Optimal. Leaf size=289 \[ -\frac {5 A b-a B}{2 a^2 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )}+\frac {(5 A b-a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{3/4}}-\frac {(5 A b-a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{3/4}}-\frac {(5 A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{3/4}}+\frac {(5 A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{3/4}} \]
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Rubi [A]
time = 0.15, antiderivative size = 289, 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 = {468, 331, 335,
303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {(5 A b-a B) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{3/4}}-\frac {(5 A b-a B) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{9/4} b^{3/4}}-\frac {(5 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^{9/4} b^{3/4}}+\frac {(5 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^{9/4} b^{3/4}}-\frac {5 A b-a B}{2 a^2 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 331
Rule 335
Rule 468
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^{3/2} \left (a+b x^2\right )^2} \, dx &=\frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )}+\frac {\left (\frac {5 A b}{2}-\frac {a B}{2}\right ) \int \frac {1}{x^{3/2} \left (a+b x^2\right )} \, dx}{2 a b}\\ &=-\frac {5 A b-a B}{2 a^2 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {(5 A b-a B) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{4 a^2}\\ &=-\frac {5 A b-a B}{2 a^2 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {(5 A b-a B) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^2}\\ &=-\frac {5 A b-a B}{2 a^2 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )}+\frac {(5 A b-a B) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^2 \sqrt {b}}-\frac {(5 A b-a B) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^2 \sqrt {b}}\\ &=-\frac {5 A b-a B}{2 a^2 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {(5 A b-a B) \text {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^2 b}-\frac {(5 A b-a B) \text {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^2 b}-\frac {(5 A b-a B) \text {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^{9/4} b^{3/4}}-\frac {(5 A b-a B) \text {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^{9/4} b^{3/4}}\\ &=-\frac {5 A b-a B}{2 a^2 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {(5 A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{3/4}}+\frac {(5 A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{3/4}}-\frac {(5 A b-a B) \text {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^{9/4} b^{3/4}}+\frac {(5 A b-a B) \text {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^{9/4} b^{3/4}}\\ &=-\frac {5 A b-a B}{2 a^2 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )}+\frac {(5 A b-a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{3/4}}-\frac {(5 A b-a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{3/4}}-\frac {(5 A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{3/4}}+\frac {(5 A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.52, size = 162, normalized size = 0.56 \begin {gather*} \frac {\frac {4 \sqrt [4]{a} \left (-4 a A-5 A b x^2+a B x^2\right )}{\sqrt {x} \left (a+b x^2\right )}+\frac {\sqrt {2} (5 A b-a B) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{b^{3/4}}+\frac {\sqrt {2} (5 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{b^{3/4}}}{8 a^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 153, normalized size = 0.53
method | result | size |
derivativedivides | \(-\frac {2 \left (\frac {\left (\frac {A b}{4}-\frac {B a}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {5 A b}{4}-\frac {B a}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{2}}-\frac {2 A}{a^{2} \sqrt {x}}\) | \(153\) |
default | \(-\frac {2 \left (\frac {\left (\frac {A b}{4}-\frac {B a}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {5 A b}{4}-\frac {B a}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{2}}-\frac {2 A}{a^{2} \sqrt {x}}\) | \(153\) |
risch | \(-\frac {2 A}{a^{2} \sqrt {x}}-\frac {x^{\frac {3}{2}} A b}{2 a^{2} \left (b \,x^{2}+a \right )}+\frac {x^{\frac {3}{2}} B}{2 a \left (b \,x^{2}+a \right )}-\frac {5 \sqrt {2}\, A \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )}{16 a^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {5 \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 a^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {5 \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 a^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, B \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )}{16 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(323\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 222, normalized size = 0.77 \begin {gather*} \frac {{\left (B a - 5 \, A b\right )} x^{2} - 4 \, A a}{2 \, {\left (a^{2} b x^{\frac {5}{2}} + a^{3} \sqrt {x}\right )}} + \frac {{\left (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 )}}{16 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 920 vs.
