Integrand size = 29, antiderivative size = 99 \[ \int \frac {\left (-3 b+2 a x^2\right ) \left (b^2+a^2 x^2\right )^{3/4}}{x} \, dx=\frac {2 \left (b^2+a^2 x^2\right )^{3/4} \left (-7 a b+2 b^2+2 a^2 x^2\right )}{7 a}-3 b^{5/2} \arctan \left (\frac {\sqrt [4]{b^2+a^2 x^2}}{\sqrt {b}}\right )+3 b^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{b^2+a^2 x^2}}{\sqrt {b}}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {457, 81, 52, 65, 304, 209, 212} \[ \int \frac {\left (-3 b+2 a x^2\right ) \left (b^2+a^2 x^2\right )^{3/4}}{x} \, dx=-3 b^{5/2} \arctan \left (\frac {\sqrt [4]{a^2 x^2+b^2}}{\sqrt {b}}\right )+3 b^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{a^2 x^2+b^2}}{\sqrt {b}}\right )-2 b \left (a^2 x^2+b^2\right )^{3/4}+\frac {4 \left (a^2 x^2+b^2\right )^{7/4}}{7 a} \]
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Rule 52
Rule 65
Rule 81
Rule 209
Rule 212
Rule 304
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(-3 b+2 a x) \left (b^2+a^2 x\right )^{3/4}}{x} \, dx,x,x^2\right ) \\ & = \frac {4 \left (b^2+a^2 x^2\right )^{7/4}}{7 a}-\frac {1}{2} (3 b) \text {Subst}\left (\int \frac {\left (b^2+a^2 x\right )^{3/4}}{x} \, dx,x,x^2\right ) \\ & = -2 b \left (b^2+a^2 x^2\right )^{3/4}+\frac {4 \left (b^2+a^2 x^2\right )^{7/4}}{7 a}-\frac {1}{2} \left (3 b^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt [4]{b^2+a^2 x}} \, dx,x,x^2\right ) \\ & = -2 b \left (b^2+a^2 x^2\right )^{3/4}+\frac {4 \left (b^2+a^2 x^2\right )^{7/4}}{7 a}-\frac {\left (6 b^3\right ) \text {Subst}\left (\int \frac {x^2}{-\frac {b^2}{a^2}+\frac {x^4}{a^2}} \, dx,x,\sqrt [4]{b^2+a^2 x^2}\right )}{a^2} \\ & = -2 b \left (b^2+a^2 x^2\right )^{3/4}+\frac {4 \left (b^2+a^2 x^2\right )^{7/4}}{7 a}+\left (3 b^3\right ) \text {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt [4]{b^2+a^2 x^2}\right )-\left (3 b^3\right ) \text {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt [4]{b^2+a^2 x^2}\right ) \\ & = -2 b \left (b^2+a^2 x^2\right )^{3/4}+\frac {4 \left (b^2+a^2 x^2\right )^{7/4}}{7 a}-3 b^{5/2} \arctan \left (\frac {\sqrt [4]{b^2+a^2 x^2}}{\sqrt {b}}\right )+3 b^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{b^2+a^2 x^2}}{\sqrt {b}}\right ) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-3 b+2 a x^2\right ) \left (b^2+a^2 x^2\right )^{3/4}}{x} \, dx=\frac {2 \left (b^2+a^2 x^2\right )^{3/4} \left (-7 a b+2 b^2+2 a^2 x^2\right )}{7 a}-3 b^{5/2} \arctan \left (\frac {\sqrt [4]{b^2+a^2 x^2}}{\sqrt {b}}\right )+3 b^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{b^2+a^2 x^2}}{\sqrt {b}}\right ) \]
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Time = 2.39 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.86
method | result | size |
pseudoelliptic | \(\frac {21 \left (\operatorname {arctanh}\left (\frac {\left (a^{2} x^{2}+b^{2}\right )^{\frac {1}{4}}}{\sqrt {b}}\right )-\arctan \left (\frac {\left (a^{2} x^{2}+b^{2}\right )^{\frac {1}{4}}}{\sqrt {b}}\right )\right ) a \,b^{\frac {5}{2}}+2 \left (a^{2} x^{2}+b^{2}\right )^{\frac {3}{4}} \left (2 a^{2} x^{2}-7 a b +2 b^{2}\right )}{7 a}\) | \(85\) |
