Optimal. Leaf size=316 \[ -\frac {5 (A b-9 a B) \sqrt {x}}{16 a b^3}+\frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b-9 a B) x^{5/2}}{16 a b^2 \left (a+b x^2\right )}-\frac {5 (A b-9 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{13/4}}+\frac {5 (A b-9 a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{13/4}}-\frac {5 (A b-9 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{13/4}}+\frac {5 (A b-9 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{13/4}} \]
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Rubi [A]
time = 0.17, antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {468, 294,
327, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {5 (A b-9 a B) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{13/4}}+\frac {5 (A b-9 a B) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{3/4} b^{13/4}}-\frac {5 (A b-9 a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{13/4}}+\frac {5 (A b-9 a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{13/4}}-\frac {5 \sqrt {x} (A b-9 a B)}{16 a b^3}+\frac {x^{5/2} (A b-9 a B)}{16 a b^2 \left (a+b x^2\right )}+\frac {x^{9/2} (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 210
Rule 217
Rule 294
Rule 327
Rule 335
Rule 468
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx &=\frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {\left (-\frac {A b}{2}+\frac {9 a B}{2}\right ) \int \frac {x^{7/2}}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b-9 a B) x^{5/2}}{16 a b^2 \left (a+b x^2\right )}-\frac {(5 (A b-9 a B)) \int \frac {x^{3/2}}{a+b x^2} \, dx}{32 a b^2}\\ &=-\frac {5 (A b-9 a B) \sqrt {x}}{16 a b^3}+\frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b-9 a B) x^{5/2}}{16 a b^2 \left (a+b x^2\right )}+\frac {(5 (A b-9 a B)) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{32 b^3}\\ &=-\frac {5 (A b-9 a B) \sqrt {x}}{16 a b^3}+\frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b-9 a B) x^{5/2}}{16 a b^2 \left (a+b x^2\right )}+\frac {(5 (A b-9 a B)) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{16 b^3}\\ &=-\frac {5 (A b-9 a B) \sqrt {x}}{16 a b^3}+\frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b-9 a B) x^{5/2}}{16 a b^2 \left (a+b x^2\right )}+\frac {(5 (A b-9 a B)) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 \sqrt {a} b^3}+\frac {(5 (A b-9 a B)) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 \sqrt {a} b^3}\\ &=-\frac {5 (A b-9 a B) \sqrt {x}}{16 a b^3}+\frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b-9 a B) x^{5/2}}{16 a b^2 \left (a+b x^2\right )}+\frac {(5 (A b-9 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 )}{64 \sqrt {a} b^{7/2}}+\frac {(5 (A b-9 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 )}{64 \sqrt {a} b^{7/2}}-\frac {(5 (A b-9 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 )}{64 \sqrt {2} a^{3/4} b^{13/4}}-\frac {(5 (A b-9 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 )}{64 \sqrt {2} a^{3/4} b^{13/4}}\\ &=-\frac {5 (A b-9 a B) \sqrt {x}}{16 a b^3}+\frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b-9 a B) x^{5/2}}{16 a b^2 \left (a+b x^2\right )}-\frac {5 (A b-9 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{13/4}}+\frac {5 (A b-9 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{13/4}}+\frac {(5 (A b-9 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 )}{32 \sqrt {2} a^{3/4} b^{13/4}}-\frac {(5 (A b-9 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 )}{32 \sqrt {2} a^{3/4} b^{13/4}}\\ &=-\frac {5 (A b-9 a B) \sqrt {x}}{16 a b^3}+\frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b-9 a B) x^{5/2}}{16 a b^2 \left (a+b x^2\right )}-\frac {5 (A b-9 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{13/4}}+\frac {5 (A b-9 a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{13/4}}-\frac {5 (A b-9 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{13/4}}+\frac {5 (A b-9 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{13/4}}\\ \end {align*}
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Mathematica [A]
