Optimal. Leaf size=237 \[ \frac {(A b-a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {(A b-a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} \sqrt [4]{b}}+\frac {(A b-a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {(A b-a B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {2 A}{3 a x^{3/2}} \]
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Rubi [A] time = 0.18, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {453, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {(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} a^{7/4} \sqrt [4]{b}}-\frac {(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} a^{7/4} \sqrt [4]{b}}+\frac {(A b-a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {(A b-a B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {2 A}{3 a x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 329
Rule 453
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )} \, dx &=-\frac {2 A}{3 a x^{3/2}}-\frac {\left (2 \left (\frac {3 A b}{2}-\frac {3 a B}{2}\right )\right ) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{3 a}\\ &=-\frac {2 A}{3 a x^{3/2}}-\frac {\left (4 \left (\frac {3 A b}{2}-\frac {3 a B}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{3 a}\\ &=-\frac {2 A}{3 a x^{3/2}}-\frac {(A b-a B) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^{3/2}}-\frac {(A b-a B) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^{3/2}}\\ &=-\frac {2 A}{3 a x^{3/2}}-\frac {(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 a^{3/2} \sqrt {b}}-\frac {(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 a^{3/2} \sqrt {b}}+\frac {(A b-a B) \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} a^{7/4} \sqrt [4]{b}}+\frac {(A b-a B) \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} a^{7/4} \sqrt [4]{b}}\\ &=-\frac {2 A}{3 a x^{3/2}}+\frac {(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} a^{7/4} \sqrt [4]{b}}-\frac {(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} a^{7/4} \sqrt [4]{b}}-\frac {(A b-a B) \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} a^{7/4} \sqrt [4]{b}}+\frac {(A b-a B) \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} a^{7/4} \sqrt [4]{b}}\\ &=-\frac {2 A}{3 a x^{3/2}}+\frac {(A b-a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {(A b-a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} \sqrt [4]{b}}+\frac {(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} a^{7/4} \sqrt [4]{b}}-\frac {(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} a^{7/4} \sqrt [4]{b}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 168, normalized size = 0.71 \begin {gather*} \frac {(A b-a B) \left (\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )-2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )\right )}{2 \sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {2 A}{3 a x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 136, normalized size = 0.57 \begin {gather*} -\frac {(a B-A b) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt {2} a^{7/4} \sqrt [4]{b}}+\frac {(a B-A b) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {2 A}{3 a x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.41, size = 653, normalized size = 2.76 \begin {gather*} -\frac {12 \, a x^{2} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{4} \sqrt {-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}} + {\left (B^{2} a^{2} - 2 \, A B a b + A^{2} b^{2}\right )} x} a^{5} b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {3}{4}} + {\left (B a^{6} b - A a^{5} b^{2}\right )} \sqrt {x} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {3}{4}}}{B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}\right ) + 3 \, a x^{2} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} \log \left (a^{2} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} \sqrt {x}\right ) - 3 \, a x^{2} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} \log \left (-a^{2} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} \sqrt {x}\right ) + 4 \, A \sqrt {x}}{6 \, a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 251, normalized size = 1.06 \begin {gather*} \frac {\sqrt {2} {\left (\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 )}{2 \, a^{2} b} + \frac {\sqrt {2} {\left (\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 )}{2 \, a^{2} b} + \frac {\sqrt {2} {\left (\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 )}{4 \, a^{2} b} - \frac {\sqrt {2} {\left (\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 )}{4 \, a^{2} b} - \frac {2 \, A}{3 \, a x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 280, normalized size = 1.18 \begin {gather*} -\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 a^{2}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 a^{2}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A b \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 a^{2}}+\frac {\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 )}{2 a}+\frac {\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 )}{2 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \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 a}-\frac {2 A}{3 a \,x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.37, size = 218, normalized size = 0.92 \begin {gather*} \frac {\frac {2 \, \sqrt {2} {\left (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 (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 (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 (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}}}}{4 \, a} - \frac {2 \, A}{3 \, a x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 811, normalized size = 3.42 \begin {gather*} -\frac {2\,A}{3\,a\,x^{3/2}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,a^3\,b^5-32\,A\,B\,a^4\,b^4+16\,B^2\,a^5\,b^3\right )-\frac {\left (A\,b-B\,a\right )\,\left (32\,A\,a^5\,b^4-32\,B\,a^6\,b^3\right )}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}+\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,a^3\,b^5-32\,A\,B\,a^4\,b^4+16\,B^2\,a^5\,b^3\right )+\frac {\left (A\,b-B\,a\right )\,\left (32\,A\,a^5\,b^4-32\,B\,a^6\,b^3\right )}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}}{\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,a^3\,b^5-32\,A\,B\,a^4\,b^4+16\,B^2\,a^5\,b^3\right )-\frac {\left (A\,b-B\,a\right )\,\left (32\,A\,a^5\,b^4-32\,B\,a^6\,b^3\right )}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}\right )}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}-\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,a^3\,b^5-32\,A\,B\,a^4\,b^4+16\,B^2\,a^5\,b^3\right )+\frac {\left (A\,b-B\,a\right )\,\left (32\,A\,a^5\,b^4-32\,B\,a^6\,b^3\right )}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}\right )}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}}\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{{\left (-a\right )}^{7/4}\,b^{1/4}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,a^3\,b^5-32\,A\,B\,a^4\,b^4+16\,B^2\,a^5\,b^3\right )-\frac {\left (A\,b-B\,a\right )\,\left (32\,A\,a^5\,b^4-32\,B\,a^6\,b^3\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}\right )}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}+\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,a^3\,b^5-32\,A\,B\,a^4\,b^4+16\,B^2\,a^5\,b^3\right )+\frac {\left (A\,b-B\,a\right )\,\left (32\,A\,a^5\,b^4-32\,B\,a^6\,b^3\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}\right )}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}}{\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,a^3\,b^5-32\,A\,B\,a^4\,b^4+16\,B^2\,a^5\,b^3\right )-\frac {\left (A\,b-B\,a\right )\,\left (32\,A\,a^5\,b^4-32\,B\,a^6\,b^3\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}-\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,a^3\,b^5-32\,A\,B\,a^4\,b^4+16\,B^2\,a^5\,b^3\right )+\frac {\left (A\,b-B\,a\right )\,\left (32\,A\,a^5\,b^4-32\,B\,a^6\,b^3\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{7/4}\,b^{1/4}}}\right )\,\left (A\,b-B\,a\right )}{{\left (-a\right )}^{7/4}\,b^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 31.26, size = 364, normalized size = 1.54 \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}{7 x^{\frac {7}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{b} & \text {for}\: a = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} + 2 B \sqrt {x}}{a} & \text {for}\: b = 0 \\- \frac {2 A}{3 a x^{\frac {3}{2}}} + \frac {\sqrt [4]{-1} A b \sqrt [4]{\frac {1}{b}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 a^{\frac {7}{4}}} - \frac {\sqrt [4]{-1} A b \sqrt [4]{\frac {1}{b}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 a^{\frac {7}{4}}} + \frac {\sqrt [4]{-1} A b \sqrt [4]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{a^{\frac {7}{4}}} - \frac {\sqrt [4]{-1} B \sqrt [4]{\frac {1}{b}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 a^{\frac {3}{4}}} + \frac {\sqrt [4]{-1} B \sqrt [4]{\frac {1}{b}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 a^{\frac {3}{4}}} - \frac {\sqrt [4]{-1} B \sqrt [4]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{a^{\frac {3}{4}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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