Optimal. Leaf size=230 \[ -\frac {c^{9/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} b^{13/4}}+\frac {c^{9/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} b^{13/4}}+\frac {c^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{13/4}}-\frac {c^{9/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} b^{13/4}}-\frac {2 c^2}{b^3 \sqrt {x}}+\frac {2 c}{5 b^2 x^{5/2}}-\frac {2}{9 b x^{9/2}} \]
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Rubi [A] time = 0.22, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {1584, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac {2 c^2}{b^3 \sqrt {x}}-\frac {c^{9/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} b^{13/4}}+\frac {c^{9/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} b^{13/4}}+\frac {c^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{13/4}}-\frac {c^{9/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} b^{13/4}}+\frac {2 c}{5 b^2 x^{5/2}}-\frac {2}{9 b x^{9/2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 325
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1584
Rubi steps
\begin {align*} \int \frac {1}{x^{7/2} \left (b x^2+c x^4\right )} \, dx &=\int \frac {1}{x^{11/2} \left (b+c x^2\right )} \, dx\\ &=-\frac {2}{9 b x^{9/2}}-\frac {c \int \frac {1}{x^{7/2} \left (b+c x^2\right )} \, dx}{b}\\ &=-\frac {2}{9 b x^{9/2}}+\frac {2 c}{5 b^2 x^{5/2}}+\frac {c^2 \int \frac {1}{x^{3/2} \left (b+c x^2\right )} \, dx}{b^2}\\ &=-\frac {2}{9 b x^{9/2}}+\frac {2 c}{5 b^2 x^{5/2}}-\frac {2 c^2}{b^3 \sqrt {x}}-\frac {c^3 \int \frac {\sqrt {x}}{b+c x^2} \, dx}{b^3}\\ &=-\frac {2}{9 b x^{9/2}}+\frac {2 c}{5 b^2 x^{5/2}}-\frac {2 c^2}{b^3 \sqrt {x}}-\frac {\left (2 c^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=-\frac {2}{9 b x^{9/2}}+\frac {2 c}{5 b^2 x^{5/2}}-\frac {2 c^2}{b^3 \sqrt {x}}+\frac {c^{5/2} \operatorname {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{b^3}-\frac {c^{5/2} \operatorname {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=-\frac {2}{9 b x^{9/2}}+\frac {2 c}{5 b^2 x^{5/2}}-\frac {2 c^2}{b^3 \sqrt {x}}-\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{2 b^3}-\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{2 b^3}-\frac {c^{9/4} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} b^{13/4}}-\frac {c^{9/4} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} b^{13/4}}\\ &=-\frac {2}{9 b x^{9/2}}+\frac {2 c}{5 b^2 x^{5/2}}-\frac {2 c^2}{b^3 \sqrt {x}}-\frac {c^{9/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} b^{13/4}}+\frac {c^{9/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} b^{13/4}}-\frac {c^{9/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{13/4}}+\frac {c^{9/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{13/4}}\\ &=-\frac {2}{9 b x^{9/2}}+\frac {2 c}{5 b^2 x^{5/2}}-\frac {2 c^2}{b^3 \sqrt {x}}+\frac {c^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{13/4}}-\frac {c^{9/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{13/4}}-\frac {c^{9/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} b^{13/4}}+\frac {c^{9/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} b^{13/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 29, normalized size = 0.13 \[ -\frac {2 \, _2F_1\left (-\frac {9}{4},1;-\frac {5}{4};-\frac {c x^2}{b}\right )}{9 b x^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 204, normalized size = 0.89 \[ \frac {180 \, b^{3} x^{5} \left (-\frac {c^{9}}{b^{13}}\right )^{\frac {1}{4}} \arctan \left (-\frac {b^{3} c^{7} \sqrt {x} \left (-\frac {c^{9}}{b^{13}}\right )^{\frac {1}{4}} - \sqrt {-b^{7} c^{9} \sqrt {-\frac {c^{9}}{b^{13}}} + c^{14} x} b^{3} \left (-\frac {c^{9}}{b^{13}}\right )^{\frac {1}{4}}}{c^{9}}\right ) - 45 \, b^{3} x^{5} \left (-\frac {c^{9}}{b^{13}}\right )^{\frac {1}{4}} \log \left (b^{10} \left (-\frac {c^{9}}{b^{13}}\right )^{\frac {3}{4}} + c^{7} \sqrt {x}\right ) + 45 \, b^{3} x^{5} \left (-\frac {c^{9}}{b^{13}}\right )^{\frac {1}{4}} \log \left (-b^{10} \left (-\frac {c^{9}}{b^{13}}\right )^{\frac {3}{4}} + c^{7} \sqrt {x}\right ) - 4 \, {\left (45 \, c^{2} x^{4} - 9 \, b c x^{2} + 5 \, b^{2}\right )} \sqrt {x}}{90 \, b^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 199, normalized size = 0.87 \[ -\frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, b^{4}} - \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, b^{4}} + \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, b^{4}} - \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, b^{4}} - \frac {2 \, {\left (45 \, c^{2} x^{4} - 9 \, b c x^{2} + 5 \, b^{2}\right )}}{45 \, b^{3} x^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 169, normalized size = 0.73 \[ -\frac {\sqrt {2}\, c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{2 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{3}}-\frac {\sqrt {2}\, c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{2 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{3}}-\frac {\sqrt {2}\, c^{2} \ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}\right )}{4 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{3}}-\frac {2 c^{2}}{b^{3} \sqrt {x}}+\frac {2 c}{5 b^{2} x^{\frac {5}{2}}}-\frac {2}{9 b \,x^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.09, size = 209, normalized size = 0.91 \[ -\frac {c^{3} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{4 \, b^{3}} - \frac {2 \, {\left (45 \, c^{2} x^{4} - 9 \, b c x^{2} + 5 \, b^{2}\right )}}{45 \, b^{3} x^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.47, size = 77, normalized size = 0.33 \[ \frac {{\left (-c\right )}^{9/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{b^{13/4}}-\frac {{\left (-c\right )}^{9/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{b^{13/4}}-\frac {\frac {2}{9\,b}-\frac {2\,c\,x^2}{5\,b^2}+\frac {2\,c^2\,x^4}{b^3}}{x^{9/2}} \]
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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