time = 0.62, size = 183, normalized size = 0.58 \begin {gather*} \frac {\frac {4 \sqrt [4]{b} \sqrt {x} \left (-5 a A b+45 a^2 B-9 A b^2 x^2+81 a b B x^2+32 b^2 B x^4\right )}{\left (a+b x^2\right )^2}+\frac {5 \sqrt {2} (-A b+9 a B) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{3/4}}+\frac {5 \sqrt {2} (A b-9 a B) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{3/4}}}{64 b^{13/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 172, normalized size = 0.54
method | result | size |
derivativedivides | \(\frac {2 B \sqrt {x}}{b^{3}}+\frac {\frac {2 \left (\left (-\frac {9}{32} b^{2} A +\frac {17}{32} a b B \right ) x^{\frac {5}{2}}-\frac {a \left (5 A b -13 B a \right ) \sqrt {x}}{32}\right )}{\left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (A b -9 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a}}{b^{3}}\) | \(172\) |
default | \(\frac {2 B \sqrt {x}}{b^{3}}+\frac {\frac {2 \left (\left (-\frac {9}{32} b^{2} A +\frac {17}{32} a b B \right ) x^{\frac {5}{2}}-\frac {a \left (5 A b -13 B a \right ) \sqrt {x}}{32}\right )}{\left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (A b -9 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a}}{b^{3}}\) | \(172\) |
risch | \(\frac {2 B \sqrt {x}}{b^{3}}-\frac {9 x^{\frac {5}{2}} A}{16 b \left (b \,x^{2}+a \right )^{2}}+\frac {17 x^{\frac {5}{2}} a B}{16 b^{2} \left (b \,x^{2}+a \right )^{2}}-\frac {5 A \sqrt {x}\, a}{16 b^{2} \left (b \,x^{2}+a \right )^{2}}+\frac {13 B \sqrt {x}\, a^{2}}{16 b^{3} \left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{64 b^{2} a}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{64 b^{2} a}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )}{128 b^{2} a}-\frac {45 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{64 b^{3}}-\frac {45 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{64 b^{3}}-\frac {45 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )}{128 b^{3}}\) | \(363\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 283, normalized size = 0.90 \begin {gather*} \frac {{\left (17 \, B a b - 9 \, A b^{2}\right )} x^{\frac {5}{2}} + {\left (13 \, B a^{2} - 5 \, A a b\right )} \sqrt {x}}{16 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac {2 \, B \sqrt {x}}{b^{3}} - \frac {5 \, {\left (\frac {2 \, \sqrt {2} {\left (9 \, B a - 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 - 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 - 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 - 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 )}}{128 \, b^{3}} \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. 793 vs.
\(2 (238) = 476\).
time = 0.96, size = 793, normalized size = 2.51 \begin {gather*} \frac {20 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \left (-\frac {6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{2} b^{6} \sqrt {-\frac {6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}} + {\left (81 \, B^{2} a^{2} - 18 \, A B a b + A^{2} b^{2}\right )} x} a^{2} b^{10} \left (-\frac {6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac {3}{4}} + {\left (9 \, B a^{3} b^{10} - A a^{2} b^{11}\right )} \sqrt {x} \left (-\frac {6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac {3}{4}}}{6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}\right ) + 5 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \left (-\frac {6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac {1}{4}} \log \left (5 \, a b^{3} \left (-\frac {6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac {1}{4}} - 5 \, {\left (9 \, B a - A b\right )} \sqrt {x}\right ) - 5 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \left (-\frac {6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac {1}{4}} \log \left (-5 \, a b^{3} \left (-\frac {6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac {1}{4}} - 5 \, {\left (9 \, B a - A b\right )} \sqrt {x}\right ) + 4 \, {\left (32 \, B b^{2} x^{4} + 45 \, B a^{2} - 5 \, A a b + 9 \, {\left (9 \, B a b - A b^{2}\right )} x^{2}\right )} \sqrt {x}}{64 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [A]
