Optimal. Leaf size=251 \[ -\frac {45 \sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4}}+\frac {45 \sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4}}+\frac {45 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4}}-\frac {45 \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{13/4}}-\frac {45}{16 a^3 \sqrt {x}}+\frac {9}{16 a^2 \sqrt {x} \left (a+b x^2\right )}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.19, antiderivative size = 251, 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} \[ \frac {9}{16 a^2 \sqrt {x} \left (a+b x^2\right )}-\frac {45 \sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4}}+\frac {45 \sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4}}+\frac {45 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4}}-\frac {45 \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{13/4}}-\frac {45}{16 a^3 \sqrt {x}}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2} \]
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^{3/2} \left (a+b x^2\right )^3} \, dx &=\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}+\frac {9 \int \frac {1}{x^{3/2} \left (a+b x^2\right )^2} \, dx}{8 a}\\ &=\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}+\frac {9}{16 a^2 \sqrt {x} \left (a+b x^2\right )}+\frac {45 \int \frac {1}{x^{3/2} \left (a+b x^2\right )} \, dx}{32 a^2}\\ &=-\frac {45}{16 a^3 \sqrt {x}}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}+\frac {9}{16 a^2 \sqrt {x} \left (a+b x^2\right )}-\frac {(45 b) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{32 a^3}\\ &=-\frac {45}{16 a^3 \sqrt {x}}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}+\frac {9}{16 a^2 \sqrt {x} \left (a+b x^2\right )}-\frac {(45 b) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{16 a^3}\\ &=-\frac {45}{16 a^3 \sqrt {x}}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}+\frac {9}{16 a^2 \sqrt {x} \left (a+b x^2\right )}+\frac {\left (45 \sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^3}-\frac {\left (45 \sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^3}\\ &=-\frac {45}{16 a^3 \sqrt {x}}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}+\frac {9}{16 a^2 \sqrt {x} \left (a+b x^2\right )}-\frac {45 \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 )}{64 a^3}-\frac {45 \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 )}{64 a^3}-\frac {\left (45 \sqrt [4]{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 )}{64 \sqrt {2} a^{13/4}}-\frac {\left (45 \sqrt [4]{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 )}{64 \sqrt {2} a^{13/4}}\\ &=-\frac {45}{16 a^3 \sqrt {x}}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}+\frac {9}{16 a^2 \sqrt {x} \left (a+b x^2\right )}-\frac {45 \sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4}}+\frac {45 \sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4}}-\frac {\left (45 \sqrt [4]{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 )}{32 \sqrt {2} a^{13/4}}+\frac {\left (45 \sqrt [4]{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 )}{32 \sqrt {2} a^{13/4}}\\ &=-\frac {45}{16 a^3 \sqrt {x}}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}+\frac {9}{16 a^2 \sqrt {x} \left (a+b x^2\right )}+\frac {45 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4}}-\frac {45 \sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4}}-\frac {45 \sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4}}+\frac {45 \sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 27, normalized size = 0.11 \[ -\frac {2 \, _2F_1\left (-\frac {1}{4},3;\frac {3}{4};-\frac {b x^2}{a}\right )}{a^3 \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 263, normalized size = 1.05 \[ \frac {180 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{4}} \arctan \left (-\frac {91125 \, a^{3} b \sqrt {x} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{4}} - \sqrt {-8303765625 \, a^{7} b \sqrt {-\frac {b}{a^{13}}} + 8303765625 \, b^{2} x} a^{3} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{4}}}{91125 \, b}\right ) - 45 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{4}} \log \left (91125 \, a^{10} \left (-\frac {b}{a^{13}}\right )^{\frac {3}{4}} + 91125 \, b \sqrt {x}\right ) + 45 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{4}} \log \left (-91125 \, a^{10} \left (-\frac {b}{a^{13}}\right )^{\frac {3}{4}} + 91125 \, b \sqrt {x}\right ) - 4 \, {\left (45 \, b^{2} x^{4} + 81 \, a b x^{2} + 32 \, a^{2}\right )} \sqrt {x}}{64 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 220, normalized size = 0.88 \[ -\frac {2}{a^{3} \sqrt {x}} - \frac {45 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{4} b^{2}} - \frac {45 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{4} b^{2}} + \frac {45 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{4} b^{2}} - \frac {45 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{4} b^{2}} - \frac {13 \, b^{2} x^{\frac {7}{2}} + 17 \, a b x^{\frac {3}{2}}}{16 \, {\left (b x^{2} + a\right )}^{2} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 178, normalized size = 0.71 \[ -\frac {13 b^{2} x^{\frac {7}{2}}}{16 \left (b \,x^{2}+a \right )^{2} a^{3}}-\frac {17 b \,x^{\frac {3}{2}}}{16 \left (b \,x^{2}+a \right )^{2} a^{2}}-\frac {45 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{64 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3}}-\frac {45 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{64 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3}}-\frac {45 \sqrt {2}\, \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3}}-\frac {2}{a^{3} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 230, normalized size = 0.92 \[ -\frac {45 \, b^{2} x^{4} + 81 \, a b x^{2} + 32 \, a^{2}}{16 \, {\left (a^{3} b^{2} x^{\frac {9}{2}} + 2 \, a^{4} b x^{\frac {5}{2}} + a^{5} \sqrt {x}\right )}} - \frac {45 \, b {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{128 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 99, normalized size = 0.39 \[ \frac {45\,{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )}{32\,a^{13/4}}-\frac {45\,{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )}{32\,a^{13/4}}-\frac {\frac {2}{a}+\frac {81\,b\,x^2}{16\,a^2}+\frac {45\,b^2\,x^4}{16\,a^3}}{a^2\,\sqrt {x}+b^2\,x^{9/2}+2\,a\,b\,x^{5/2}} \]
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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