Optimal. Leaf size=74 \[ -\frac {2 b p}{15 a x^3}+\frac {2 b^2 p}{5 a^2 x}+\frac {2 b^{5/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{5 a^{5/2}}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2505, 331, 211}
\begin {gather*} \frac {2 b^{5/2} p \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{5 a^{5/2}}+\frac {2 b^2 p}{5 a^2 x}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5}-\frac {2 b p}{15 a x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 331
Rule 2505
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^6} \, dx &=-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5}+\frac {1}{5} (2 b p) \int \frac {1}{x^4 \left (a+b x^2\right )} \, dx\\ &=-\frac {2 b p}{15 a x^3}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5}-\frac {\left (2 b^2 p\right ) \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{5 a}\\ &=-\frac {2 b p}{15 a x^3}+\frac {2 b^2 p}{5 a^2 x}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5}+\frac {\left (2 b^3 p\right ) \int \frac {1}{a+b x^2} \, dx}{5 a^2}\\ &=-\frac {2 b p}{15 a x^3}+\frac {2 b^2 p}{5 a^2 x}+\frac {2 b^{5/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{5 a^{5/2}}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.00, size = 49, normalized size = 0.66 \begin {gather*} -\frac {2 b p \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\frac {b x^2}{a}\right )}{15 a x^3}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.36, size = 236, normalized size = 3.19
method | result | size |
risch | \(-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{5 x^{5}}-\frac {-6 \left (\munderset {\textit {\_R} =\RootOf \left (a^{5} \textit {\_Z}^{2}+b^{5} p^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{5}+2 b^{5} p^{2}\right ) x -a^{3} b^{2} p \textit {\_R} \right )\right ) a^{2} x^{5}+3 i \pi \,a^{2} \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}-3 i \pi \,a^{2} \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-3 i \pi \,a^{2} \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}+3 i \pi \,a^{2} \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )-12 b^{2} p \,x^{4}+4 a b p \,x^{2}+6 \ln \left (c \right ) a^{2}}{30 a^{2} x^{5}}\) | \(236\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.48, size = 62, normalized size = 0.84 \begin {gather*} \frac {2}{15} \, b p {\left (\frac {3 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {3 \, b x^{2} - a}{a^{2} x^{3}}\right )} - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 170, normalized size = 2.30 \begin {gather*} \left [\frac {3 \, b^{2} p x^{5} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 6 \, b^{2} p x^{4} - 2 \, a b p x^{2} - 3 \, a^{2} p \log \left (b x^{2} + a\right ) - 3 \, a^{2} \log \left (c\right )}{15 \, a^{2} x^{5}}, \frac {6 \, b^{2} p x^{5} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 6 \, b^{2} p x^{4} - 2 \, a b p x^{2} - 3 \, a^{2} p \log \left (b x^{2} + a\right ) - 3 \, a^{2} \log \left (c\right )}{15 \, a^{2} x^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 566 vs.
\(2 (70) = 140\).
time = 169.06, size = 566, normalized size = 7.65 \begin {gather*} \begin {cases} - \frac {\log {\left (0^{p} c \right )}}{5 x^{5}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {\log {\left (a^{p} c \right )}}{5 x^{5}} & \text {for}\: b = 0 \\- \frac {2 p}{25 x^{5}} - \frac {\log {\left (c \left (b x^{2}\right )^{p} \right )}}{5 x^{5}} & \text {for}\: a = 0 \\- \frac {3 a^{3} \sqrt {- \frac {a}{b}} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{15 a^{3} x^{5} \sqrt {- \frac {a}{b}} + 15 a^{2} b x^{7} \sqrt {- \frac {a}{b}}} - \frac {2 a^{2} p x^{2} \sqrt {- \frac {a}{b}}}{\frac {15 a^{3} x^{5} \sqrt {- \frac {a}{b}}}{b} + 15 a^{2} x^{7} \sqrt {- \frac {a}{b}}} - \frac {3 a^{2} x^{2} \sqrt {- \frac {a}{b}} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{\frac {15 a^{3} x^{5} \sqrt {- \frac {a}{b}}}{b} + 15 a^{2} x^{7} \sqrt {- \frac {a}{b}}} + \frac {6 a b p x^{5} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{\frac {15 a^{3} x^{5} \sqrt {- \frac {a}{b}}}{b} + 15 a^{2} x^{7} \sqrt {- \frac {a}{b}}} + \frac {4 a b p x^{4} \sqrt {- \frac {a}{b}}}{\frac {15 a^{3} x^{5} \sqrt {- \frac {a}{b}}}{b} + 15 a^{2} x^{7} \sqrt {- \frac {a}{b}}} - \frac {3 a b x^{5} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{\frac {15 a^{3} x^{5} \sqrt {- \frac {a}{b}}}{b} + 15 a^{2} x^{7} \sqrt {- \frac {a}{b}}} + \frac {6 b^{2} p x^{7} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{\frac {15 a^{3} x^{5} \sqrt {- \frac {a}{b}}}{b} + 15 a^{2} x^{7} \sqrt {- \frac {a}{b}}} + \frac {6 b^{2} p x^{6} \sqrt {- \frac {a}{b}}}{\frac {15 a^{3} x^{5} \sqrt {- \frac {a}{b}}}{b} + 15 a^{2} x^{7} \sqrt {- \frac {a}{b}}} - \frac {3 b^{2} x^{7} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{\frac {15 a^{3} x^{5} \sqrt {- \frac {a}{b}}}{b} + 15 a^{2} x^{7} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.53, size = 71, normalized size = 0.96 \begin {gather*} \frac {2 \, b^{3} p \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{5 \, \sqrt {a b} a^{2}} - \frac {p \log \left (b x^{2} + a\right )}{5 \, x^{5}} + \frac {6 \, b^{2} p x^{4} - 2 \, a b p x^{2} - 3 \, a^{2} \log \left (c\right )}{15 \, a^{2} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.26, size = 61, normalized size = 0.82 \begin {gather*} \frac {2\,b^{5/2}\,p\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{5\,a^{5/2}}-\frac {\frac {2\,b\,p}{3\,a}-\frac {2\,b^2\,p\,x^2}{a^2}}{5\,x^3}-\frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________