Integrand size = 25, antiderivative size = 230 \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=-\frac {b e^2 n x}{3 d^4 \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{9 d^3 x^3}+\frac {23 b e n \sqrt {d+e x^2}}{9 d^4 x}-\frac {16 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^4}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}} \]
-16/3*b*e^(3/2)*n*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/d^4+1/3*(-a-b*ln(c*x^ n))/d/x^3/(e*x^2+d)^(3/2)+2*e*(a+b*ln(c*x^n))/d^2/x/(e*x^2+d)^(3/2)+8/3*e^ 2*x*(a+b*ln(c*x^n))/d^3/(e*x^2+d)^(3/2)-1/3*b*e^2*n*x/d^4/(e*x^2+d)^(1/2)+ 16/3*e^2*x*(a+b*ln(c*x^n))/d^4/(e*x^2+d)^(1/2)-1/9*b*n*(e*x^2+d)^(1/2)/d^3 /x^3+23/9*b*e*n*(e*x^2+d)^(1/2)/d^4/x
Time = 0.15 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.79 \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\frac {-3 a d^3-b d^3 n+18 a d^2 e x^2+21 b d^2 e n x^2+72 a d e^2 x^4+42 b d e^2 n x^4+48 a e^3 x^6+20 b e^3 n x^6+3 b \left (-d^3+6 d^2 e x^2+24 d e^2 x^4+16 e^3 x^6\right ) \log \left (c x^n\right )-48 b e^{3/2} n x^3 \left (d+e x^2\right )^{3/2} \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{9 d^4 x^3 \left (d+e x^2\right )^{3/2}} \]
(-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 + 20*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))
Time = 0.57 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2792, 27, 2336, 25, 1588, 27, 358, 224, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2792 |
| \(\displaystyle -b n \int -\frac {-16 e^3 x^6-24 d e^2 x^4-6 d^2 e x^2+d^3}{3 d^4 x^4 \left (e x^2+d\right )^{3/2}}dx+\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b n \int \frac {-16 e^3 x^6-24 d e^2 x^4-6 d^2 e x^2+d^3}{x^4 \left (e x^2+d\right )^{3/2}}dx}{3 d^4}+\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {b n \left (-\frac {\int -\frac {-16 d e^2 x^4-7 d^2 e x^2+d^3}{x^4 \sqrt {e x^2+d}}dx}{d}-\frac {e^2 x}{\sqrt {d+e x^2}}\right )}{3 d^4}+\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b n \left (\frac {\int \frac {-16 d e^2 x^4-7 d^2 e x^2+d^3}{x^4 \sqrt {e x^2+d}}dx}{d}-\frac {e^2 x}{\sqrt {d+e x^2}}\right )}{3 d^4}+\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1588 |
| \(\displaystyle \frac {b n \left (\frac {-\frac {\int \frac {d^2 e \left (48 e x^2+23 d\right )}{x^2 \sqrt {e x^2+d}}dx}{3 d}-\frac {d^2 \sqrt {d+e x^2}}{3 x^3}}{d}-\frac {e^2 x}{\sqrt {d+e x^2}}\right )}{3 d^4}+\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b n \left (\frac {-\frac {1}{3} d e \int \frac {48 e x^2+23 d}{x^2 \sqrt {e x^2+d}}dx-\frac {d^2 \sqrt {d+e x^2}}{3 x^3}}{d}-\frac {e^2 x}{\sqrt {d+e x^2}}\right )}{3 d^4}+\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 358 |
| \(\displaystyle \frac {b n \left (\frac {-\frac {1}{3} d e \left (48 e \int \frac {1}{\sqrt {e x^2+d}}dx-\frac {23 \sqrt {d+e x^2}}{x}\right )-\frac {d^2 \sqrt {d+e x^2}}{3 x^3}}{d}-\frac {e^2 x}{\sqrt {d+e x^2}}\right )}{3 d^4}+\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b n \left (\frac {-\frac {1}{3} d e \left (48 e \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}-\frac {23 \sqrt {d+e x^2}}{x}\right )-\frac {d^2 \sqrt {d+e x^2}}{3 x^3}}{d}-\frac {e^2 x}{\sqrt {d+e x^2}}\right )}{3 d^4}+\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {b n \left (\frac {-\frac {1}{3} d e \left (48 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {23 \sqrt {d+e x^2}}{x}\right )-\frac {d^2 \sqrt {d+e x^2}}{3 x^3}}{d}-\frac {e^2 x}{\sqrt {d+e x^2}}\right )}{3 d^4}\) |
(b*n*(-((e^2*x)/Sqrt[d + e*x^2]) + (-1/3*(d^2*Sqrt[d + e*x^2])/x^3 - (d*e* ((-23*Sqrt[d + e*x^2])/x + 48*Sqrt[e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] ))/3)/d))/(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])
3.4.8.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x_ Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + S imp[d/e^2 Int[(e*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e , m, p}, x] && NeQ[b*c - a*d, 0] && EqQ[Simplify[m + 2*p + 3], 0] && NeQ[m, -1]
Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c _.)*(x_)^4)^(p_.), x_Symbol] :> With[{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] + Simp[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] && Ne Q[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]
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[PolynomialRema inder[(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] + Simp[1/(2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x] }, Simp[(a + b*Log[c*x^n]) u, x] - Simp[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])
\[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x^{4} \left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
Time = 0.38 (sec) , antiderivative size = 520, normalized size of antiderivative = 2.26 \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\left [\frac {24 \, {\left (b e^{3} n x^{7} + 2 \, b d e^{2} n x^{5} + b d^{2} e n x^{3}\right )} \sqrt {e} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + {\left (4 \, {\left (5 \, b e^{3} n + 12 \, a e^{3}\right )} x^{6} - b d^{3} n + 6 \, {\left (7 \, b d e^{2} n + 12 \, a d e^{2}\right )} x^{4} - 3 \, a d^{3} + 3 \, {\left (7 \, b d^{2} e n + 6 \, a d^{2} e\right )} x^{2} + 3 \, {\left (16 \, b e^{3} x^{6} + 24 \, b d e^{2} x^{4} + 6 \, b d^{2} e x^{2} - b d^{3}\right )} \log \left (c\right ) + 3 \, {\left (16 \, b e^{3} n x^{6} + 24 \, b d e^{2} n x^{4} + 6 \, b d^{2} e n x^{2} - b d^{3} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{9 \, {\left (d^{4} e^{2} x^{7} + 2 \, d^{5} e x^{5} + d^{6} x^{3}\right )}}, \frac {48 \, {\left (b e^{3} n x^{7} + 2 \, b d e^{2} n x^{5} + b d^{2} e n x^{3}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (4 \, {\left (5 \, b e^{3} n + 12 \, a e^{3}\right )} x^{6} - b d^{3} n + 6 \, {\left (7 \, b d e^{2} n + 12 \, a d e^{2}\right )} x^{4} - 3 \, a d^{3} + 3 \, {\left (7 \, b d^{2} e n + 6 \, a d^{2} e\right )} x^{2} + 3 \, {\left (16 \, b e^{3} x^{6} + 24 \, b d e^{2} x^{4} + 6 \, b d^{2} e x^{2} - b d^{3}\right )} \log \left (c\right ) + 3 \, {\left (16 \, b e^{3} n x^{6} + 24 \, b d e^{2} n x^{4} + 6 \, b d^{2} e n x^{2} - b d^{3} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{9 \, {\left (d^{4} e^{2} x^{7} + 2 \, d^{5} e x^{5} + d^{6} x^{3}\right )}}\right ] \]
[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)*lo g(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)*l og(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/sq rt(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*(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)]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^4\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \]