Optimal. Leaf size=298 \[ \frac {(3 a B+5 A b) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{9/4} b^{7/4}}-\frac {(3 a B+5 A b) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{9/4} b^{7/4}}-\frac {(3 a B+5 A b) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} b^{7/4}}+\frac {(3 a B+5 A b) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{9/4} b^{7/4}}+\frac {x^{3/2} (3 a B+5 A b)}{16 a^2 b \left (a+b x^2\right )}+\frac {x^{3/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.22, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {457, 290, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {(3 a B+5 A b) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{9/4} b^{7/4}}-\frac {(3 a B+5 A b) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{9/4} b^{7/4}}-\frac {(3 a B+5 A b) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} b^{7/4}}+\frac {(3 a B+5 A b) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{9/4} b^{7/4}}+\frac {x^{3/2} (3 a B+5 A b)}{16 a^2 b \left (a+b x^2\right )}+\frac {x^{3/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 290
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 )^3} \, dx &=\frac {(A b-a B) x^{3/2}}{4 a b \left (a+b x^2\right )^2}+\frac {\left (\frac {5 A b}{2}+\frac {3 a B}{2}\right ) \int \frac {\sqrt {x}}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac {(A b-a B) x^{3/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(5 A b+3 a B) x^{3/2}}{16 a^2 b \left (a+b x^2\right )}+\frac {(5 A b+3 a B) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{32 a^2 b}\\ &=\frac {(A b-a B) x^{3/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(5 A b+3 a B) x^{3/2}}{16 a^2 b \left (a+b x^2\right )}+\frac {(5 A b+3 a B) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{16 a^2 b}\\ &=\frac {(A b-a B) x^{3/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(5 A b+3 a B) x^{3/2}}{16 a^2 b \left (a+b x^2\right )}-\frac {(5 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 )}{32 a^2 b^{3/2}}+\frac {(5 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 )}{32 a^2 b^{3/2}}\\ &=\frac {(A b-a B) x^{3/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(5 A b+3 a B) x^{3/2}}{16 a^2 b \left (a+b x^2\right )}+\frac {(5 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 )}{64 a^2 b^2}+\frac {(5 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 )}{64 a^2 b^2}+\frac {(5 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 )}{64 \sqrt {2} a^{9/4} b^{7/4}}+\frac {(5 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 )}{64 \sqrt {2} a^{9/4} b^{7/4}}\\ &=\frac {(A b-a B) x^{3/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(5 A b+3 a B) x^{3/2}}{16 a^2 b \left (a+b x^2\right )}+\frac {(5 A b+3 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{9/4} b^{7/4}}-\frac {(5 A b+3 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{9/4} b^{7/4}}+\frac {(5 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 )}{32 \sqrt {2} a^{9/4} b^{7/4}}-\frac {(5 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 )}{32 \sqrt {2} a^{9/4} b^{7/4}}\\ &=\frac {(A b-a B) x^{3/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(5 A b+3 a B) x^{3/2}}{16 a^2 b \left (a+b x^2\right )}-\frac {(5 A b+3 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} b^{7/4}}+\frac {(5 A b+3 a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} b^{7/4}}+\frac {(5 A b+3 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{9/4} b^{7/4}}-\frac {(5 A b+3 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{9/4} b^{7/4}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 62, normalized size = 0.21 \begin {gather*} \frac {2 x^{3/2} \left ((A b-a B) \, _2F_1\left (\frac {3}{4},3;\frac {7}{4};-\frac {b x^2}{a}\right )+a B \, _2F_1\left (\frac {3}{4},2;\frac {7}{4};-\frac {b x^2}{a}\right )\right )}{3 a^3 b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.72, size = 182, normalized size = 0.61 \begin {gather*} -\frac {(3 a B+5 A b) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{32 \sqrt {2} a^{9/4} b^{7/4}}-\frac {(3 a B+5 A b) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{32 \sqrt {2} a^{9/4} b^{7/4}}-\frac {x^{3/2} \left (a^2 B-9 a A b-3 a b B x^2-5 A b^2 x^2\right )}{16 a^2 b \left (a+b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 1005, normalized size = 3.37 \begin {gather*} -\frac {4 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \left (-\frac {81 \, B^{4} a^{4} + 540 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 1500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{7}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {{\left (729 \, B^{6} a^{6} + 7290 \, A B^{5} a^{5} b + 30375 \, A^{2} B^{4} a^{4} b^{2} + 67500 \, A^{3} B^{3} a^{3} b^{3} + 84375 \, A^{4} B^{2} a^{2} b^{4} + 56250 \, A^{5} B a b^{5} + 15625 \, A^{6} b^{6}\right )} x - {\left (81 \, B^{4} a^{9} b^{3} + 540 \, A B^{3} a^{8} b^{4} + 1350 \, A^{2} B^{2} a^{7} b^{5} + 1500 \, A^{3} B a^{6} b^{6} + 625 \, A^{4} a^{5} b^{7}\right )} \sqrt {-\frac {81 \, B^{4} a^{4} + 540 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 1500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{7}}}} a^{2} b^{2} \left (-\frac {81 \, B^{4} a^{4} + 540 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 1500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{7}}\right )^{\frac {1}{4}} - {\left (27 \, B^{3} a^{5} b^{2} + 135 \, A B^{2} a^{4} b^{3} + 