Optimal. Leaf size=261 \[ \frac {(3 a B+A b) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{7/4}}-\frac {(3 a B+A b) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{7/4}}-\frac {(3 a B+A b) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{7/4}}+\frac {(3 a B+A b) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{5/4} b^{7/4}}+\frac {x^{3/2} (A b-a B)}{2 a b \left (a+b x^2\right )} \]
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Rubi [A] time = 0.18, antiderivative size = 261, 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 = {457, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac {(3 a B+A b) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{7/4}}-\frac {(3 a B+A b) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{7/4}}-\frac {(3 a B+A b) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{7/4}}+\frac {(3 a B+A b) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{5/4} b^{7/4}}+\frac {x^{3/2} (A b-a B)}{2 a b \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 329
Rule 457
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx &=\frac {(A b-a B) x^{3/2}}{2 a b \left (a+b x^2\right )}+\frac {\left (\frac {A b}{2}+\frac {3 a B}{2}\right ) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{2 a b}\\ &=\frac {(A b-a B) x^{3/2}}{2 a b \left (a+b x^2\right )}+\frac {\left (\frac {A b}{2}+\frac {3 a B}{2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a b}\\ &=\frac {(A b-a B) x^{3/2}}{2 a b \left (a+b x^2\right )}-\frac {(A b+3 a B) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a b^{3/2}}+\frac {(A b+3 a B) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a b^{3/2}}\\ &=\frac {(A b-a B) x^{3/2}}{2 a b \left (a+b x^2\right )}+\frac {(A b+3 a B) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a b^2}+\frac {(A b+3 a B) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a b^2}+\frac {(A b+3 a B) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{5/4} b^{7/4}}+\frac {(A b+3 a B) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{5/4} b^{7/4}}\\ &=\frac {(A b-a B) x^{3/2}}{2 a b \left (a+b x^2\right )}+\frac {(A b+3 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{7/4}}-\frac {(A b+3 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{7/4}}+\frac {(A b+3 a B) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{7/4}}-\frac {(A b+3 a B) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{7/4}}\\ &=\frac {(A b-a B) x^{3/2}}{2 a b \left (a+b x^2\right )}-\frac {(A b+3 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{7/4}}+\frac {(A b+3 a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{7/4}}+\frac {(A b+3 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{7/4}}-\frac {(A b+3 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{7/4}}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 95, normalized size = 0.36 \[ \frac {2 x^{3/2} (A b-a B) \, _2F_1\left (\frac {3}{4},2;\frac {7}{4};-\frac {b x^2}{a}\right )}{3 a^2 b}+\frac {B \left (\tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{-a}}\right )+\tanh ^{-1}\left (\frac {a \sqrt [4]{b} \sqrt {x}}{(-a)^{5/4}}\right )\right )}{\sqrt [4]{-a} b^{7/4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 912, normalized size = 3.49 \[ -\frac {4 \, {\left (B a - A b\right )} x^{\frac {3}{2}} + 4 \, {\left (a b^{2} x^{2} + a^{2} b\right )} \left (-\frac {81 \, B^{4} a^{4} + 108 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 12 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{7}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {{\left (729 \, B^{6} a^{6} + 1458 \, A B^{5} a^{5} b + 1215 \, A^{2} B^{4} a^{4} b^{2} + 540 \, A^{3} B^{3} a^{3} b^{3} + 135 \, A^{4} B^{2} a^{2} b^{4} + 18 \, A^{5} B a b^{5} + A^{6} b^{6}\right )} x - {\left (81 \, B^{4} a^{7} b^{3} + 108 \, A B^{3} a^{6} b^{4} + 54 \, A^{2} B^{2} a^{5} b^{5} + 12 \, A^{3} B a^{4} b^{6} + A^{4} a^{3} b^{7}\right )} \sqrt {-\frac {81 \, B^{4} a^{4} + 108 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 12 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{7}}}} a b^{2} \left (-\frac {81 \, B^{4} a^{4} + 108 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 12 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{7}}\right )^{\frac {1}{4}} - {\left (27 \, B^{3} a^{4} b^{2} + 27 \, A B^{2} a^{3} b^{3} + 9 \, A^{2} B a^{2} b^{4} + A^{3} a b^{5}\right )} \sqrt {x} \left (-\frac {81 \, B^{4} a^{4} + 108 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 12 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{7}}\right )^{\frac {1}{4}}}{81 \, B^{4} a^{4} + 108 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 12 \, A^{3} B a b^{3} + A^{4} b^{4}}\right ) - {\left (a b^{2} x^{2} + a^{2} b\right )} \left (-\frac {81 \, B^{4} a^{4} + 108 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 12 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{7}}\right )^{\frac {1}{4}} \log \left (a^{4} b^{5} \left (-\frac {81 \, B^{4} a^{4} + 108 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 12 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{7}}\right )^{\frac {3}{4}} + {\left (27 \, B^{3} a^{3} + 27 \, A B^{2} a^{2} b + 9 \, A^{2} B a b^{2} + A^{3} b^{3}\right )} \sqrt {x}\right ) + {\left (a b^{2} x^{2} + a^{2} b\right )} \left (-\frac {81 \, B^{4} a^{4} + 108 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 12 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{7}}\right )^{\frac {1}{4}} \log \left (-a^{4} b^{5} \left (-\frac {81 \, B^{4} a^{4} + 108 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 12 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{7}}\right )^{\frac {3}{4}} + {\left (27 \, B^{3} a^{3} + 27 \, A B^{2} a^{2} b + 9 \, A^{2} B a b^{2} + A^{3} b^{3}\right )} \sqrt {x}\right )}{8 \, {\left (a b^{2} x^{2} + a^{2} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 273, normalized size = 1.05 \[ -\frac {B a x^{\frac {3}{2}} - A b x^{\frac {3}{2}}}{2 \, {\left (b x^{2} + a\right )} a b} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{4}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{4}} - \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2} b^{4}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 305, normalized size = 1.17 \[ \frac {\sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} a b}+\frac {\sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} a b}+\frac {\sqrt {2}\, A \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} a b}+\frac {3 \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {3 \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {3 \sqrt {2}\, B \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {\left (A b -B a \right ) x^{\frac {3}{2}}}{2 \left (b \,x^{2}+a \right ) a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.28, size = 217, normalized size = 0.83 \[ -\frac {{\left (B a - A b\right )} x^{\frac {3}{2}}}{2 \, {\left (a b^{2} x^{2} + a^{2} b\right )}} + \frac {{\left (3 \, B a + A b\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 91, normalized size = 0.35 \[ \frac {\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (A\,b+3\,B\,a\right )}{4\,{\left (-a\right )}^{5/4}\,b^{7/4}}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (A\,b+3\,B\,a\right )}{4\,{\left (-a\right )}^{5/4}\,b^{7/4}}+\frac {x^{3/2}\,\left (A\,b-B\,a\right )}{2\,a\,b\,\left (b\,x^2+a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 32.13, size = 162, normalized size = 0.62 \[ \frac {2 A x^{\frac {3}{2}}}{4 a^{2} + 4 a b x^{2}} + 2 A \operatorname {RootSum} {\left (65536 t^{4} a^{5} b^{3} + 1, \left (t \mapsto t \log {\left (4096 t^{3} a^{4} b^{2} + \sqrt {x} \right )} \right )\right )} - \frac {2 B a x^{\frac {3}{2}}}{4 a^{2} b + 4 a b^{2} x^{2}} - \frac {2 B a \operatorname {RootSum} {\left (65536 t^{4} a^{5} b^{3} + 1, \left (t \mapsto t \log {\left (4096 t^{3} a^{4} b^{2} + \sqrt {x} \right )} \right )\right )}}{b} + \frac {2 B \operatorname {RootSum} {\left (256 t^{4} a b^{3} + 1, \left (t \mapsto t \log {\left (64 t^{3} a b^{2} + \sqrt {x} \right )} \right )\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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