Integrand size = 25, antiderivative size = 230 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {8 b e^3 n \sqrt {d+e x^2}}{105 d^3 x}-\frac {8 b e^2 n \left (d+e x^2\right )^{3/2}}{315 d^3 x^3}-\frac {b n \left (d+e x^2\right )^{5/2}}{49 d^2 x^7}+\frac {38 b e n \left (d+e x^2\right )^{5/2}}{1225 d^3 x^5}+\frac {8 b e^{7/2} n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{105 d^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3} \]
-1/49*b*n*(e*x^2+d)^(3/2)/d/x^7+13/1225*b*e*n*(e*x^2+d)^(3/2)/d^2/x^5+62/1 1025*b*e^2*n*(e*x^2+d)^(3/2)/d^3/x^3+8/105*b*e^(7/2)*n*arctanh(x*e^(1/2)/( e*x^2+d)^(1/2))/d^3-1/7*(e*x^2+d)^(3/2)*(a+b*ln(c*x^n))/d/x^7+4/35*e*(e*x^ 2+d)^(3/2)*(a+b*ln(c*x^n))/d^2/x^5-8/105*e^2*(e*x^2+d)^(3/2)*(a+b*ln(c*x^n ))/d^3/x^3-8/105*b*e^3*n*(e*x^2+d)^(1/2)/d^3/x
Time = 0.16 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {\sqrt {d+e x^2} \left (105 a \left (15 d^3+3 d^2 e x^2-4 d e^2 x^4+8 e^3 x^6\right )+b n \left (225 d^3+108 d^2 e x^2-179 d e^2 x^4+778 e^3 x^6\right )\right )+105 b \sqrt {d+e x^2} \left (15 d^3+3 d^2 e x^2-4 d e^2 x^4+8 e^3 x^6\right ) \log \left (c x^n\right )-840 b e^{7/2} n x^7 \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{11025 d^3 x^7} \]
-1/11025*(Sqrt[d + e*x^2]*(105*a*(15*d^3 + 3*d^2*e*x^2 - 4*d*e^2*x^4 + 8*e ^3*x^6) + b*n*(225*d^3 + 108*d^2*e*x^2 - 179*d*e^2*x^4 + 778*e^3*x^6)) + 1 05*b*Sqrt[d + e*x^2]*(15*d^3 + 3*d^2*e*x^2 - 4*d*e^2*x^4 + 8*e^3*x^6)*Log[ c*x^n] - 840*b*e^(7/2)*n*x^7*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(d^3*x^7)
Time = 0.43 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2792, 27, 1588, 27, 358, 247, 247, 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.
| \(\displaystyle \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx\) |
| \(\Big \downarrow \) 2792 |
| \(\displaystyle -b n \int -\frac {\left (e x^2+d\right )^{3/2} \left (8 e^2 x^4-12 d e x^2+15 d^2\right )}{105 d^3 x^8}dx-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b n \int \frac {\left (e x^2+d\right )^{3/2} \left (8 e^2 x^4-12 d e x^2+15 d^2\right )}{x^8}dx}{105 d^3}-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 1588 |
| \(\displaystyle \frac {b n \left (-\frac {\int \frac {2 d e \left (57 d-28 e x^2\right ) \left (e x^2+d\right )^{3/2}}{x^6}dx}{7 d}-\frac {15 d \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{105 d^3}-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b n \left (-\frac {2}{7} e \int \frac {\left (57 d-28 e x^2\right ) \left (e x^2+d\right )^{3/2}}{x^6}dx-\frac {15 d \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{105 d^3}-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 358 |
| \(\displaystyle \frac {b n \left (-\frac {2}{7} e \left (-28 e \int \frac {\left (e x^2+d\right )^{3/2}}{x^4}dx-\frac {57 \left (d+e x^2\right )^{5/2}}{5 x^5}\right )-\frac {15 d \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{105 d^3}-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle \frac {b n \left (-\frac {2}{7} e \left (-28 e \left (e \int \frac {\sqrt {e x^2+d}}{x^2}dx-\frac {\left (d+e x^2\right )^{3/2}}{3 x^3}\right )-\frac {57 \left (d+e x^2\right )^{5/2}}{5 x^5}\right )-\frac {15 d \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{105 d^3}-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle \frac {b n \left (-\frac {2}{7} e \left (-28 e \left (e \left (e \int \frac {1}{\sqrt {e x^2+d}}dx-\frac {\sqrt {d+e x^2}}{x}\right )-\frac {\left (d+e x^2\right )^{3/2}}{3 x^3}\right )-\frac {57 \left (d+e x^2\right )^{5/2}}{5 x^5}\right )-\frac {15 d \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{105 d^3}-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b n \left (-\frac {2}{7} e \left (-28 e \left (e \left (e \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}-\frac {\sqrt {d+e x^2}}{x}\right )-\frac {\left (d+e x^2\right )^{3/2}}{3 x^3}\right )-\frac {57 \left (d+e x^2\right )^{5/2}}{5 x^5}\right )-\frac {15 d \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{105 d^3}-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {b n \left (-\frac {2}{7} e \left (-28 e \left (e \left (\sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {\sqrt {d+e x^2}}{x}\right )-\frac {\left (d+e x^2\right )^{3/2}}{3 x^3}\right )-\frac {57 \left (d+e x^2\right )^{5/2}}{5 x^5}\right )-\frac {15 d \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{105 d^3}\) |
(b*n*((-15*d*(d + e*x^2)^(5/2))/(7*x^7) - (2*e*((-57*(d + e*x^2)^(5/2))/(5 *x^5) - 28*e*(-1/3*(d + e*x^2)^(3/2)/x^3 + e*(-(Sqrt[d + e*x^2]/x) + Sqrt[ e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]))))/7))/(105*d^3) - ((d + e*x^2)^( 3/2)*(a + b*Log[c*x^n]))/(7*d*x^7) + (4*e*(d + e*x^2)^(3/2)*(a + b*Log[c*x ^n]))/(35*d^2*x^5) - (8*e^2*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(105*d^3 *x^3)
3.3.62.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
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[((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])
\[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {e \,x^{2}+d}}{x^{8}}d x\]
Time = 0.38 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.85 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\left [\frac {420 \, b e^{\frac {7}{2}} n x^{7} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - {\left (2 \, {\left (389 \, b e^{3} n + 420 \, a e^{3}\right )} x^{6} + 225 \, b d^{3} n - {\left (179 \, b d e^{2} n + 420 \, a d e^{2}\right )} x^{4} + 1575 \, a d^{3} + 9 \, {\left (12 \, b d^{2} e n + 35 \, a d^{2} e\right )} x^{2} + 105 \, {\left (8 \, b e^{3} x^{6} - 4 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + 15 \, b d^{3}\right )} \log \left (c\right ) + 105 \, {\left (8 \, b e^{3} n x^{6} - 4 \, b d e^{2} n x^{4} + 3 \, b d^{2} e n x^{2} + 15 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{11025 \, d^{3} x^{7}}, -\frac {840 \, b \sqrt {-e} e^{3} n x^{7} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (2 \, {\left (389 \, b e^{3} n + 420 \, a e^{3}\right )} x^{6} + 225 \, b d^{3} n - {\left (179 \, b d e^{2} n + 420 \, a d e^{2}\right )} x^{4} + 1575 \, a d^{3} + 9 \, {\left (12 \, b d^{2} e n + 35 \, a d^{2} e\right )} x^{2} + 105 \, {\left (8 \, b e^{3} x^{6} - 4 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + 15 \, b d^{3}\right )} \log \left (c\right ) + 105 \, {\left (8 \, b e^{3} n x^{6} - 4 \, b d e^{2} n x^{4} + 3 \, b d^{2} e n x^{2} + 15 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{11025 \, d^{3} x^{7}}\right ] \]
[1/11025*(420*b*e^(7/2)*n*x^7*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) - (2*(389*b*e^3*n + 420*a*e^3)*x^6 + 225*b*d^3*n - (179*b*d*e^2*n + 42 0*a*d*e^2)*x^4 + 1575*a*d^3 + 9*(12*b*d^2*e*n + 35*a*d^2*e)*x^2 + 105*(8*b *e^3*x^6 - 4*b*d*e^2*x^4 + 3*b*d^2*e*x^2 + 15*b*d^3)*log(c) + 105*(8*b*e^3 *n*x^6 - 4*b*d*e^2*n*x^4 + 3*b*d^2*e*n*x^2 + 15*b*d^3*n)*log(x))*sqrt(e*x^ 2 + d))/(d^3*x^7), -1/11025*(840*b*sqrt(-e)*e^3*n*x^7*arctan(sqrt(-e)*x/sq rt(e*x^2 + d)) + (2*(389*b*e^3*n + 420*a*e^3)*x^6 + 225*b*d^3*n - (179*b*d *e^2*n + 420*a*d*e^2)*x^4 + 1575*a*d^3 + 9*(12*b*d^2*e*n + 35*a*d^2*e)*x^2 + 105*(8*b*e^3*x^6 - 4*b*d*e^2*x^4 + 3*b*d^2*e*x^2 + 15*b*d^3)*log(c) + 1 05*(8*b*e^3*n*x^6 - 4*b*d*e^2*n*x^4 + 3*b*d^2*e*n*x^2 + 15*b*d^3*n)*log(x) )*sqrt(e*x^2 + d))/(d^3*x^7)]
\[ \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \sqrt {d + e x^{2}}}{x^{8}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\int { \frac {\sqrt {e x^{2} + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{8}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^8} \,d x \]