225 \, A^{2} B a^{3} b^{4} + 125 \, A^{3} a^{2} b^{5}\right )} \sqrt {x} \left (-\frac {81 \, B^{4} a^{4} + 540 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 1500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{7}}\right )^{\frac {1}{4}}}{81 \, B^{4} a^{4} + 540 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 1500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}\right ) - {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \left (-\frac {81 \, B^{4} a^{4} + 540 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 1500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{7}}\right )^{\frac {1}{4}} \log \left (a^{7} b^{5} \left (-\frac {81 \, B^{4} a^{4} + 540 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 1500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{7}}\right )^{\frac {3}{4}} + {\left (27 \, B^{3} a^{3} + 135 \, A B^{2} a^{2} b + 225 \, A^{2} B a b^{2} + 125 \, A^{3} b^{3}\right )} \sqrt {x}\right ) + {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \left (-\frac {81 \, B^{4} a^{4} + 540 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 1500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{7}}\right )^{\frac {1}{4}} \log \left (-a^{7} b^{5} \left (-\frac {81 \, B^{4} a^{4} + 540 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 1500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{7}}\right )^{\frac {3}{4}} + {\left (27 \, B^{3} a^{3} + 135 \, A B^{2} a^{2} b + 225 \, A^{2} B a b^{2} + 125 \, A^{3} b^{3}\right )} \sqrt {x}\right ) - 4 \, {\left ({\left (3 \, B a b + 5 \, A b^{2}\right )} x^{3} - {\left (B a^{2} - 9 \, A a b\right )} x\right )} \sqrt {x}}{64 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.52, size = 298, normalized size = 1.00 \begin {gather*} \frac {3 \, B a b x^{\frac {7}{2}} + 5 \, A b^{2} x^{\frac {7}{2}} - B a^{2} x^{\frac {3}{2}} + 9 \, A a b x^{\frac {3}{2}}}{16 \, {\left (b x^{2} + a\right )}^{2} a^{2} b} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + 5 \, \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 )}{64 \, a^{3} b^{4}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + 5 \, \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 )}{64 \, a^{3} b^{4}} - \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + 5 \, \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 )}{128 \, a^{3} b^{4}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + 5 \, \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 )}{128 \, a^{3} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 335, normalized size = 1.12 \begin {gather*} \frac {5 \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{64 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b}+\frac {5 \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{64 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b}+\frac {5 \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 )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b}+\frac {3 \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{64 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \,b^{2}}+\frac {3 \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{64 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \,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 )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \,b^{2}}+\frac {\frac {\left (5 A b +3 B a \right ) x^{\frac {7}{2}}}{16 a^{2}}+\frac {\left (9 A b -B a \right ) x^{\frac {3}{2}}}{16 a b}}{\left (b \,x^{2}+a \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.43, size = 253, normalized size = 0.85 \begin {gather*} \frac {{\left (3 \, B a b + 5 \, A b^{2}\right )} x^{\frac {7}{2}} - {\left (B a^{2} - 9 \, A a b\right )} x^{\frac {3}{2}}}{16 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} + \frac {{\left (3 \, B a + 5 \, 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 )}}{128 \, a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 124, normalized size = 0.42 \begin {gather*} \frac {\frac {x^{7/2}\,\left (5\,A\,b+3\,B\,a\right )}{16\,a^2}+\frac {x^{3/2}\,\left (9\,A\,b-B\,a\right )}{16\,a\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4}+\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (5\,A\,b+3\,B\,a\right )}{32\,{\left (-a\right )}^{9/4}\,b^{7/4}}-\frac {\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (5\,A\,b+3\,B\,a\right )}{32\,{\left (-a\right )}^{9/4}\,b^{7/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 152.84, size = 299, normalized size = 1.00 \begin {gather*} \frac {18 A a x^{\frac {3}{2}}}{32 a^{4} + 64 a^{3} b x^{2} + 32 a^{2} b^{2} x^{4}} + \frac {10 A b x^{\frac {7}{2}}}{32 a^{4} + 64 a^{3} b x^{2} + 32 a^{2} b^{2} x^{4}} + 2 A \operatorname {RootSum} {\left (268435456 t^{4} a^{9} b^{3} + 625, \left (t \mapsto t \log {\left (\frac {2097152 t^{3} a^{7} b^{2}}{125} + \sqrt {x} \right )} \right )\right )} - \frac {18 B a^{2} x^{\frac {3}{2}}}{32 a^{4} b + 64 a^{3} b^{2} x^{2} + 32 a^{2} b^{3} x^{4}} - \frac {10 B a x^{\frac {7}{2}}}{32 a^{4} + 64 a^{3} b x^{2} + 32 a^{2} b^{2} x^{4}} - \frac {2 B a \operatorname {RootSum} {\left (268435456 t^{4} a^{9} b^{3} + 625, \left (t \mapsto t \log {\left (\frac {2097152 t^{3} a^{7} b^{2}}{125} + \sqrt {x} \right )} \right )\right )}}{b} + \frac {2 B x^{\frac {3}{2}}}{4 a^{2} b + 4 a b^{2} x^{2}} + \frac {2 B \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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