Optimal. Leaf size=298 \[ \frac {(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(3 A b+5 a B) \sqrt {x}}{16 a b^2 \left (a+b x^2\right )}-\frac {(3 A b+5 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(3 A b+5 a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{9/4}}-\frac {(3 A b+5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(3 A b+5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}} \]
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Rubi [A]
time = 0.14, 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 = {468, 294, 335,
217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {(5 a B+3 A b) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(5 a B+3 A b) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{7/4} b^{9/4}}-\frac {(5 a B+3 A b) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(5 a B+3 A b) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}}-\frac {\sqrt {x} (5 a B+3 A b)}{16 a b^2 \left (a+b x^2\right )}+\frac {x^{5/2} (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 335
Rule 468
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx &=\frac {(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}+\frac {\left (\frac {3 A b}{2}+\frac {5 a B}{2}\right ) \int \frac {x^{3/2}}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac {(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(3 A b+5 a B) \sqrt {x}}{16 a b^2 \left (a+b x^2\right )}+\frac {(3 A b+5 a B) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{32 a b^2}\\ &=\frac {(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(3 A b+5 a B) \sqrt {x}}{16 a b^2 \left (a+b x^2\right )}+\frac {(3 A b+5 a B) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{16 a b^2}\\ &=\frac {(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(3 A b+5 a B) \sqrt {x}}{16 a b^2 \left (a+b x^2\right )}+\frac {(3 A b+5 a B) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^{3/2} b^2}+\frac {(3 A b+5 a B) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^{3/2} b^2}\\ &=\frac {(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(3 A b+5 a B) \sqrt {x}}{16 a b^2 \left (a+b x^2\right )}+\frac {(3 A b+5 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 a^{3/2} b^{5/2}}+\frac {(3 A b+5 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 a^{3/2} b^{5/2}}-\frac {(3 A b+5 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^{7/4} b^{9/4}}-\frac {(3 A b+5 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^{7/4} b^{9/4}}\\ &=\frac {(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(3 A b+5 a B) \sqrt {x}}{16 a b^2 \left (a+b x^2\right )}-\frac {(3 A b+5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(3 A b+5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(3 A b+5 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^{7/4} b^{9/4}}-\frac {(3 A b+5 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^{7/4} b^{9/4}}\\ &=\frac {(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(3 A b+5 a B) \sqrt {x}}{16 a b^2 \left (a+b x^2\right )}-\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^{7/4} b^{9/4}}+\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^{7/4} b^{9/4}}-\frac {(3 A b+5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(3 A b+5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}}\\ \end {align*}
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Mathematica [A]
time = 0.60, size = 173, normalized size = 0.58 \begin {gather*} \frac {-\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (5 a^2 B-A b^2 x^2+3 a b \left (A+3 B x^2\right )\right )}{\left (a+b x^2\right )^2}-\sqrt {2} (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 )+\sqrt {2} (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 )}{64 a^{7/4} b^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 167, normalized size = 0.56
method | result | size |
derivativedivides | \(\frac {\frac {\left (A b -9 B a \right ) x^{\frac {5}{2}}}{16 a b}-\frac {\left (3 A b +5 B a \right ) \sqrt {x}}{16 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (3 A b +5 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b^{2} a^{2}}\) | \(167\) |
default | \(\frac {\frac {\left (A b -9 B a \right ) x^{\frac {5}{2}}}{16 a b}-\frac {\left (3 A b +5 B a \right ) \sqrt {x}}{16 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (3 A b +5 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b^{2} a^{2}}\) | \(167\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 280, normalized size = 0.94 \begin {gather*} -\frac {{\left (9 \, B a b - A b^{2}\right )} x^{\frac {5}{2}} + {\left (5 \, B a^{2} + 3 \, A a b\right )} \sqrt {x}}{16 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (5 \, B a + 3 \, 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 (5 \, B a + 3 \, 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 (5 \, B a + 3 \, 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 (5 \, B a + 3 \, 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}}}}{128 \, a b^{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. 806 vs.
\(2 (218) = 436\).
time = 0.70, size = 806, normalized size = 2.70 \begin {gather*} \frac {4 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{4} b^{4} \sqrt {-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}} + {\left (25 \, B^{2} a^{2} + 30 \, A B a b + 9 \, A^{2} b^{2}\right )} x} a^{5} b^{7} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {3}{4}} - {\left (5 \, B a^{6} b^{7} + 3 \, A a^{5} b^{8}\right )} \sqrt {x} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {3}{4}}}{625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}\right ) + {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} \log \left (a^{2} b^{2} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} + {\left (5 \, B a + 3 \, A b\right )} \sqrt {x}\right ) - {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} \log \left (-a^{2} b^{2} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} + {\left (5 \, B a + 3 \, A b\right )} \sqrt {x}\right ) - 4 \, {\left (5 \, B a^{2} + 3 \, A a b + {\left (9 \, B a b - A b^{2}\right )} x^{2}\right )} \sqrt {x}}{64 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1445 vs.
\(2 (287) = 574\).
time = 159.96, size = 1445, normalized size = 4.85 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {9}{2}}}{9}}{a^{3}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{b^{3}} & \text {for}\: a = 0 \\- \frac {12 A a^{2} b \sqrt {x}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} - \frac {3 A a^{2} b \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} + \frac {3 A a^{2} b \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} + \frac {6 A a^{2} b \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} + \frac {4 A a b^{2} x^{\frac {5}{2}}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} - \frac {6 A a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} + \frac {6 A a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} + \frac {12 A a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} - \frac {3 A b^{3} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} + \frac {3 A b^{3} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} + \frac {6 A b^{3} x^{4} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} - \frac {20 B a^{3} \sqrt {x}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} - \frac {5 B a^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} + \frac {5 B a^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} + \frac {10 B a^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} - \frac {36 B a^{2} b x^{\frac {5}{2}}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} - \frac {10 B a^{2} b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} + \frac {10 B a^{2} b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} + \frac {20 B a^{2} b x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} - \frac {5 B a b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} + \frac {5 B a b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} + \frac {10 B a b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b^{2} + 128 a^{3} b^{3} x^{2} + 64 a^{2} b^{4} x^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.58, size = 298, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \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^{2} b^{3}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \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^{2} b^{3}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \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^{2} b^{3}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \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^{2} b^{3}} - \frac {9 \, B a b x^{\frac {5}{2}} - A b^{2} x^{\frac {5}{2}} + 5 \, B a^{2} \sqrt {x} + 3 \, A a b \sqrt {x}}{16 \, {\left (b x^{2} + a\right )}^{2} a b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.26, size = 799, normalized size = 2.68 \begin {gather*} -\frac {\frac {\sqrt {x}\,\left (3\,A\,b+5\,B\,a\right )}{16\,b^2}-\frac {x^{5/2}\,\left (A\,b-9\,B\,a\right )}{16\,a\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4}+\frac {\mathrm {atan}\left (\frac {\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}-\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}+\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}+\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}}{\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}-\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}-\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}+\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}}\right )\,\left (3\,A\,b+5\,B\,a\right )\,1{}\mathrm {i}}{32\,{\left (-a\right )}^{7/4}\,b^{9/4}}+\frac {\mathrm {atan}\left (\frac {\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}-\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}+\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}+\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}}{\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}-\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}-\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}+\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}}\right )\,\left (3\,A\,b+5\,B\,a\right )}{32\,{\left (-a\right )}^{7/4}\,b^{9/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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