Optimal. Leaf size=230 \[ -\frac {8 b e^3 n \sqrt {d+e x^2}}{105 d^3 x}-\frac {8 b e^2 n \left (d+e x^2\right )^{3/2}}{315 d^3 x^3}-\frac {b n \left (d+e x^2\right )^{5/2}}{49 d^2 x^7}+\frac {38 b e n \left (d+e x^2\right )^{5/2}}{1225 d^3 x^5}+\frac {8 b e^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{105 d^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {277, 270, 2392,
12, 1279, 462, 283, 223, 212} \begin {gather*} -\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 {8 b e^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{105 d^3}-\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 {38 b e n \left (d+e x^2\right )^{5/2}}{1225 d^3 x^5}-\frac {b n \left (d+e x^2\right )^{5/2}}{49 d^2 x^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 212
Rule 223
Rule 270
Rule 277
Rule 283
Rule 462
Rule 1279
Rule 2392
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx &=-\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}-(b n) \int \frac {\left (d+e x^2\right )^{3/2} \left (-15 d^2+12 d e x^2-8 e^2 x^4\right )}{105 d^3 x^8} \, dx\\ &=-\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}-\frac {(b n) \int \frac {\left (d+e x^2\right )^{3/2} \left (-15 d^2+12 d e x^2-8 e^2 x^4\right )}{x^8} \, dx}{105 d^3}\\ &=-\frac {b n \left (d+e x^2\right )^{5/2}}{49 d^2 x^7}-\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}+\frac {(b n) \int \frac {\left (d+e x^2\right )^{3/2} \left (-114 d^2 e+56 d e^2 x^2\right )}{x^6} \, dx}{735 d^4}\\ &=-\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 {\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}+\frac {\left (8 b e^2 n\right ) \int \frac {\left (d+e x^2\right )^{3/2}}{x^4} \, dx}{105 d^3}\\ &=-\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 {\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}+\frac {\left (8 b e^3 n\right ) \int \frac {\sqrt {d+e x^2}}{x^2} \, dx}{105 d^3}\\ &=-\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 {\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}+\frac {\left (8 b e^4 n\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{105 d^3}\\ &=-\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 {\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}+\frac {\left (8 b e^4 n\right ) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{105 d^3}\\ &=-\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 \tanh ^{-1}\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}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.15, size = 180, normalized size = 0.78 \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {e \,x^{2}+d}}{x^{8}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.42, size = 206, normalized size = 0.90 \begin {gather*} \frac {420 \, b n x^{7} e^{\frac {7}{2}} \log \left (-2 \, x^{2} e - 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} - d\right ) - {\left (2 \, {\left (389 \, b n + 420 \, a\right )} x^{6} e^{3} - {\left (179 \, b d n + 420 \, a d\right )} x^{4} e^{2} + 225 \, b d^{3} n + 1575 \, a d^{3} + 9 \, {\left (12 \, b d^{2} n + 35 \, a d^{2}\right )} x^{2} e + 105 \, {\left (8 \, b x^{6} e^{3} - 4 \, b d x^{4} e^{2} + 3 \, b d^{2} x^{2} e + 15 \, b d^{3}\right )} \log \left (c\right ) + 105 \, {\left (8 \, b n x^{6} e^{3} - 4 \, b d n x^{4} e^{2} + 3 \, b d^{2} n x^{2} e + 15 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}}{11025 \, d^{3} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \sqrt {d + e x^{2}}}{x^{8}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^8} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________