Optimal. Leaf size=186 \[ -\frac {8 a^3 \sqrt {a x+b x^3}}{231 b^2}+\frac {8 a^2 x^2 \sqrt {a x+b x^3}}{385 b}+\frac {4}{55} a x^4 \sqrt {a x+b x^3}+\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2}+\frac {4 a^{15/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a x+b x^3}} \]
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Rubi [A]
time = 0.16, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2046, 2049,
2036, 335, 226} \begin {gather*} \frac {4 a^{15/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a x+b x^3}}-\frac {8 a^3 \sqrt {a x+b x^3}}{231 b^2}+\frac {8 a^2 x^2 \sqrt {a x+b x^3}}{385 b}+\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2}+\frac {4}{55} a x^4 \sqrt {a x+b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 335
Rule 2036
Rule 2046
Rule 2049
Rubi steps
\begin {align*} \int x^2 \left (a x+b x^3\right )^{3/2} \, dx &=\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2}+\frac {1}{5} (2 a) \int x^3 \sqrt {a x+b x^3} \, dx\\ &=\frac {4}{55} a x^4 \sqrt {a x+b x^3}+\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2}+\frac {1}{55} \left (4 a^2\right ) \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx\\ &=\frac {8 a^2 x^2 \sqrt {a x+b x^3}}{385 b}+\frac {4}{55} a x^4 \sqrt {a x+b x^3}+\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2}-\frac {\left (4 a^3\right ) \int \frac {x^2}{\sqrt {a x+b x^3}} \, dx}{77 b}\\ &=-\frac {8 a^3 \sqrt {a x+b x^3}}{231 b^2}+\frac {8 a^2 x^2 \sqrt {a x+b x^3}}{385 b}+\frac {4}{55} a x^4 \sqrt {a x+b x^3}+\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2}+\frac {\left (4 a^4\right ) \int \frac {1}{\sqrt {a x+b x^3}} \, dx}{231 b^2}\\ &=-\frac {8 a^3 \sqrt {a x+b x^3}}{231 b^2}+\frac {8 a^2 x^2 \sqrt {a x+b x^3}}{385 b}+\frac {4}{55} a x^4 \sqrt {a x+b x^3}+\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2}+\frac {\left (4 a^4 \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x^2}} \, dx}{231 b^2 \sqrt {a x+b x^3}}\\ &=-\frac {8 a^3 \sqrt {a x+b x^3}}{231 b^2}+\frac {8 a^2 x^2 \sqrt {a x+b x^3}}{385 b}+\frac {4}{55} a x^4 \sqrt {a x+b x^3}+\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2}+\frac {\left (8 a^4 \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{231 b^2 \sqrt {a x+b x^3}}\\ &=-\frac {8 a^3 \sqrt {a x+b x^3}}{231 b^2}+\frac {8 a^2 x^2 \sqrt {a x+b x^3}}{385 b}+\frac {4}{55} a x^4 \sqrt {a x+b x^3}+\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2}+\frac {4 a^{15/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a x+b x^3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.04, size = 94, normalized size = 0.51 \begin {gather*} \frac {2 \sqrt {x \left (a+b x^2\right )} \left (-\left (\left (5 a-11 b x^2\right ) \left (a+b x^2\right )^2 \sqrt {1+\frac {b x^2}{a}}\right )+5 a^3 \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^2}{a}\right )\right )}{165 b^2 \sqrt {1+\frac {b x^2}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 188, normalized size = 1.01
method | result | size |
risch | \(-\frac {2 \left (-77 b^{3} x^{6}-119 a \,b^{2} x^{4}-12 a^{2} b \,x^{2}+20 a^{3}\right ) x \left (b \,x^{2}+a \right )}{1155 b^{2} \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {4 a^{4} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{231 b^{3} \sqrt {b \,x^{3}+a x}}\) | \(169\) |
default | \(\frac {2 b \,x^{6} \sqrt {b \,x^{3}+a x}}{15}+\frac {34 a \,x^{4} \sqrt {b \,x^{3}+a x}}{165}+\frac {8 a^{2} x^{2} \sqrt {b \,x^{3}+a x}}{385 b}-\frac {8 a^{3} \sqrt {b \,x^{3}+a x}}{231 b^{2}}+\frac {4 a^{4} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{231 b^{3} \sqrt {b \,x^{3}+a x}}\) | \(188\) |
elliptic | \(\frac {2 b \,x^{6} \sqrt {b \,x^{3}+a x}}{15}+\frac {34 a \,x^{4} \sqrt {b \,x^{3}+a x}}{165}+\frac {8 a^{2} x^{2} \sqrt {b \,x^{3}+a x}}{385 b}-\frac {8 a^{3} \sqrt {b \,x^{3}+a x}}{231 b^{2}}+\frac {4 a^{4} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{231 b^{3} \sqrt {b \,x^{3}+a x}}\) | \(188\) |
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 = 0.66, size = 70, normalized size = 0.38 \begin {gather*} \frac {2 \, {\left (20 \, a^{4} \sqrt {b} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (77 \, b^{4} x^{6} + 119 \, a b^{3} x^{4} + 12 \, a^{2} b^{2} x^{2} - 20 \, a^{3} b\right )} \sqrt {b x^{3} + a x}\right )}}{1155 \, b^{3}} \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 x^{2} \left (x \left (a + b 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 = -0.01 \begin {gather*} \int x^2\,{\left (b\,x^3+a\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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