time = 0.68, size = 304, normalized size = 0.96 \begin {gather*} \frac {2 \, B \sqrt {x}}{b^{3}} - \frac {5 \, \sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - \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 )}{64 \, a b^{4}} - \frac {5 \, \sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - \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 )}{64 \, a b^{4}} - \frac {5 \, \sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - \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 )}{128 \, a b^{4}} + \frac {5 \, \sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - \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 )}{128 \, a b^{4}} + \frac {17 \, B a b x^{\frac {5}{2}} - 9 \, A b^{2} x^{\frac {5}{2}} + 13 \, B a^{2} \sqrt {x} - 5 \, A a b \sqrt {x}}{16 \, {\left (b x^{2} + a\right )}^{2} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 760, normalized size = 2.41 \begin {gather*} \frac {\sqrt {x}\,\left (\frac {13\,B\,a^2}{16}-\frac {5\,A\,a\,b}{16}\right )-x^{5/2}\,\left (\frac {9\,A\,b^2}{16}-\frac {17\,B\,a\,b}{16}\right )}{a^2\,b^3+2\,a\,b^4\,x^2+b^5\,x^4}+\frac {2\,B\,\sqrt {x}}{b^3}-\frac {\mathrm {atan}\left (\frac {\frac {\left (A\,b-9\,B\,a\right )\,\left (\frac {25\,\sqrt {x}\,\left (A^2\,b^2-18\,A\,B\,a\,b+81\,B^2\,a^2\right )}{64\,b^3}-\frac {5\,\left (45\,B\,a^2-5\,A\,a\,b\right )\,\left (A\,b-9\,B\,a\right )}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )\,5{}\mathrm {i}}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}+\frac {\left (A\,b-9\,B\,a\right )\,\left (\frac {25\,\sqrt {x}\,\left (A^2\,b^2-18\,A\,B\,a\,b+81\,B^2\,a^2\right )}{64\,b^3}+\frac {5\,\left (45\,B\,a^2-5\,A\,a\,b\right )\,\left (A\,b-9\,B\,a\right )}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )\,5{}\mathrm {i}}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}}{\frac {5\,\left (A\,b-9\,B\,a\right )\,\left (\frac {25\,\sqrt {x}\,\left (A^2\,b^2-18\,A\,B\,a\,b+81\,B^2\,a^2\right )}{64\,b^3}-\frac {5\,\left (45\,B\,a^2-5\,A\,a\,b\right )\,\left (A\,b-9\,B\,a\right )}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}-\frac {5\,\left (A\,b-9\,B\,a\right )\,\left (\frac {25\,\sqrt {x}\,\left (A^2\,b^2-18\,A\,B\,a\,b+81\,B^2\,a^2\right )}{64\,b^3}+\frac {5\,\left (45\,B\,a^2-5\,A\,a\,b\right )\,\left (A\,b-9\,B\,a\right )}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}}\right )\,\left (A\,b-9\,B\,a\right )\,5{}\mathrm {i}}{32\,{\left (-a\right )}^{3/4}\,b^{13/4}}-\frac {5\,\mathrm {atan}\left (\frac {\frac {5\,\left (A\,b-9\,B\,a\right )\,\left (\frac {25\,\sqrt {x}\,\left (A^2\,b^2-18\,A\,B\,a\,b+81\,B^2\,a^2\right )}{64\,b^3}-\frac {\left (45\,B\,a^2-5\,A\,a\,b\right )\,\left (A\,b-9\,B\,a\right )\,5{}\mathrm {i}}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}+\frac {5\,\left (A\,b-9\,B\,a\right )\,\left (\frac {25\,\sqrt {x}\,\left (A^2\,b^2-18\,A\,B\,a\,b+81\,B^2\,a^2\right )}{64\,b^3}+\frac {\left (45\,B\,a^2-5\,A\,a\,b\right )\,\left (A\,b-9\,B\,a\right )\,5{}\mathrm {i}}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}}{\frac {\left (A\,b-9\,B\,a\right )\,\left (\frac {25\,\sqrt {x}\,\left (A^2\,b^2-18\,A\,B\,a\,b+81\,B^2\,a^2\right )}{64\,b^3}-\frac {\left (45\,B\,a^2-5\,A\,a\,b\right )\,\left (A\,b-9\,B\,a\right )\,5{}\mathrm {i}}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )\,5{}\mathrm {i}}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}-\frac {\left (A\,b-9\,B\,a\right )\,\left (\frac {25\,\sqrt {x}\,\left (A^2\,b^2-18\,A\,B\,a\,b+81\,B^2\,a^2\right )}{64\,b^3}+\frac {\left (45\,B\,a^2-5\,A\,a\,b\right )\,\left (A\,b-9\,B\,a\right )\,5{}\mathrm {i}}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )\,5{}\mathrm {i}}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}}\right )\,\left (A\,b-9\,B\,a\right )}{32\,{\left (-a\right )}^{3/4}\,b^{13/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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