Optimal. Leaf size=293 \[ \frac {(A b-a B) \sqrt {x}}{4 a b \left (a+b x^2\right )^2}+\frac {(7 A b+a B) \sqrt {x}}{16 a^2 b \left (a+b x^2\right )}-\frac {3 (7 A b+a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} b^{5/4}}+\frac {3 (7 A b+a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} b^{5/4}}-\frac {3 (7 A b+a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {3 (7 A b+a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{11/4} b^{5/4}} \]
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Rubi [A]
time = 0.15, antiderivative size = 293, 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, 296, 335,
217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {3 (a B+7 A b) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} b^{5/4}}+\frac {3 (a B+7 A b) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{11/4} b^{5/4}}-\frac {3 (a B+7 A b) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {3 (a B+7 A b) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\sqrt {x} (a B+7 A b)}{16 a^2 b \left (a+b x^2\right )}+\frac {\sqrt {x} (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 296
Rule 335
Rule 468
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )^3} \, dx &=\frac {(A b-a B) \sqrt {x}}{4 a b \left (a+b x^2\right )^2}+\frac {\left (\frac {7 A b}{2}+\frac {a B}{2}\right ) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac {(A b-a B) \sqrt {x}}{4 a b \left (a+b x^2\right )^2}+\frac {(7 A b+a B) \sqrt {x}}{16 a^2 b \left (a+b x^2\right )}+\frac {(3 (7 A b+a B)) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{32 a^2 b}\\ &=\frac {(A b-a B) \sqrt {x}}{4 a b \left (a+b x^2\right )^2}+\frac {(7 A b+a B) \sqrt {x}}{16 a^2 b \left (a+b x^2\right )}+\frac {(3 (7 A b+a B)) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{16 a^2 b}\\ &=\frac {(A b-a B) \sqrt {x}}{4 a b \left (a+b x^2\right )^2}+\frac {(7 A b+a B) \sqrt {x}}{16 a^2 b \left (a+b x^2\right )}+\frac {(3 (7 A b+a B)) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^{5/2} b}+\frac {(3 (7 A b+a B)) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^{5/2} b}\\ &=\frac {(A b-a B) \sqrt {x}}{4 a b \left (a+b x^2\right )^2}+\frac {(7 A b+a B) \sqrt {x}}{16 a^2 b \left (a+b x^2\right )}+\frac {(3 (7 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 )}{64 a^{5/2} b^{3/2}}+\frac {(3 (7 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 )}{64 a^{5/2} b^{3/2}}-\frac {(3 (7 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 )}{64 \sqrt {2} a^{11/4} b^{5/4}}-\frac {(3 (7 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 )}{64 \sqrt {2} a^{11/4} b^{5/4}}\\ &=\frac {(A b-a B) \sqrt {x}}{4 a b \left (a+b x^2\right )^2}+\frac {(7 A b+a B) \sqrt {x}}{16 a^2 b \left (a+b x^2\right )}-\frac {3 (7 A b+a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {3 (7 A b+a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {(3 (7 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 )}{32 \sqrt {2} a^{11/4} b^{5/4}}-\frac {(3 (7 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 )}{32 \sqrt {2} a^{11/4} b^{5/4}}\\ &=\frac {(A b-a B) \sqrt {x}}{4 a b \left (a+b x^2\right )^2}+\frac {(7 A b+a B) \sqrt {x}}{16 a^2 b \left (a+b x^2\right )}-\frac {3 (7 A b+a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} b^{5/4}}+\frac {3 (7 A b+a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} b^{5/4}}-\frac {3 (7 A b+a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {3 (7 A b+a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}\\ \end {align*}
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Mathematica [A]
time = 0.61, size = 172, normalized size = 0.59 \begin {gather*} \frac {\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (11 a A b-3 a^2 B+7 A b^2 x^2+a b B x^2\right )}{\left (a+b x^2\right )^2}-3 \sqrt {2} (7 A b+a B) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+3 \sqrt {2} (7 A b+a B) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{64 a^{11/4} b^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 166, normalized size = 0.57
method | result | size |
derivativedivides | \(\frac {\frac {\left (7 A b +B a \right ) x^{\frac {5}{2}}}{16 a^{2}}+\frac {\left (11 A b -3 B a \right ) \sqrt {x}}{16 a b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 \left (7 A b +B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a^{3} b}\) | \(166\) |
default | \(\frac {\frac {\left (7 A b +B a \right ) x^{\frac {5}{2}}}{16 a^{2}}+\frac {\left (11 A b -3 B a \right ) \sqrt {x}}{16 a b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 \left (7 A b +B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a^{3} b}\) | \(166\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 276, normalized size = 0.94 \begin {gather*} \frac {{\left (B a b + 7 \, A b^{2}\right )} x^{\frac {5}{2}} - {\left (3 \, B a^{2} - 11 \, A a b\right )} \sqrt {x}}{16 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} {\left (B a + 7 \, 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 (B a + 7 \, 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 (B a + 7 \, 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 (B a + 7 \, 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 \, a^{2} b} \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.
