Optimal. Leaf size=243 \[ \frac {9 b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}}-\frac {9 b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}}-\frac {9 b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4}}+\frac {9 b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{13/4}}+\frac {9 b}{2 a^3 \sqrt {x}}-\frac {9}{10 a^2 x^{5/2}}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.19, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac {9 b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}}-\frac {9 b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}}-\frac {9 b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4}}+\frac {9 b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{13/4}}+\frac {9 b}{2 a^3 \sqrt {x}}-\frac {9}{10 a^2 x^{5/2}}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 204
Rule 290
Rule 297
Rule 325
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{x^{7/2} \left (a+b x^2\right )^2} \, dx &=\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}+\frac {9 \int \frac {1}{x^{7/2} \left (a+b x^2\right )} \, dx}{4 a}\\ &=-\frac {9}{10 a^2 x^{5/2}}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}-\frac {(9 b) \int \frac {1}{x^{3/2} \left (a+b x^2\right )} \, dx}{4 a^2}\\ &=-\frac {9}{10 a^2 x^{5/2}}+\frac {9 b}{2 a^3 \sqrt {x}}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}+\frac {\left (9 b^2\right ) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{4 a^3}\\ &=-\frac {9}{10 a^2 x^{5/2}}+\frac {9 b}{2 a^3 \sqrt {x}}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}+\frac {\left (9 b^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^3}\\ &=-\frac {9}{10 a^2 x^{5/2}}+\frac {9 b}{2 a^3 \sqrt {x}}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}-\frac {\left (9 b^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^3}+\frac {\left (9 b^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^3}\\ &=-\frac {9}{10 a^2 x^{5/2}}+\frac {9 b}{2 a^3 \sqrt {x}}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}+\frac {(9 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 )}{8 a^3}+\frac {(9 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 )}{8 a^3}+\frac {\left (9 b^{5/4}\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 )}{8 \sqrt {2} a^{13/4}}+\frac {\left (9 b^{5/4}\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 )}{8 \sqrt {2} a^{13/4}}\\ &=-\frac {9}{10 a^2 x^{5/2}}+\frac {9 b}{2 a^3 \sqrt {x}}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}+\frac {9 b^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}}-\frac {9 b^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}}+\frac {\left (9 b^{5/4}\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 )}{4 \sqrt {2} a^{13/4}}-\frac {\left (9 b^{5/4}\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 )}{4 \sqrt {2} a^{13/4}}\\ &=-\frac {9}{10 a^2 x^{5/2}}+\frac {9 b}{2 a^3 \sqrt {x}}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}-\frac {9 b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4}}+\frac {9 b^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4}}+\frac {9 b^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}}-\frac {9 b^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 29, normalized size = 0.12 \[ -\frac {2 \, _2F_1\left (-\frac {5}{4},2;-\frac {1}{4};-\frac {b x^2}{a}\right )}{5 a^2 x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 251, normalized size = 1.03 \[ -\frac {180 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{4}} \arctan \left (-\frac {729 \, a^{3} b^{4} \sqrt {x} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{4}} - \sqrt {-531441 \, a^{7} b^{5} \sqrt {-\frac {b^{5}}{a^{13}}} + 531441 \, b^{8} x} a^{3} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{4}}}{729 \, b^{5}}\right ) - 45 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{4}} \log \left (729 \, a^{10} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{4}} + 729 \, b^{4} \sqrt {x}\right ) + 45 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{4}} \log \left (-729 \, a^{10} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{4}} + 729 \, b^{4} \sqrt {x}\right ) - 4 \, {\left (45 \, b^{2} x^{4} + 36 \, a b x^{2} - 4 \, a^{2}\right )} \sqrt {x}}{40 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.66, size = 220, normalized size = 0.91 \[ \frac {b^{2} x^{\frac {3}{2}}}{2 \, {\left (b x^{2} + a\right )} a^{3}} + \frac {9 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{4} b} + \frac {9 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{4} b} - \frac {9 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{4} b} + \frac {9 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{4} b} + \frac {2 \, {\left (10 \, b x^{2} - a\right )}}{5 \, a^{3} x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 172, normalized size = 0.71 \[ \frac {b^{2} x^{\frac {3}{2}}}{2 \left (b \,x^{2}+a \right ) a^{3}}+\frac {9 \sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3}}+\frac {9 \sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3}}+\frac {9 \sqrt {2}\, b \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3}}+\frac {4 b}{a^{3} \sqrt {x}}-\frac {2}{5 a^{2} x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.04, size = 221, normalized size = 0.91 \[ \frac {45 \, b^{2} x^{4} + 36 \, a b x^{2} - 4 \, a^{2}}{10 \, {\left (a^{3} b x^{\frac {9}{2}} + a^{4} x^{\frac {5}{2}}\right )}} + \frac {9 \, b^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.69, size = 87, normalized size = 0.36 \[ \frac {\frac {18\,b\,x^2}{5\,a^2}-\frac {2}{5\,a}+\frac {9\,b^2\,x^4}{2\,a^3}}{a\,x^{5/2}+b\,x^{9/2}}-\frac {9\,{\left (-b\right )}^{5/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )}{4\,a^{13/4}}+\frac {9\,{\left (-b\right )}^{5/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )}{4\,a^{13/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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