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Time = 0.27 (sec) , antiderivative size = 299, normalized size of antiderivative = 3.02 \[ \int \frac {\left (-3 b+2 a x^2\right ) \left (b^2+a^2 x^2\right )^{3/4}}{x} \, dx=\left [-\frac {42 \, a b^{\frac {5}{2}} \arctan \left (\frac {{\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}}}{\sqrt {b}}\right ) - 21 \, a b^{\frac {5}{2}} \log \left (\frac {a^{2} x^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}} b^{\frac {3}{2}} + 2 \, \sqrt {a^{2} x^{2} + b^{2}} b + 2 \, {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {3}{4}} \sqrt {b}}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} x^{2} - 7 \, a b + 2 \, b^{2}\right )} {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {3}{4}}}{14 \, a}, -\frac {42 \, a \sqrt {-b} b^{2} \arctan \left (\frac {{\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}} \sqrt {-b}}{b}\right ) - 21 \, a \sqrt {-b} b^{2} \log \left (\frac {a^{2} x^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}} \sqrt {-b} b - 2 \, \sqrt {a^{2} x^{2} + b^{2}} b - 2 \, {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {3}{4}} \sqrt {-b}}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} x^{2} - 7 \, a b + 2 \, b^{2}\right )} {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {3}{4}}}{14 \, a}\right ] \]
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Time = 8.42 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.39 \[ \int \frac {\left (-3 b+2 a x^2\right ) \left (b^2+a^2 x^2\right )^{3/4}}{x} \, dx=- a \left (\begin {cases} - x^{2} \left (b^{2}\right )^{\frac {3}{4}} & \text {for}\: a^{2} = 0 \\- \frac {4 \left (a^{2} x^{2} + b^{2}\right )^{\frac {7}{4}}}{7 a^{2}} & \text {otherwise} \end {cases}\right ) - \frac {3 b \left (\begin {cases} 2 b^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt [4]{a^{2} x^{2} + b^{2}}}{\sqrt {b}} \right )} + \frac {2 b^{2} \operatorname {atan}{\left (\frac {\sqrt [4]{a^{2} x^{2} + b^{2}}}{\sqrt {- b}} \right )}}{\sqrt {- b}} + \frac {4 \left (a^{2} x^{2} + b^{2}\right )^{\frac {3}{4}}}{3} & \text {for}\: a^{2} \neq 0 \\- \left (b^{2}\right )^{\frac {3}{4}} \log {\left (\frac {1}{x^{2}} \right )} & \text {otherwise} \end {cases}\right )}{2} \]
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Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.08 \[ \int \frac {\left (-3 b+2 a x^2\right ) \left (b^2+a^2 x^2\right )^{3/4}}{x} \, dx=-\frac {1}{2} \, {\left (6 \, b^{\frac {3}{2}} \arctan \left (\frac {{\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}}}{\sqrt {b}}\right ) + 3 \, b^{\frac {3}{2}} \log \left (-\frac {\sqrt {b} - {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}}}{\sqrt {b} + {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}}}\right ) + 4 \, {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {3}{4}}\right )} b + \frac {4 \, {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {7}{4}}}{7 \, a} \]
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Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.98 \[ \int \frac {\left (-3 b+2 a x^2\right ) \left (b^2+a^2 x^2\right )^{3/4}}{x} \, dx=-\frac {3 \, b^{3} \arctan \left (\frac {{\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}}}{\sqrt {-b}}\right )}{\sqrt {-b}} - 3 \, b^{\frac {5}{2}} \arctan \left (\frac {{\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}}}{\sqrt {b}}\right ) - \frac {2 \, {\left (7 \, {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {3}{4}} a^{7} b - 2 \, {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {7}{4}} a^{6}\right )}}{7 \, a^{7}} \]
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Time = 6.39 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.82 \[ \int \frac {\left (-3 b+2 a x^2\right ) \left (b^2+a^2 x^2\right )^{3/4}}{x} \, dx=3\,b^{5/2}\,\mathrm {atanh}\left (\frac {{\left (a^2\,x^2+b^2\right )}^{1/4}}{\sqrt {b}}\right )-3\,b^{5/2}\,\mathrm {atan}\left (\frac {{\left (a^2\,x^2+b^2\right )}^{1/4}}{\sqrt {b}}\right )-2\,b\,{\left (a^2\,x^2+b^2\right )}^{3/4}+\frac {4\,{\left (a^2\,x^2+b^2\right )}^{7/4}}{7\,a} \]
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