Optimal. Leaf size=257 \[ -\frac {a^{3/4} (A b-a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{11/4}}+\frac {a^{3/4} (A b-a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{11/4}}+\frac {a^{3/4} (A b-a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{11/4}}-\frac {a^{3/4} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} b^{11/4}}+\frac {2 x^{3/2} (A b-a B)}{3 b^2}+\frac {2 B x^{7/2}}{7 b} \]
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Rubi [A] time = 0.21, antiderivative size = 257, 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 = {459, 321, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac {a^{3/4} (A b-a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{11/4}}+\frac {a^{3/4} (A b-a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{11/4}}+\frac {a^{3/4} (A b-a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{11/4}}-\frac {a^{3/4} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} b^{11/4}}+\frac {2 x^{3/2} (A b-a B)}{3 b^2}+\frac {2 B x^{7/2}}{7 b} \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 321
Rule 329
Rule 459
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^{5/2} \left (A+B x^2\right )}{a+b x^2} \, dx &=\frac {2 B x^{7/2}}{7 b}-\frac {\left (2 \left (-\frac {7 A b}{2}+\frac {7 a B}{2}\right )\right ) \int \frac {x^{5/2}}{a+b x^2} \, dx}{7 b}\\ &=\frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {(a (A b-a B)) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{b^2}\\ &=\frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {(2 a (A b-a B)) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=\frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{7/2}}{7 b}+\frac {(a (A b-a B)) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^{5/2}}-\frac {(a (A b-a B)) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^{5/2}}\\ &=\frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {(a (A b-a B)) \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 )}{2 b^3}-\frac {(a (A b-a B)) \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 )}{2 b^3}-\frac {\left (a^{3/4} (A b-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 )}{2 \sqrt {2} b^{11/4}}-\frac {\left (a^{3/4} (A b-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 )}{2 \sqrt {2} b^{11/4}}\\ &=\frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {a^{3/4} (A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{11/4}}+\frac {a^{3/4} (A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{11/4}}-\frac {\left (a^{3/4} (A b-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 )}{\sqrt {2} b^{11/4}}+\frac {\left (a^{3/4} (A b-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 )}{\sqrt {2} b^{11/4}}\\ &=\frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{7/2}}{7 b}+\frac {a^{3/4} (A b-a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{11/4}}-\frac {a^{3/4} (A b-a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{11/4}}-\frac {a^{3/4} (A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{11/4}}+\frac {a^{3/4} (A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{11/4}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 110, normalized size = 0.43 \[ \frac {2 b^{3/4} x^{3/2} \left (-7 a B+7 A b+3 b B x^2\right )-21 (-a)^{3/4} (a B-A b) \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{-a}}\right )+21 (-a)^{3/4} (a B-A b) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{-a}}\right )}{21 b^{11/4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 899, normalized size = 3.50 \[ \frac {84 \, b^{2} \left (-\frac {B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {{\left (B^{6} a^{10} - 6 \, A B^{5} a^{9} b + 15 \, A^{2} B^{4} a^{8} b^{2} - 20 \, A^{3} B^{3} a^{7} b^{3} + 15 \, A^{4} B^{2} a^{6} b^{4} - 6 \, A^{5} B a^{5} b^{5} + A^{6} a^{4} b^{6}\right )} x - {\left (B^{4} a^{7} b^{5} - 4 \, A B^{3} a^{6} b^{6} + 6 \, A^{2} B^{2} a^{5} b^{7} - 4 \, A^{3} B a^{4} b^{8} + A^{4} a^{3} b^{9}\right )} \sqrt {-\frac {B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}}} b^{3} \left (-\frac {B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac {1}{4}} + {\left (B^{3} a^{5} b^{3} - 3 \, A B^{2} a^{4} b^{4} + 3 \, A^{2} B a^{3} b^{5} - A^{3} a^{2} b^{6}\right )} \sqrt {x} \left (-\frac {B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac {1}{4}}}{B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}\right ) - 21 \, b^{2} \left (-\frac {B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac {1}{4}} \log \left (b^{8} \left (-\frac {B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac {3}{4}} - {\left (B^{3} a^{5} - 3 \, A B^{2} a^{4} b + 3 \, A^{2} B a^{3} b^{2} - A^{3} a^{2} b^{3}\right )} \sqrt {x}\right ) + 21 \, b^{2} \left (-\frac {B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac {1}{4}} \log \left (-b^{8} \left (-\frac {B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac {3}{4}} - {\left (B^{3} a^{5} - 3 \, A B^{2} a^{4} b + 3 \, A^{2} B a^{3} b^{2} - A^{3} a^{2} b^{3}\right )} \sqrt {x}\right ) + 4 \, {\left (3 \, B b x^{3} - 7 \, {\left (B a - A b\right )} x\right )} \sqrt {x}}{42 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 264, normalized size = 1.03 \[ \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} B a - \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, b^{5}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} B a - \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, b^{5}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} B a - \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, b^{5}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} B a - \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, b^{5}} + \frac {2 \, {\left (3 \, B b^{6} x^{\frac {7}{2}} - 7 \, B a b^{5} x^{\frac {3}{2}} + 7 \, A b^{6} x^{\frac {3}{2}}\right )}}{21 \, b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 308, normalized size = 1.20 \[ \frac {2 B \,x^{\frac {7}{2}}}{7 b}+\frac {2 A \,x^{\frac {3}{2}}}{3 b}-\frac {2 B a \,x^{\frac {3}{2}}}{3 b^{2}}-\frac {\sqrt {2}\, A a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}-\frac {\sqrt {2}\, A a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}-\frac {\sqrt {2}\, A a \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {\sqrt {2}\, B \,a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}}+\frac {\sqrt {2}\, B \,a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}}+\frac {\sqrt {2}\, B \,a^{2} \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.41, size = 214, normalized size = 0.83 \[ \frac {{\left (B a^{2} - A a b\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, b^{2}} + \frac {2 \, {\left (3 \, B b x^{\frac {7}{2}} - 7 \, {\left (B a - A b\right )} x^{\frac {3}{2}}\right )}}{21 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 92, normalized size = 0.36 \[ x^{3/2}\,\left (\frac {2\,A}{3\,b}-\frac {2\,B\,a}{3\,b^2}\right )+\frac {2\,B\,x^{7/2}}{7\,b}+\frac {{\left (-a\right )}^{3/4}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (A\,b-B\,a\right )}{b^{11/4}}+\frac {{\left (-a\right )}^{3/4}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}}\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{b^{11/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 54.69, size = 502, normalized size = 1.95 \[ \begin {cases} \tilde {\infty } \left (\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {7}{2}}}{7}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {7}{2}}}{7} + \frac {2 B x^{\frac {11}{2}}}{11}}{a} & \text {for}\: b = 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {7}{2}}}{7}}{b} & \text {for}\: a = 0 \\\frac {\left (-1\right )^{\frac {3}{4}} A a^{\frac {3}{4}} \left (\frac {1}{b}\right )^{\frac {3}{4}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 b} - \frac {\left (-1\right )^{\frac {3}{4}} A a^{\frac {3}{4}} \left (\frac {1}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 b} + \frac {\left (-1\right )^{\frac {3}{4}} A a^{\frac {3}{4}} \left (\frac {1}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{b} - \frac {2 \left (-1\right )^{\frac {3}{4}} A a^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{b^{2} \sqrt [4]{\frac {1}{b}}} + \frac {2 A x^{\frac {3}{2}}}{3 b} - \frac {\left (-1\right )^{\frac {3}{4}} B a^{\frac {7}{4}} \left (\frac {1}{b}\right )^{\frac {3}{4}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 b^{2}} + \frac {\left (-1\right )^{\frac {3}{4}} B a^{\frac {7}{4}} \left (\frac {1}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 b^{2}} - \frac {\left (-1\right )^{\frac {3}{4}} B a^{\frac {7}{4}} \left (\frac {1}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{b^{2}} + \frac {2 \left (-1\right )^{\frac {3}{4}} B a^{\frac {7}{4}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{b^{3} \sqrt [4]{\frac {1}{b}}} - \frac {2 B a x^{\frac {3}{2}}}{3 b^{2}} + \frac {2 B x^{\frac {7}{2}}}{7 b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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