\(2 (204) = 408\).
time = 1.67, size = 920, normalized size = 3.18 \begin {gather*} \frac {4 \, {\left (a^{2} b x^{3} + a^{3} x\right )} \left (-\frac {B^{4} a^{4} - 20 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {{\left (B^{6} a^{6} - 30 \, A B^{5} a^{5} b + 375 \, A^{2} B^{4} a^{4} b^{2} - 2500 \, A^{3} B^{3} a^{3} b^{3} + 9375 \, A^{4} B^{2} a^{2} b^{4} - 18750 \, A^{5} B a b^{5} + 15625 \, A^{6} b^{6}\right )} x - {\left (B^{4} a^{9} b - 20 \, A B^{3} a^{8} b^{2} + 150 \, A^{2} B^{2} a^{7} b^{3} - 500 \, A^{3} B a^{6} b^{4} + 625 \, A^{4} a^{5} b^{5}\right )} \sqrt {-\frac {B^{4} a^{4} - 20 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{3}}}} a^{2} b \left (-\frac {B^{4} a^{4} - 20 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{3}}\right )^{\frac {1}{4}} + {\left (B^{3} a^{5} b - 15 \, A B^{2} a^{4} b^{2} + 75 \, A^{2} B a^{3} b^{3} - 125 \, A^{3} a^{2} b^{4}\right )} \sqrt {x} \left (-\frac {B^{4} a^{4} - 20 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{3}}\right )^{\frac {1}{4}}}{B^{4} a^{4} - 20 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}\right ) - {\left (a^{2} b x^{3} + a^{3} x\right )} \left (-\frac {B^{4} a^{4} - 20 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{3}}\right )^{\frac {1}{4}} \log \left (a^{7} b^{2} \left (-\frac {B^{4} a^{4} - 20 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{3}}\right )^{\frac {3}{4}} - {\left (B^{3} a^{3} - 15 \, A B^{2} a^{2} b + 75 \, A^{2} B a b^{2} - 125 \, A^{3} b^{3}\right )} \sqrt {x}\right ) + {\left (a^{2} b x^{3} + a^{3} x\right )} \left (-\frac {B^{4} a^{4} - 20 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{3}}\right )^{\frac {1}{4}} \log \left (-a^{7} b^{2} \left (-\frac {B^{4} a^{4} - 20 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{3}}\right )^{\frac {3}{4}} - {\left (B^{3} a^{3} - 15 \, A B^{2} a^{2} b + 75 \, A^{2} B a b^{2} - 125 \, A^{3} b^{3}\right )} \sqrt {x}\right ) + 4 \, {\left ({\left (B a - 5 \, A b\right )} x^{2} - 4 \, A a\right )} \sqrt {x}}{8 \, {\left (a^{2} b x^{3} + a^{3} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 79.38, size = 573, normalized size = 1.98 \begin {gather*} A \left (\begin {cases} \frac {\tilde {\infty }}{x^{\frac {9}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{9 b^{2} x^{\frac {9}{2}}} & \text {for}\: a = 0 \\- \frac {2}{a^{2} \sqrt {x}} & \text {for}\: b = 0 \\- \frac {5 a \sqrt {x} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} + \frac {5 a \sqrt {x} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} - \frac {10 a \sqrt {x} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} - \frac {16 a \sqrt [4]{- \frac {a}{b}}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} - \frac {5 b x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} + \frac {5 b x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} - \frac {10 b x^{\frac {5}{2}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} - \frac {20 b x^{2} \sqrt [4]{- \frac {a}{b}}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} & \text {otherwise} \end {cases}\right ) + \frac {2 B x^{\frac {3}{2}}}{4 a^{2} + 4 a b x^{2}} + 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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 278, normalized size = 0.96 \begin {gather*} \frac {B a x^{2} - 5 \, A b x^{2} - 4 \, A a}{2 \, {\left (b x^{\frac {5}{2}} + a \sqrt {x}\right )} a^{2}} + \frac {\sqrt {2} {\left (\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 )}{8 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (\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 )}{8 \, a^{3} b^{3}} - \frac {\sqrt {2} {\left (\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 )}{16 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (\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 )}{16 \, a^{3} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 104, normalized size = 0.36 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (5\,A\,b-B\,a\right )}{4\,{\left (-a\right )}^{9/4}\,b^{3/4}}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (5\,A\,b-B\,a\right )}{4\,{\left (-a\right )}^{9/4}\,b^{3/4}}-\frac {\frac {2\,A}{a}+\frac {x^2\,\left (5\,A\,b-B\,a\right )}{2\,a^2}}{a\,\sqrt {x}+b\,x^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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