Optimal. Leaf size=364 \[ \frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {110 \left (a+b x^3\right )^5 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {55 \left (a+b x^3\right )^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1343, 199, 200, 31, 634, 617, 204, 628} \[ \frac {55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {55 \left (a+b x^3\right )^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {110 \left (a+b x^3\right )^5 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 199
Rule 200
Rule 204
Rule 617
Rule 628
Rule 634
Rule 1343
Rubi steps
\begin {align*} \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx &=\frac {\left (2 a b+2 b^2 x^3\right )^5 \int \frac {1}{\left (2 a b+2 b^2 x^3\right )^5} \, dx}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ &=\frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {\left (11 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac {1}{\left (2 a b+2 b^2 x^3\right )^4} \, dx}{24 a b \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ &=\frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {\left (11 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac {1}{\left (2 a b+2 b^2 x^3\right )^3} \, dx}{54 a^2 b^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ &=\frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac {1}{\left (2 a b+2 b^2 x^3\right )^2} \, dx}{648 a^3 b^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ &=\frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac {1}{2 a b+2 b^2 x^3} \, dx}{1944 a^4 b^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ &=\frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac {1}{\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b}+\sqrt [3]{2} b^{2/3} x} \, dx}{5832\ 2^{2/3} a^{14/3} b^{14/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac {2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b}-\sqrt [3]{2} b^{2/3} x}{2^{2/3} a^{2/3} b^{2/3}-2^{2/3} \sqrt [3]{a} b x+2^{2/3} b^{4/3} x^2} \, dx}{5832\ 2^{2/3} a^{14/3} b^{14/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ &=\frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac {-2^{2/3} \sqrt [3]{a} b+2\ 2^{2/3} b^{4/3} x}{2^{2/3} a^{2/3} b^{2/3}-2^{2/3} \sqrt [3]{a} b x+2^{2/3} b^{4/3} x^2} \, dx}{23328 a^{14/3} b^{16/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac {1}{2^{2/3} a^{2/3} b^{2/3}-2^{2/3} \sqrt [3]{a} b x+2^{2/3} b^{4/3} x^2} \, dx}{3888 \sqrt [3]{2} a^{13/3} b^{13/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ &=\frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {55 \left (a+b x^3\right )^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3888 a^{14/3} b^{16/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ &=\frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {110 \left (a+b x^3\right )^5 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {55 \left (a+b x^3\right )^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 211, normalized size = 0.58 \[ \frac {\left (a+b x^3\right ) \left (-\frac {220 \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+660 a^{2/3} x \left (a+b x^3\right )^3+396 a^{5/3} x \left (a+b x^3\right )^2+297 a^{8/3} x \left (a+b x^3\right )+243 a^{11/3} x+\frac {440 \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac {440 \sqrt {3} \left (a+b x^3\right )^4 \tan ^{-1}\left (\frac {2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{b}}\right )}{2916 a^{14/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.73, size = 719, normalized size = 1.98 \[ \left [\frac {660 \, a^{2} b^{4} x^{10} + 2376 \, a^{3} b^{3} x^{7} + 3069 \, a^{4} b^{2} x^{4} + 1596 \, a^{5} b x + 660 \, \sqrt {\frac {1}{3}} {\left (a b^{5} x^{12} + 4 \, a^{2} b^{4} x^{9} + 6 \, a^{3} b^{3} x^{6} + 4 \, a^{4} b^{2} x^{3} + a^{5} b\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) - 220 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 440 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{2916 \, {\left (a^{6} b^{5} x^{12} + 4 \, a^{7} b^{4} x^{9} + 6 \, a^{8} b^{3} x^{6} + 4 \, a^{9} b^{2} x^{3} + a^{10} b\right )}}, \frac {660 \, a^{2} b^{4} x^{10} + 2376 \, a^{3} b^{3} x^{7} + 3069 \, a^{4} b^{2} x^{4} + 1596 \, a^{5} b x + 1320 \, \sqrt {\frac {1}{3}} {\left (a b^{5} x^{12} + 4 \, a^{2} b^{4} x^{9} + 6 \, a^{3} b^{3} x^{6} + 4 \, a^{4} b^{2} x^{3} + a^{5} b\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - 220 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 440 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{2916 \, {\left (a^{6} b^{5} x^{12} + 4 \, a^{7} b^{4} x^{9} + 6 \, a^{8} b^{3} x^{6} + 4 \, a^{9} b^{2} x^{3} + a^{10} b\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 519, normalized size = 1.43 \[ \frac {\left (-440 \sqrt {3}\, b^{4} x^{12} \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+440 b^{4} x^{12} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )-220 b^{4} x^{12} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )-1760 \sqrt {3}\, a \,b^{3} x^{9} \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+1760 a \,b^{3} x^{9} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )-880 a \,b^{3} x^{9} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )+660 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4} x^{10}-2640 \sqrt {3}\, a^{2} b^{2} x^{6} \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+2640 a^{2} b^{2} x^{6} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )-1320 a^{2} b^{2} x^{6} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )+2376 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{3} x^{7}-1760 \sqrt {3}\, a^{3} b \,x^{3} \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+1760 a^{3} b \,x^{3} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )-880 a^{3} b \,x^{3} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )+3069 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b^{2} x^{4}-440 \sqrt {3}\, a^{4} \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+440 a^{4} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )-220 a^{4} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )+1596 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3} b x \right ) \left (b \,x^{3}+a \right )}{2916 \left (\frac {a}{b}\right )^{\frac {2}{3}} \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}} a^{4} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.15, size = 189, normalized size = 0.52 \[ \frac {220 \, b^{3} x^{10} + 792 \, a b^{2} x^{7} + 1023 \, a^{2} b x^{4} + 532 \, a^{3} x}{972 \, {\left (a^{4} b^{4} x^{12} + 4 \, a^{5} b^{3} x^{9} + 6 \, a^{6} b^{2} x^{6} + 4 \, a^{7} b x^{3} + a^{8}\right )}} + \frac {110 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {55 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{729 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {110 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{729 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________