Optimal. Leaf size=151 \[ \frac {3 b^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}{\sqrt {a x^2-b}-\sqrt {b}}\right )}{\sqrt {2}}+\frac {3 b^{7/4} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^2-b}}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b}}{\sqrt {2}}}{\sqrt [4]{a x^2-b}}\right )}{\sqrt {2}}+\frac {2}{7} \left (a x^2-b\right )^{3/4} \left (2 a x^2+5 b\right ) \]
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Rubi [A] time = 0.24, antiderivative size = 230, normalized size of antiderivative = 1.52, number of steps used = 13, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {446, 80, 50, 63, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {3 b^{7/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}+\sqrt {a x^2-b}+\sqrt {b}\right )}{2 \sqrt {2}}+\frac {3 b^{7/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}+\sqrt {a x^2-b}+\sqrt {b}\right )}{2 \sqrt {2}}+\frac {3 b^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2}}-\frac {3 b^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2}}+2 b \left (a x^2-b\right )^{3/4}+\frac {4}{7} \left (a x^2-b\right )^{7/4} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 204
Rule 297
Rule 446
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {\left (-b+a x^2\right )^{3/4} \left (3 b+2 a x^2\right )}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(-b+a x)^{3/4} (3 b+2 a x)}{x} \, dx,x,x^2\right )\\ &=\frac {4}{7} \left (-b+a x^2\right )^{7/4}+\frac {1}{2} (3 b) \operatorname {Subst}\left (\int \frac {(-b+a x)^{3/4}}{x} \, dx,x,x^2\right )\\ &=2 b \left (-b+a x^2\right )^{3/4}+\frac {4}{7} \left (-b+a x^2\right )^{7/4}-\frac {1}{2} \left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{-b+a x}} \, dx,x,x^2\right )\\ &=2 b \left (-b+a x^2\right )^{3/4}+\frac {4}{7} \left (-b+a x^2\right )^{7/4}-\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{a}\\ &=2 b \left (-b+a x^2\right )^{3/4}+\frac {4}{7} \left (-b+a x^2\right )^{7/4}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{a}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{a}\\ &=2 b \left (-b+a x^2\right )^{3/4}+\frac {4}{7} \left (-b+a x^2\right )^{7/4}-\frac {\left (3 b^{7/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{2 \sqrt {2}}-\frac {\left (3 b^{7/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{2 \sqrt {2}}-\frac {1}{2} \left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^2}\right )-\frac {1}{2} \left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^2}\right )\\ &=2 b \left (-b+a x^2\right )^{3/4}+\frac {4}{7} \left (-b+a x^2\right )^{7/4}-\frac {3 b^{7/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}+\sqrt {-b+a x^2}\right )}{2 \sqrt {2}}+\frac {3 b^{7/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}+\sqrt {-b+a x^2}\right )}{2 \sqrt {2}}-\frac {\left (3 b^{7/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )}{\sqrt {2}}+\frac {\left (3 b^{7/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )}{\sqrt {2}}\\ &=2 b \left (-b+a x^2\right )^{3/4}+\frac {4}{7} \left (-b+a x^2\right )^{7/4}+\frac {3 b^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )}{\sqrt {2}}-\frac {3 b^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )}{\sqrt {2}}-\frac {3 b^{7/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}+\sqrt {-b+a x^2}\right )}{2 \sqrt {2}}+\frac {3 b^{7/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}+\sqrt {-b+a x^2}\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 91, normalized size = 0.60 \begin {gather*} \frac {2}{7} \left (a x^2-b\right )^{3/4} \left (2 a x^2+5 b\right )+3 b (-b)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{-b}}\right )+3 (-b)^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{-b}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 150, normalized size = 0.99 \begin {gather*} \frac {2}{7} \left (-b+a x^2\right )^{3/4} \left (5 b+2 a x^2\right )-\frac {3 b^{7/4} \tan ^{-1}\left (\frac {-\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^2}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {2}}+\frac {3 b^{7/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {b}+\sqrt {-b+a x^2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 173, normalized size = 1.15 \begin {gather*} \frac {2}{7} \, {\left (2 \, a x^{2} + 5 \, b\right )} {\left (a x^{2} - b\right )}^{\frac {3}{4}} + 6 \, \left (-b^{7}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (-b^{7}\right )^{\frac {1}{4}} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{5} - \sqrt {\sqrt {a x^{2} - b} b^{10} - \sqrt {-b^{7}} b^{7}} \left (-b^{7}\right )^{\frac {1}{4}}}{b^{7}}\right ) - \frac {3}{2} \, \left (-b^{7}\right )^{\frac {1}{4}} \log \left (27 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{5} + 27 \, \left (-b^{7}\right )^{\frac {3}{4}}\right ) + \frac {3}{2} \, \left (-b^{7}\right )^{\frac {1}{4}} \log \left (27 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{5} - 27 \, \left (-b^{7}\right )^{\frac {3}{4}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 189, normalized size = 1.25 \begin {gather*} -\frac {3}{2} \, \sqrt {2} b^{\frac {7}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right ) - \frac {3}{2} \, \sqrt {2} b^{\frac {7}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right ) + \frac {3}{4} \, \sqrt {2} b^{\frac {7}{4}} \log \left (\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right ) - \frac {3}{4} \, \sqrt {2} b^{\frac {7}{4}} \log \left (-\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right ) + \frac {4}{7} \, {\left (a x^{2} - b\right )}^{\frac {7}{4}} + 2 \, {\left (a x^{2} - b\right )}^{\frac {3}{4}} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{2}-b \right )^{\frac {3}{4}} \left (2 a \,x^{2}+3 b \right )}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 195, normalized size = 1.29 \begin {gather*} -\frac {1}{4} \, {\left (3 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}}\right )} b - 8 \, {\left (a x^{2} - b\right )}^{\frac {3}{4}}\right )} b + \frac {4}{7} \, {\left (a x^{2} - b\right )}^{\frac {7}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 78, normalized size = 0.52 \begin {gather*} \frac {4\,{\left (a\,x^2-b\right )}^{7/4}}{7}-3\,{\left (-b\right )}^{7/4}\,\mathrm {atan}\left (\frac {{\left (a\,x^2-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )+3\,{\left (-b\right )}^{7/4}\,\mathrm {atanh}\left (\frac {{\left (a\,x^2-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )+2\,b\,{\left (a\,x^2-b\right )}^{3/4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.02, size = 80, normalized size = 0.53 \begin {gather*} - \frac {3 a^{\frac {3}{4}} 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 e^{2 i \pi }}{a x^{2}}} \right )}}{2 \Gamma \left (\frac {1}{4}\right )} + 2 a \left (\begin {cases} \frac {x^{2} \left (- b\right )^{\frac {3}{4}}}{2} & \text {for}\: a = 0 \\\frac {2 \left (a x^{2} - b\right )^{\frac {7}{4}}}{7 a} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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