Optimal. Leaf size=310 \[ \frac {\sqrt [4]{a} (5 A b-9 a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} b^{13/4}}-\frac {\sqrt [4]{a} (5 A b-9 a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} b^{13/4}}+\frac {\sqrt [4]{a} (5 A b-9 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{13/4}}-\frac {\sqrt [4]{a} (5 A b-9 a B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} b^{13/4}}+\frac {\sqrt {x} (5 A b-9 a B)}{2 b^3}-\frac {x^{5/2} (5 A b-9 a B)}{10 a b^2}+\frac {x^{9/2} (A b-a B)}{2 a b \left (a+b x^2\right )} \]
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Rubi [A] time = 0.24, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {457, 321, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {x^{5/2} (5 A b-9 a B)}{10 a b^2}+\frac {\sqrt {x} (5 A b-9 a B)}{2 b^3}+\frac {\sqrt [4]{a} (5 A b-9 a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} b^{13/4}}-\frac {\sqrt [4]{a} (5 A b-9 a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} b^{13/4}}+\frac {\sqrt [4]{a} (5 A b-9 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{13/4}}-\frac {\sqrt [4]{a} (5 A b-9 a B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} b^{13/4}}+\frac {x^{9/2} (A b-a B)}{2 a b \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 321
Rule 329
Rule 457
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx &=\frac {(A b-a B) x^{9/2}}{2 a b \left (a+b x^2\right )}+\frac {\left (-\frac {5 A b}{2}+\frac {9 a B}{2}\right ) \int \frac {x^{7/2}}{a+b x^2} \, dx}{2 a b}\\ &=-\frac {(5 A b-9 a B) x^{5/2}}{10 a b^2}+\frac {(A b-a B) x^{9/2}}{2 a b \left (a+b x^2\right )}+\frac {(5 A b-9 a B) \int \frac {x^{3/2}}{a+b x^2} \, dx}{4 b^2}\\ &=\frac {(5 A b-9 a B) \sqrt {x}}{2 b^3}-\frac {(5 A b-9 a B) x^{5/2}}{10 a b^2}+\frac {(A b-a B) x^{9/2}}{2 a b \left (a+b x^2\right )}-\frac {(a (5 A b-9 a B)) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{4 b^3}\\ &=\frac {(5 A b-9 a B) \sqrt {x}}{2 b^3}-\frac {(5 A b-9 a B) x^{5/2}}{10 a b^2}+\frac {(A b-a B) x^{9/2}}{2 a b \left (a+b x^2\right )}-\frac {(a (5 A b-9 a B)) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 b^3}\\ &=\frac {(5 A b-9 a B) \sqrt {x}}{2 b^3}-\frac {(5 A b-9 a B) x^{5/2}}{10 a b^2}+\frac {(A b-a B) x^{9/2}}{2 a b \left (a+b x^2\right )}-\frac {\left (\sqrt {a} (5 A b-9 a B)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 b^3}-\frac {\left (\sqrt {a} (5 A b-9 a B)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 b^3}\\ &=\frac {(5 A b-9 a B) \sqrt {x}}{2 b^3}-\frac {(5 A b-9 a B) x^{5/2}}{10 a b^2}+\frac {(A b-a B) x^{9/2}}{2 a b \left (a+b x^2\right )}-\frac {\left (\sqrt {a} (5 A b-9 a B)\right ) \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 b^{7/2}}-\frac {\left (\sqrt {a} (5 A b-9 a B)\right ) \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 b^{7/2}}+\frac {\left (\sqrt [4]{a} (5 A b-9 a B)\right ) \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} b^{13/4}}+\frac {\left (\sqrt [4]{a} (5 A b-9 a B)\right ) \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} b^{13/4}}\\ &=\frac {(5 A b-9 a B) \sqrt {x}}{2 b^3}-\frac {(5 A b-9 a B) x^{5/2}}{10 a b^2}+\frac {(A b-a B) x^{9/2}}{2 a b \left (a+b x^2\right )}+\frac {\sqrt [4]{a} (5 A b-9 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{13/4}}-\frac {\sqrt [4]{a} (5 A b-9 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{13/4}}-\frac {\left (\sqrt [4]{a} (5 A b-9 a B)\right ) \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} b^{13/4}}+\frac {\left (\sqrt [4]{a} (5 A b-9 a B)\right ) \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} b^{13/4}}\\ &=\frac {(5 A b-9 a B) \sqrt {x}}{2 b^3}-\frac {(5 A b-9 a B) x^{5/2}}{10 a b^2}+\frac {(A b-a B) x^{9/2}}{2 a b \left (a+b x^2\right )}+\frac {\sqrt [4]{a} (5 A b-9 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{13/4}}-\frac {\sqrt [4]{a} (5 A b-9 a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{13/4}}+\frac {\sqrt [4]{a} (5 A b-9 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{13/4}}-\frac {\sqrt [4]{a} (5 A b-9 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{13/4}}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 385, normalized size = 1.24 \begin {gather*} \frac {-45 \sqrt {2} a^{5/4} B \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+45 \sqrt {2} a^{5/4} B \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-\frac {40 a^2 \sqrt [4]{b} B \sqrt {x}}{a+b x^2}+\frac {40 a A b^{5/4} \sqrt {x}}{a+b x^2}-10 \sqrt {2} \sqrt [4]{a} (9 a B-5 A b) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )+10 \sqrt {2} \sqrt [4]{a} (9 a B-5 A b) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )+25 \sqrt {2} \sqrt [4]{a} A b \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-25 \sqrt {2} \sqrt [4]{a} A b \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-320 a \sqrt [4]{b} B \sqrt {x}+160 A b^{5/4} \sqrt {x}+32 b^{5/4} B x^{5/2}}{80 b^{13/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.62, size = 208, normalized size = 0.67 \begin {gather*} -\frac {\left (9 a^{5/4} B-5 \sqrt [4]{a} A b\right ) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{4 \sqrt {2} b^{13/4}}+\frac {\left (9 a^{5/4} B-5 \sqrt [4]{a} A b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{4 \sqrt {2} b^{13/4}}+\frac {-45 a^2 B \sqrt {x}+25 a A b \sqrt {x}-36 a b B x^{5/2}+20 A b^2 x^{5/2}+4 b^2 B x^{9/2}}{10 b^3 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.61, size = 748, normalized size = 2.41 \begin {gather*} -\frac {20 \, {\left (b^{4} x^{2} + a b^{3}\right )} \left (-\frac {6561 \, B^{4} a^{5} - 14580 \, A B^{3} a^{4} b + 12150 \, A^{2} B^{2} a^{3} b^{2} - 4500 \, A^{3} B a^{2} b^{3} + 625 \, A^{4} a b^{4}}{b^{13}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {b^{6} \sqrt {-\frac {6561 \, B^{4} a^{5} - 14580 \, A B^{3} a^{4} b + 12150 \, A^{2} B^{2} a^{3} b^{2} - 4500 \, A^{3} B a^{2} b^{3} + 625 \, A^{4} a b^{4}}{b^{13}}} + {\left (81 \, B^{2} a^{2} - 90 \, A B a b + 25 \, A^{2} b^{2}\right )} x} b^{10} \left (-\frac {6561 \, B^{4} a^{5} - 14580 \, A B^{3} a^{4} b + 12150 \, A^{2} B^{2} a^{3} b^{2} - 4500 \, A^{3} B a^{2} b^{3} + 625 \, A^{4} a b^{4}}{b^{13}}\right )^{\frac {3}{4}} + {\left (9 \, B a b^{10} - 5 \, A b^{11}\right )} \sqrt {x} \left (-\frac {6561 \, B^{4} a^{5} - 14580 \, A B^{3} a^{4} b + 12150 \, A^{2} B^{2} a^{3} b^{2} - 4500 \, A^{3} B a^{2} b^{3} + 625 \, A^{4} a b^{4}}{b^{13}}\right )^{\frac {3}{4}}}{6561 \, B^{4} a^{5} - 14580 \, A B^{3} a^{4} b + 12150 \, A^{2} B^{2} a^{3} b^{2} - 4500 \, A^{3} B a^{2} b^{3} + 625 \, A^{4} a b^{4}}\right ) + 5 \, {\left (b^{4} x^{2} + a b^{3}\right )} \left (-\frac {6561 \, B^{4} a^{5} - 14580 \, A B^{3} a^{4} b + 12150 \, A^{2} B^{2} a^{3} b^{2} - 