Optimal. Leaf size=203 \[ \frac {4 c^{15/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{231 b^{9/4} \sqrt {b x^2+c x^4}}+\frac {8 c^3 \sqrt {b x^2+c x^4}}{231 b^2 x^{5/2}}-\frac {8 c^2 \sqrt {b x^2+c x^4}}{385 b x^{9/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{55 x^{13/2}}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}} \]
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Rubi [A] time = 0.30, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2020, 2025, 2032, 329, 220} \[ \frac {8 c^3 \sqrt {b x^2+c x^4}}{231 b^2 x^{5/2}}+\frac {4 c^{15/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{231 b^{9/4} \sqrt {b x^2+c x^4}}-\frac {8 c^2 \sqrt {b x^2+c x^4}}{385 b x^{9/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{55 x^{13/2}}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2020
Rule 2025
Rule 2032
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{23/2}} \, dx &=-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}}+\frac {1}{5} (2 c) \int \frac {\sqrt {b x^2+c x^4}}{x^{15/2}} \, dx\\ &=-\frac {4 c \sqrt {b x^2+c x^4}}{55 x^{13/2}}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}}+\frac {1}{55} \left (4 c^2\right ) \int \frac {1}{x^{7/2} \sqrt {b x^2+c x^4}} \, dx\\ &=-\frac {4 c \sqrt {b x^2+c x^4}}{55 x^{13/2}}-\frac {8 c^2 \sqrt {b x^2+c x^4}}{385 b x^{9/2}}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}}-\frac {\left (4 c^3\right ) \int \frac {1}{x^{3/2} \sqrt {b x^2+c x^4}} \, dx}{77 b}\\ &=-\frac {4 c \sqrt {b x^2+c x^4}}{55 x^{13/2}}-\frac {8 c^2 \sqrt {b x^2+c x^4}}{385 b x^{9/2}}+\frac {8 c^3 \sqrt {b x^2+c x^4}}{231 b^2 x^{5/2}}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}}+\frac {\left (4 c^4\right ) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{231 b^2}\\ &=-\frac {4 c \sqrt {b x^2+c x^4}}{55 x^{13/2}}-\frac {8 c^2 \sqrt {b x^2+c x^4}}{385 b x^{9/2}}+\frac {8 c^3 \sqrt {b x^2+c x^4}}{231 b^2 x^{5/2}}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}}+\frac {\left (4 c^4 x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{231 b^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {4 c \sqrt {b x^2+c x^4}}{55 x^{13/2}}-\frac {8 c^2 \sqrt {b x^2+c x^4}}{385 b x^{9/2}}+\frac {8 c^3 \sqrt {b x^2+c x^4}}{231 b^2 x^{5/2}}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}}+\frac {\left (8 c^4 x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{231 b^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {4 c \sqrt {b x^2+c x^4}}{55 x^{13/2}}-\frac {8 c^2 \sqrt {b x^2+c x^4}}{385 b x^{9/2}}+\frac {8 c^3 \sqrt {b x^2+c x^4}}{231 b^2 x^{5/2}}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}}+\frac {4 c^{15/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{231 b^{9/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 58, normalized size = 0.29 \[ -\frac {2 b \sqrt {x^2 \left (b+c x^2\right )} \, _2F_1\left (-\frac {15}{4},-\frac {3}{2};-\frac {11}{4};-\frac {c x^2}{b}\right )}{15 x^{17/2} \sqrt {\frac {c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} {\left (c x^{2} + b\right )}}{x^{\frac {19}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{x^{\frac {23}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 167, normalized size = 0.82 \[ \frac {2 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (20 c^{4} x^{8}+10 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \sqrt {-b c}\, c^{3} x^{7} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+8 b \,c^{3} x^{6}-131 b^{2} c^{2} x^{4}-196 b^{3} c \,x^{2}-77 b^{4}\right )}{1155 \left (c \,x^{2}+b \right )^{2} b^{2} x^{\frac {21}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{x^{\frac {23}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^{23/2}} \,d x \]
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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