Optimal. Leaf size=203 \[ \frac {4 b^{15/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{231 c^{9/4} \sqrt {b x^2+c x^4}}-\frac {8 b^3 \sqrt {b x^2+c x^4}}{231 c^2 \sqrt {x}}+\frac {8 b^2 x^{3/2} \sqrt {b x^2+c x^4}}{385 c}+\frac {4}{55} b x^{7/2} \sqrt {b x^2+c x^4}+\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2} \]
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Rubi [A] time = 0.29, 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 = {2021, 2024, 2032, 329, 220} \[ -\frac {8 b^3 \sqrt {b x^2+c x^4}}{231 c^2 \sqrt {x}}+\frac {4 b^{15/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{231 c^{9/4} \sqrt {b x^2+c x^4}}+\frac {8 b^2 x^{3/2} \sqrt {b x^2+c x^4}}{385 c}+\frac {4}{55} b x^{7/2} \sqrt {b x^2+c x^4}+\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2021
Rule 2024
Rule 2032
Rubi steps
\begin {align*} \int \sqrt {x} \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2}+\frac {1}{5} (2 b) \int x^{5/2} \sqrt {b x^2+c x^4} \, dx\\ &=\frac {4}{55} b x^{7/2} \sqrt {b x^2+c x^4}+\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2}+\frac {1}{55} \left (4 b^2\right ) \int \frac {x^{9/2}}{\sqrt {b x^2+c x^4}} \, dx\\ &=\frac {8 b^2 x^{3/2} \sqrt {b x^2+c x^4}}{385 c}+\frac {4}{55} b x^{7/2} \sqrt {b x^2+c x^4}+\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2}-\frac {\left (4 b^3\right ) \int \frac {x^{5/2}}{\sqrt {b x^2+c x^4}} \, dx}{77 c}\\ &=-\frac {8 b^3 \sqrt {b x^2+c x^4}}{231 c^2 \sqrt {x}}+\frac {8 b^2 x^{3/2} \sqrt {b x^2+c x^4}}{385 c}+\frac {4}{55} b x^{7/2} \sqrt {b x^2+c x^4}+\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2}+\frac {\left (4 b^4\right ) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{231 c^2}\\ &=-\frac {8 b^3 \sqrt {b x^2+c x^4}}{231 c^2 \sqrt {x}}+\frac {8 b^2 x^{3/2} \sqrt {b x^2+c x^4}}{385 c}+\frac {4}{55} b x^{7/2} \sqrt {b x^2+c x^4}+\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2}+\frac {\left (4 b^4 x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{231 c^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {8 b^3 \sqrt {b x^2+c x^4}}{231 c^2 \sqrt {x}}+\frac {8 b^2 x^{3/2} \sqrt {b x^2+c x^4}}{385 c}+\frac {4}{55} b x^{7/2} \sqrt {b x^2+c x^4}+\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2}+\frac {\left (8 b^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 c^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {8 b^3 \sqrt {b x^2+c x^4}}{231 c^2 \sqrt {x}}+\frac {8 b^2 x^{3/2} \sqrt {b x^2+c x^4}}{385 c}+\frac {4}{55} b x^{7/2} \sqrt {b x^2+c x^4}+\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2}+\frac {4 b^{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 c^{9/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 101, normalized size = 0.50 \[ \frac {2 \sqrt {x^2 \left (b+c x^2\right )} \left (5 b^3 \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^2}{b}\right )-\left (5 b-11 c x^2\right ) \left (b+c x^2\right )^2 \sqrt {\frac {c x^2}{b}+1}\right )}{165 c^2 \sqrt {x} \sqrt {\frac {c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} \sqrt {x}, x\right ) \]
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 \[ \int {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} \sqrt {x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 168, normalized size = 0.83 \[ \frac {2 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (77 c^{5} x^{9}+196 b \,c^{4} x^{7}+131 b^{2} c^{3} x^{5}-8 b^{3} c^{2} x^{3}-20 b^{4} c x +10 \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, b^{4} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )\right )}{1155 \left (c \,x^{2}+b \right )^{2} c^{3} x^{\frac {7}{2}}} \]
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 \[ \int {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} \sqrt {x}\,{d x} \]
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 \[ \int \sqrt {x}\,{\left (c\,x^4+b\,x^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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