Optimal. Leaf size=203 \[ -\frac {8 b^3 \sqrt {b x^2+c x^4}}{231 c^2 \sqrt {x}}+\frac {8 b^2 x^{3/2} \sqrt {b x^2+c x^4}}{385 c}+\frac {4}{55} b x^{7/2} \sqrt {b x^2+c x^4}+\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2}+\frac {4 b^{15/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{231 c^{9/4} \sqrt {b x^2+c x^4}} \]
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Rubi [A]
time = 0.18, 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 = {2046, 2049,
2057, 335, 226} \begin {gather*} \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 \text {ArcTan}\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 335
Rule 2046
Rule 2049
Rule 2057
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 ) \text {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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.05, size = 101, normalized size = 0.50 \begin {gather*} \frac {2 \sqrt {x^2 \left (b+c x^2\right )} \left (-\left (\left (5 b-11 c x^2\right ) \left (b+c x^2\right )^2 \sqrt {1+\frac {c x^2}{b}}\right )+5 b^3 \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^2}{b}\right )\right )}{165 c^2 \sqrt {x} \sqrt {1+\frac {c x^2}{b}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 168, normalized size = 0.83
method | result | size |
default | \(\frac {2 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (77 c^{5} x^{9}+196 b \,c^{4} x^{7}+10 b^{4} \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+131 b^{2} c^{3} x^{5}-8 b^{3} c^{2} x^{3}-20 b^{4} c x \right )}{1155 x^{\frac {7}{2}} \left (c \,x^{2}+b \right )^{2} c^{3}}\) | \(168\) |
risch | \(-\frac {2 \left (-77 c^{3} x^{6}-119 b \,c^{2} x^{4}-12 b^{2} c \,x^{2}+20 b^{3}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{1155 \sqrt {x}\, c^{2}}+\frac {4 b^{4} \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {x \left (c \,x^{2}+b \right )}}{231 c^{3} \sqrt {c \,x^{3}+b x}\, x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}\) | \(202\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.10, size = 79, normalized size = 0.39 \begin {gather*} \frac {2 \, {\left (20 \, b^{4} \sqrt {c} x {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) + {\left (77 \, c^{4} x^{6} + 119 \, b c^{3} x^{4} + 12 \, b^{2} c^{2} x^{2} - 20 \, b^{3} c\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{1155 \, c^{3} x} \end {gather*}
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 \begin {gather*} \int \sqrt {x} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sqrt {x}\,{\left (c\,x^4+b\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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