, \sqrt{e x^{2} + d} \sqrt{e} x - d\right ) +{\left (4 \,{\left (5 \, b e^{3} n + 12 \, a e^{3}\right )} x^{6} - b d^{3} n + 6 \,{\left (7 \, b d e^{2} n + 12 \, a d e^{2}\right )} x^{4} - 3 \, a d^{3} + 3 \,{\left (7 \, b d^{2} e n + 6 \, a d^{2} e\right )} x^{2} + 3 \,{\left (16 \, b e^{3} x^{6} + 24 \, b d e^{2} x^{4} + 6 \, b d^{2} e x^{2} - b d^{3}\right )} \log \left (c\right ) + 3 \,{\left (16 \, b e^{3} n x^{6} + 24 \, b d e^{2} n x^{4} + 6 \, b d^{2} e n x^{2} - b d^{3} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{9 \,{\left (d^{4} e^{2} x^{7} + 2 \, d^{5} e x^{5} + d^{6} x^{3}\right )}}, \frac{48 \,{\left (b e^{3} n x^{7} + 2 \, b d e^{2} n x^{5} + b d^{2} e n x^{3}\right )} \sqrt{-e} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) +{\left (4 \,{\left (5 \, b e^{3} n + 12 \, a e^{3}\right )} x^{6} - b d^{3} n + 6 \,{\left (7 \, b d e^{2} n + 12 \, a d e^{2}\right )} x^{4} - 3 \, a d^{3} + 3 \,{\left (7 \, b d^{2} e n + 6 \, a d^{2} e\right )} x^{2} + 3 \,{\left (16 \, b e^{3} x^{6} + 24 \, b d e^{2} x^{4} + 6 \, b d^{2} e x^{2} - b d^{3}\right )} \log \left (c\right ) + 3 \,{\left (16 \, b e^{3} n x^{6} + 24 \, b d e^{2} n x^{4} + 6 \, b d^{2} e n x^{2} - b d^{3} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{9 \,{\left (d^{4} e^{2} x^{7} + 2 \, d^{5} e x^{5} + d^{6} x^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^4/(e*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

[1/9*(24*(b*e^3*n*x^7 + 2*b*d*e^2*n*x^5 + b*d^2*e*n*x^3)*sqrt(e)*log(-2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x -
d) + (4*(5*b*e^3*n + 12*a*e^3)*x^6 - b*d^3*n + 6*(7*b*d*e^2*n + 12*a*d*e^2)*x^4 - 3*a*d^3 + 3*(7*b*d^2*e*n + 6
*a*d^2*e)*x^2 + 3*(16*b*e^3*x^6 + 24*b*d*e^2*x^4 + 6*b*d^2*e*x^2 - b*d^3)*log(c) + 3*(16*b*e^3*n*x^6 + 24*b*d*
e^2*n*x^4 + 6*b*d^2*e*n*x^2 - b*d^3*n)*log(x))*sqrt(e*x^2 + d))/(d^4*e^2*x^7 + 2*d^5*e*x^5 + d^6*x^3), 1/9*(48
*(b*e^3*n*x^7 + 2*b*d*e^2*n*x^5 + b*d^2*e*n*x^3)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + (4*(5*b*e^3*n +
12*a*e^3)*x^6 - b*d^3*n + 6*(7*b*d*e^2*n + 12*a*d*e^2)*x^4 - 3*a*d^3 + 3*(7*b*d^2*e*n + 6*a*d^2*e)*x^2 + 3*(1
6*b*e^3*x^6 + 24*b*d*e^2*x^4 + 6*b*d^2*e*x^2 - b*d^3)*log(c) + 3*(16*b*e^3*n*x^6 + 24*b*d*e^2*n*x^4 + 6*b*d^2*
e*n*x^2 - b*d^3*n)*log(x))*sqrt(e*x^2 + d))/(d^4*e^2*x^7 + 2*d^5*e*x^5 + d^6*x^3)]

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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**4/(e*x**2+d)**(5/2),x)

[Out]

Timed out

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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{5}{2}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^4/(e*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)/((e*x^2 + d)^(5/2)*x^4), x)