4500 \, A^{3} B a^{2} b^{3} + 625 \, A^{4} a b^{4}}{b^{13}}\right )^{\frac {1}{4}} \log \left (b^{3} \left (-\frac {6561 \, B^{4} a^{5} - 14580 \, A B^{3} a^{4} b + 12150 \, A^{2} B^{2} a^{3} b^{2} - 4500 \, A^{3} B a^{2} b^{3} + 625 \, A^{4} a b^{4}}{b^{13}}\right )^{\frac {1}{4}} - {\left (9 \, B a - 5 \, A b\right )} \sqrt {x}\right ) - 5 \, {\left (b^{4} x^{2} + a b^{3}\right )} \left (-\frac {6561 \, B^{4} a^{5} - 14580 \, A B^{3} a^{4} b + 12150 \, A^{2} B^{2} a^{3} b^{2} - 4500 \, A^{3} B a^{2} b^{3} + 625 \, A^{4} a b^{4}}{b^{13}}\right )^{\frac {1}{4}} \log \left (-b^{3} \left (-\frac {6561 \, B^{4} a^{5} - 14580 \, A B^{3} a^{4} b + 12150 \, A^{2} B^{2} a^{3} b^{2} - 4500 \, A^{3} B a^{2} b^{3} + 625 \, A^{4} a b^{4}}{b^{13}}\right )^{\frac {1}{4}} - {\left (9 \, B a - 5 \, A b\right )} \sqrt {x}\right ) - 4 \, {\left (4 \, B b^{2} x^{4} - 45 \, B a^{2} + 25 \, A a b - 4 \, {\left (9 \, B a b - 5 \, A b^{2}\right )} x^{2}\right )} \sqrt {x}}{40 \, {\left (b^{4} x^{2} + a b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 298, normalized size = 0.96 \begin {gather*} \frac {\sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac {1}{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 \, b^{4}} + \frac {\sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac {1}{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 \, b^{4}} + \frac {\sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac {1}{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 \, b^{4}} - \frac {\sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac {1}{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 \, b^{4}} - \frac {B a^{2} \sqrt {x} - A a b \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} b^{3}} + \frac {2 \, {\left (B b^{8} x^{\frac {5}{2}} - 10 \, B a b^{7} \sqrt {x} + 5 \, A b^{8} \sqrt {x}\right )}}{5 \, b^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 339, normalized size = 1.09 \begin {gather*} \frac {2 B \,x^{\frac {5}{2}}}{5 b^{2}}+\frac {A a \sqrt {x}}{2 \left (b \,x^{2}+a \right ) b^{2}}-\frac {B \,a^{2} \sqrt {x}}{2 \left (b \,x^{2}+a \right ) b^{3}}-\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 b^{2}}-\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 b^{2}}-\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{4}} \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 b^{2}}+\frac {9 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 b^{3}}+\frac {9 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 b^{3}}+\frac {9 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B 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 b^{3}}+\frac {2 A \sqrt {x}}{b^{2}}-\frac {4 B a \sqrt {x}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.46, size = 271, normalized size = 0.87 \begin {gather*} -\frac {{\left (B a^{2} - A a b\right )} \sqrt {x}}{2 \, {\left (b^{4} x^{2} + a b^{3}\right )}} + \frac {{\left (\frac {2 \, \sqrt {2} {\left (9 \, B a - 5 \, A b\right )} \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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (9 \, B a - 5 \, A b\right )} \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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (9 \, B a - 5 \, A b\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (9 \, B a - 5 \, A b\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )} a}{16 \, b^{3}} + \frac {2 \, {\left (B b x^{\frac {5}{2}} - 5 \, {\left (2 \, B a - A b\right )} \sqrt {x}\right )}}{5 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 823, normalized size = 2.65 \begin {gather*} \sqrt {x}\,\left (\frac {2\,A}{b^2}-\frac {4\,B\,a}{b^3}\right )+\frac {2\,B\,x^{5/2}}{5\,b^2}-\frac {\sqrt {x}\,\left (\frac {B\,a^2}{2}-\frac {A\,a\,b}{2}\right )}{b^4\,x^2+a\,b^3}+\frac {{\left (-a\right )}^{1/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-a\right )}^{1/4}\,\left (\frac {\sqrt {x}\,\left (25\,A^2\,a^2\,b^2-90\,A\,B\,a^3\,b+81\,B^2\,a^4\right )}{b^3}-\frac {{\left (-a\right )}^{1/4}\,\left (5\,A\,b-9\,B\,a\right )\,\left (72\,B\,a^3-40\,A\,a^2\,b\right )}{8\,b^{13/4}}\right )\,\left (5\,A\,b-9\,B\,a\right )\,1{}\mathrm {i}}{8\,b^{13/4}}+\frac {{\left (-a\right )}^{1/4}\,\left (\frac {\sqrt {x}\,\left (25\,A^2\,a^2\,b^2-90\,A\,B\,a^3\,b+81\,B^2\,a^4\right )}{b^3}+\frac {{\left (-a\right )}^{1/4}\,\left (5\,A\,b-9\,B\,a\right )\,\left (72\,B\,a^3-40\,A\,a^2\,b\right )}{8\,b^{13/4}}\right )\,\left (5\,A\,b-9\,B\,a\right )\,1{}\mathrm {i}}{8\,b^{13/4}}}{\frac {{\left (-a\right )}^{1/4}\,\left (\frac {\sqrt {x}\,\left (25\,A^2\,a^2\,b^2-90\,A\,B\,a^3\,b+81\,B^2\,a^4\right )}{b^3}-\frac {{\left (-a\right )}^{1/4}\,\left (5\,A\,b-9\,B\,a\right )\,\left (72\,B\,a^3-40\,A\,a^2\,b\right )}{8\,b^{13/4}}\right )\,\left (5\,A\,b-9\,B\,a\right )}{8\,b^{13/4}}-\frac {{\left (-a\right )}^{1/4}\,\left (\frac {\sqrt {x}\,\left (25\,A^2\,a^2\,b^2-90\,A\,B\,a^3\,b+81\,B^2\,a^4\right )}{b^3}+\frac {{\left (-a\right )}^{1/4}\,\left (5\,A\,b-9\,B\,a\right )\,\left (72\,B\,a^3-40\,A\,a^2\,b\right )}{8\,b^{13/4}}\right )\,\left (5\,A\,b-9\,B\,a\right )}{8\,b^{13/4}}}\right )\,\left (5\,A\,b-9\,B\,a\right )\,1{}\mathrm {i}}{4\,b^{13/4}}+\frac {{\left (-a\right )}^{1/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-a\right )}^{1/4}\,\left (\frac {\sqrt {x}\,\left (25\,A^2\,a^2\,b^2-90\,A\,B\,a^3\,b+81\,B^2\,a^4\right )}{b^3}-\frac {{\left (-a\right )}^{1/4}\,\left (5\,A\,b-9\,B\,a\right )\,\left (72\,B\,a^3-40\,A\,a^2\,b\right )\,1{}\mathrm {i}}{8\,b^{13/4}}\right )\,\left (5\,A\,b-9\,B\,a\right )}{8\,b^{13/4}}+\frac {{\left (-a\right )}^{1/4}\,\left (\frac {\sqrt {x}\,\left (25\,A^2\,a^2\,b^2-90\,A\,B\,a^3\,b+81\,B^2\,a^4\right )}{b^3}+\frac {{\left (-a\right )}^{1/4}\,\left (5\,A\,b-9\,B\,a\right )\,\left (72\,B\,a^3-40\,A\,a^2\,b\right )\,1{}\mathrm {i}}{8\,b^{13/4}}\right )\,\left (5\,A\,b-9\,B\,a\right )}{8\,b^{13/4}}}{\frac {{\left (-a\right )}^{1/4}\,\left (\frac {\sqrt {x}\,\left (25\,A^2\,a^2\,b^2-90\,A\,B\,a^3\,b+81\,B^2\,a^4\right )}{b^3}-\frac {{\left (-a\right )}^{1/4}\,\left (5\,A\,b-9\,B\,a\right )\,\left (72\,B\,a^3-40\,A\,a^2\,b\right )\,1{}\mathrm {i}}{8\,b^{13/4}}\right )\,\left (5\,A\,b-9\,B\,a\right )\,1{}\mathrm {i}}{8\,b^{13/4}}-\frac {{\left (-a\right )}^{1/4}\,\left (\frac {\sqrt {x}\,\left (25\,A^2\,a^2\,b^2-90\,A\,B\,a^3\,b+81\,B^2\,a^4\right )}{b^3}+\frac {{\left (-a\right )}^{1/4}\,\left (5\,A\,b-9\,B\,a\right )\,\left (72\,B\,a^3-40\,A\,a^2\,b\right )\,1{}\mathrm {i}}{8\,b^{13/4}}\right )\,\left (5\,A\,b-9\,B\,a\right )\,1{}\mathrm {i}}{8\,b^{13/4}}}\right )\,\left (5\,A\,b-9\,B\,a\right )}{4\,b^{13/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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