Optimal. Leaf size=264 \[ -\frac {165 c^{7/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}}+\frac {165 c^{7/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}}-\frac {165 c^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{19/4}}+\frac {165 c^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} b^{19/4}}+\frac {55 c}{16 b^4 x^{3/2}}-\frac {165}{112 b^3 x^{7/2}}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.23, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {1584, 290, 325, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac {165 c^{7/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}}+\frac {165 c^{7/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}}-\frac {165 c^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{19/4}}+\frac {165 c^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} b^{19/4}}+\frac {55 c}{16 b^4 x^{3/2}}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}-\frac {165}{112 b^3 x^{7/2}}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 290
Rule 325
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {1}{x^{9/2} \left (b+c x^2\right )^3} \, dx\\ &=\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15 \int \frac {1}{x^{9/2} \left (b+c x^2\right )^2} \, dx}{8 b}\\ &=\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}+\frac {165 \int \frac {1}{x^{9/2} \left (b+c x^2\right )} \, dx}{32 b^2}\\ &=-\frac {165}{112 b^3 x^{7/2}}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}-\frac {(165 c) \int \frac {1}{x^{5/2} \left (b+c x^2\right )} \, dx}{32 b^3}\\ &=-\frac {165}{112 b^3 x^{7/2}}+\frac {55 c}{16 b^4 x^{3/2}}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}+\frac {\left (165 c^2\right ) \int \frac {1}{\sqrt {x} \left (b+c x^2\right )} \, dx}{32 b^4}\\ &=-\frac {165}{112 b^3 x^{7/2}}+\frac {55 c}{16 b^4 x^{3/2}}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}+\frac {\left (165 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{b+c x^4} \, dx,x,\sqrt {x}\right )}{16 b^4}\\ &=-\frac {165}{112 b^3 x^{7/2}}+\frac {55 c}{16 b^4 x^{3/2}}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}+\frac {\left (165 c^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^{9/2}}+\frac {\left (165 c^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^{9/2}}\\ &=-\frac {165}{112 b^3 x^{7/2}}+\frac {55 c}{16 b^4 x^{3/2}}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}+\frac {\left (165 c^{3/2}\right ) \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 )}{64 b^{9/2}}+\frac {\left (165 c^{3/2}\right ) \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 )}{64 b^{9/2}}-\frac {\left (165 c^{7/4}\right ) \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 )}{64 \sqrt {2} b^{19/4}}-\frac {\left (165 c^{7/4}\right ) \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 )}{64 \sqrt {2} b^{19/4}}\\ &=-\frac {165}{112 b^3 x^{7/2}}+\frac {55 c}{16 b^4 x^{3/2}}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}-\frac {165 c^{7/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}}+\frac {165 c^{7/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}}+\frac {\left (165 c^{7/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{19/4}}-\frac {\left (165 c^{7/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{19/4}}\\ &=-\frac {165}{112 b^3 x^{7/2}}+\frac {55 c}{16 b^4 x^{3/2}}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}-\frac {165 c^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{19/4}}+\frac {165 c^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{19/4}}-\frac {165 c^{7/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}}+\frac {165 c^{7/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 29, normalized size = 0.11 \[ -\frac {2 \, _2F_1\left (-\frac {7}{4},3;-\frac {3}{4};-\frac {c x^2}{b}\right )}{7 b^3 x^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 300, normalized size = 1.14 \[ \frac {4620 \, {\left (b^{4} c^{2} x^{8} + 2 \, b^{5} c x^{6} + b^{6} x^{4}\right )} \left (-\frac {c^{7}}{b^{19}}\right )^{\frac {1}{4}} \arctan \left (-\frac {b^{14} c^{2} \sqrt {x} \left (-\frac {c^{7}}{b^{19}}\right )^{\frac {3}{4}} - \sqrt {b^{10} \sqrt {-\frac {c^{7}}{b^{19}}} + c^{4} x} b^{14} \left (-\frac {c^{7}}{b^{19}}\right )^{\frac {3}{4}}}{c^{7}}\right ) + 1155 \, {\left (b^{4} c^{2} x^{8} + 2 \, b^{5} c x^{6} + b^{6} x^{4}\right )} \left (-\frac {c^{7}}{b^{19}}\right )^{\frac {1}{4}} \log \left (165 \, b^{5} \left (-\frac {c^{7}}{b^{19}}\right )^{\frac {1}{4}} + 165 \, c^{2} \sqrt {x}\right ) - 1155 \, {\left (b^{4} c^{2} x^{8} + 2 \, b^{5} c x^{6} + b^{6} x^{4}\right )} \left (-\frac {c^{7}}{b^{19}}\right )^{\frac {1}{4}} \log \left (-165 \, b^{5} \left (-\frac {c^{7}}{b^{19}}\right )^{\frac {1}{4}} + 165 \, c^{2} \sqrt {x}\right ) + 4 \, {\left (385 \, c^{3} x^{6} + 605 \, b c^{2} x^{4} + 160 \, b^{2} c x^{2} - 32 \, b^{3}\right )} \sqrt {x}}{448 \, {\left (b^{4} c^{2} x^{8} + 2 \, b^{5} c x^{6} + b^{6} x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 224, normalized size = 0.85 \[ \frac {165 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} c \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{5}} + \frac {165 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} c \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{5}} + \frac {165 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} c \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{5}} - \frac {165 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} c \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{5}} + \frac {23 \, c^{3} x^{\frac {5}{2}} + 27 \, b c^{2} \sqrt {x}}{16 \, {\left (c x^{2} + b\right )}^{2} b^{4}} + \frac {2 \, {\left (7 \, c x^{2} - b\right )}}{7 \, b^{4} x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 198, normalized size = 0.75 \[ \frac {23 c^{3} x^{\frac {5}{2}}}{16 \left (c \,x^{2}+b \right )^{2} b^{4}}+\frac {27 c^{2} \sqrt {x}}{16 \left (c \,x^{2}+b \right )^{2} b^{3}}+\frac {165 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{64 b^{5}}+\frac {165 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{64 b^{5}}+\frac {165 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, c^{2} \ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}\right )}{128 b^{5}}+\frac {2 c}{b^{4} x^{\frac {3}{2}}}-\frac {2}{7 b^{3} x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 246, normalized size = 0.93 \[ \frac {385 \, c^{3} x^{6} + 605 \, b c^{2} x^{4} + 160 \, b^{2} c x^{2} - 32 \, b^{3}}{112 \, {\left (b^{4} c^{2} x^{\frac {15}{2}} + 2 \, b^{5} c x^{\frac {11}{2}} + b^{6} x^{\frac {7}{2}}\right )}} + \frac {165 \, {\left (\frac {2 \, \sqrt {2} c^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {2 \, \sqrt {2} c^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {\sqrt {2} c^{\frac {7}{4}} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}}} - \frac {\sqrt {2} c^{\frac {7}{4}} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}}}\right )}}{128 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.35, size = 109, normalized size = 0.41 \[ \frac {\frac {10\,c\,x^2}{7\,b^2}-\frac {2}{7\,b}+\frac {605\,c^2\,x^4}{112\,b^3}+\frac {55\,c^3\,x^6}{16\,b^4}}{b^2\,x^{7/2}+c^2\,x^{15/2}+2\,b\,c\,x^{11/2}}+\frac {165\,{\left (-c\right )}^{7/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{32\,b^{19/4}}+\frac {165\,{\left (-c\right )}^{7/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{32\,b^{19/4}} \]
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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