\(2 (213) = 426\).
time = 0.78, size = 793, normalized size = 2.71 \begin {gather*} \frac {12 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \left (-\frac {B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{6} b^{2} \sqrt {-\frac {B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}} + {\left (B^{2} a^{2} + 14 \, A B a b + 49 \, A^{2} b^{2}\right )} x} a^{8} b^{4} \left (-\frac {B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac {3}{4}} - {\left (B a^{9} b^{4} + 7 \, A a^{8} b^{5}\right )} \sqrt {x} \left (-\frac {B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac {3}{4}}}{B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}\right ) + 3 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \left (-\frac {B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac {1}{4}} \log \left (3 \, a^{3} b \left (-\frac {B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac {1}{4}} + 3 \, {\left (B a + 7 \, A b\right )} \sqrt {x}\right ) - 3 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \left (-\frac {B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac {1}{4}} \log \left (-3 \, a^{3} b \left (-\frac {B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac {1}{4}} + 3 \, {\left (B a + 7 \, A b\right )} \sqrt {x}\right ) - 4 \, {\left (3 \, B a^{2} - 11 \, A a b - {\left (B a b + 7 \, A b^{2}\right )} x^{2}\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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1406 vs.
\(2 (287) = 574\).
time = 141.89, size = 1406, normalized size = 4.80 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{11 x^{\frac {11}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 A \sqrt {x} + \frac {2 B x^{\frac {5}{2}}}{5}}{a^{3}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{11 x^{\frac {11}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}}{b^{3}} & \text {for}\: a = 0 \\\frac {44 A a^{2} b \sqrt {x}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} - \frac {21 A a^{2} b \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} + \frac {21 A a^{2} b \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} + \frac {42 A a^{2} b \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} + \frac {28 A a b^{2} x^{\frac {5}{2}}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} - \frac {42 A a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} + \frac {42 A a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} + \frac {84 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^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} - \frac {21 A b^{3} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} + \frac {21 A b^{3} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} + \frac {42 A b^{3} x^{4} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} - \frac {12 B a^{3} \sqrt {x}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} - \frac {3 B a^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} + \frac {3 B a^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} + \frac {6 B a^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} + \frac {4 B a^{2} b x^{\frac {5}{2}}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} - \frac {6 B a^{2} b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} + \frac {6 B a^{2} b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} + \frac {12 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^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} - \frac {3 B a b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} + \frac {3 B a b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} x^{4}} + \frac {6 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^{5} b + 128 a^{4} b^{2} x^{2} + 64 a^{3} b^{3} 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 = 1.27, size = 293, normalized size = 1.00 \begin {gather*} \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a + 7 \, \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^{3} b^{2}} + \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a + 7 \, \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^{3} b^{2}} + \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a + 7 \, \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^{3} b^{2}} - \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a + 7 \, \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^{3} b^{2}} + \frac {B a b x^{\frac {5}{2}} + 7 \, A b^{2} x^{\frac {5}{2}} - 3 \, B a^{2} \sqrt {x} + 11 \, A a b \sqrt {x}}{16 \, {\left (b x^{2} + a\right )}^{2} a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 780, normalized size = 2.66 \begin {gather*} \frac {\frac {x^{5/2}\,\left (7\,A\,b+B\,a\right )}{16\,a^2}+\frac {\sqrt {x}\,\left (11\,A\,b-3\,B\,a\right )}{16\,a\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4}-\frac {\mathrm {atan}\left (\frac {\frac {\left (7\,A\,b+B\,a\right )\,\left (\frac {9\,\sqrt {x}\,\left (49\,A^2\,b^3+14\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{64\,a^4}-\frac {9\,\left (7\,A\,b+B\,a\right )\,\left (7\,A\,b^3+B\,a\,b^2\right )}{64\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )\,3{}\mathrm {i}}{64\,{\left (-a\right )}^{11/4}\,b^{5/4}}+\frac {\left (7\,A\,b+B\,a\right )\,\left (\frac {9\,\sqrt {x}\,\left (49\,A^2\,b^3+14\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{64\,a^4}+\frac {9\,\left (7\,A\,b+B\,a\right )\,\left (7\,A\,b^3+B\,a\,b^2\right )}{64\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )\,3{}\mathrm {i}}{64\,{\left (-a\right )}^{11/4}\,b^{5/4}}}{\frac {3\,\left (7\,A\,b+B\,a\right )\,\left (\frac {9\,\sqrt {x}\,\left (49\,A^2\,b^3+14\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{64\,a^4}-\frac {9\,\left (7\,A\,b+B\,a\right )\,\left (7\,A\,b^3+B\,a\,b^2\right )}{64\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )}{64\,{\left (-a\right )}^{11/4}\,b^{5/4}}-\frac {3\,\left (7\,A\,b+B\,a\right )\,\left (\frac {9\,\sqrt {x}\,\left (49\,A^2\,b^3+14\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{64\,a^4}+\frac {9\,\left (7\,A\,b+B\,a\right )\,\left (7\,A\,b^3+B\,a\,b^2\right )}{64\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )}{64\,{\left (-a\right )}^{11/4}\,b^{5/4}}}\right )\,\left (7\,A\,b+B\,a\right )\,3{}\mathrm {i}}{32\,{\left (-a\right )}^{11/4}\,b^{5/4}}-\frac {3\,\mathrm {atan}\left (\frac {\frac {3\,\left (7\,A\,b+B\,a\right )\,\left (\frac {9\,\sqrt {x}\,\left (49\,A^2\,b^3+14\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{64\,a^4}-\frac {\left (7\,A\,b+B\,a\right )\,\left (7\,A\,b^3+B\,a\,b^2\right )\,9{}\mathrm {i}}{64\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )}{64\,{\left (-a\right )}^{11/4}\,b^{5/4}}+\frac {3\,\left (7\,A\,b+B\,a\right )\,\left (\frac {9\,\sqrt {x}\,\left (49\,A^2\,b^3+14\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{64\,a^4}+\frac {\left (7\,A\,b+B\,a\right )\,\left (7\,A\,b^3+B\,a\,b^2\right )\,9{}\mathrm {i}}{64\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )}{64\,{\left (-a\right )}^{11/4}\,b^{5/4}}}{\frac {\left (7\,A\,b+B\,a\right )\,\left (\frac {9\,\sqrt {x}\,\left (49\,A^2\,b^3+14\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{64\,a^4}-\frac {\left (7\,A\,b+B\,a\right )\,\left (7\,A\,b^3+B\,a\,b^2\right )\,9{}\mathrm {i}}{64\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )\,3{}\mathrm {i}}{64\,{\left (-a\right )}^{11/4}\,b^{5/4}}-\frac {\left (7\,A\,b+B\,a\right )\,\left (\frac {9\,\sqrt {x}\,\left (49\,A^2\,b^3+14\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{64\,a^4}+\frac {\left (7\,A\,b+B\,a\right )\,\left (7\,A\,b^3+B\,a\,b^2\right )\,9{}\mathrm {i}}{64\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )\,3{}\mathrm {i}}{64\,{\left (-a\right )}^{11/4}\,b^{5/4}}}\right )\,\left (7\,A\,b+B\,a\right )}{32\,{\left (-a\right )}^{11/4}\,b^{5/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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