Optimal. Leaf size=99 \[ \frac {2 \left (a^2 x^2+b^2\right )^{3/4} \left (2 a^2 x^2-7 a b+2 b^2\right )}{7 a}-3 b^{5/2} \tan ^{-1}\left (\frac {\sqrt [4]{a^2 x^2+b^2}}{\sqrt {b}}\right )+3 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2 x^2+b^2}}{\sqrt {b}}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {446, 80, 50, 63, 298, 203, 206} \begin {gather*} -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}-3 b^{5/2} \tan ^{-1}\left (\frac {\sqrt [4]{a^2 x^2+b^2}}{\sqrt {b}}\right )+3 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2 x^2+b^2}}{\sqrt {b}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 203
Rule 206
Rule 298
Rule 446
Rubi steps
\begin {align*} \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} \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt [4]{b^2+a^2 x^2}\right )-\left (3 b^3\right ) \operatorname {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} \tan ^{-1}\left (\frac {\sqrt [4]{b^2+a^2 x^2}}{\sqrt {b}}\right )+3 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b^2+a^2 x^2}}{\sqrt {b}}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 99, normalized size = 1.00 \begin {gather*} \frac {2 \left (a^2 x^2+b^2\right )^{3/4} \left (2 a^2 x^2-7 a b+2 b^2\right )}{7 a}-3 b^{5/2} \tan ^{-1}\left (\frac {\sqrt [4]{a^2 x^2+b^2}}{\sqrt {b}}\right )+3 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2 x^2+b^2}}{\sqrt {b}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 99, normalized size = 1.00 \begin {gather*} \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} \tan ^{-1}\left (\frac {\sqrt [4]{b^2+a^2 x^2}}{\sqrt {b}}\right )+3 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b^2+a^2 x^2}}{\sqrt {b}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 299, normalized size = 3.02 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 97, normalized size = 0.98 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (2 a \,x^{2}-3 b \right ) \left (a^{2} x^{2}+b^{2}\right )^{\frac {3}{4}}}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 107, normalized size = 1.08 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 81, normalized size = 0.82 \begin {gather*} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.77, size = 88, normalized size = 0.89 \begin {gather*} \frac {3 a^{\frac {3}{2}} b x^{\frac {3}{2}} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b^{2} e^{i \pi }}{a^{2} x^{2}}} \right )}}{2 \Gamma \left (\frac {1}{4}\right )} + 2 a \left (\begin {cases} \frac {x^{2} \left (b^{2}\right )^{\frac {3}{4}}}{2} & \text {for}\: a^{2} = 0 \\\frac {2 \left (a^{2} x^{2} + b^{2}\right )^{\frac {7}{4}}}{7 a^{2}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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