Optimal. Leaf size=158 \[ \frac {8 a^2 \sqrt {a x+b x^3}}{77 b}+\frac {12}{77} a x^2 \sqrt {a x+b x^3}+\frac {2}{11} x \left (a x+b x^3\right )^{3/2}-\frac {4 a^{11/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{77 b^{5/4} \sqrt {a x+b x^3}} \]
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Rubi [A]
time = 0.09, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2029, 2046,
2049, 2036, 335, 226} \begin {gather*} -\frac {4 a^{11/4} \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 )}{77 b^{5/4} \sqrt {a x+b x^3}}+\frac {8 a^2 \sqrt {a x+b x^3}}{77 b}+\frac {2}{11} x \left (a x+b x^3\right )^{3/2}+\frac {12}{77} a x^2 \sqrt {a x+b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 335
Rule 2029
Rule 2036
Rule 2046
Rule 2049
Rubi steps
\begin {align*} \int \left (a x+b x^3\right )^{3/2} \, dx &=\frac {2}{11} x \left (a x+b x^3\right )^{3/2}+\frac {1}{11} (6 a) \int x \sqrt {a x+b x^3} \, dx\\ &=\frac {12}{77} a x^2 \sqrt {a x+b x^3}+\frac {2}{11} x \left (a x+b x^3\right )^{3/2}+\frac {1}{77} \left (12 a^2\right ) \int \frac {x^2}{\sqrt {a x+b x^3}} \, dx\\ &=\frac {8 a^2 \sqrt {a x+b x^3}}{77 b}+\frac {12}{77} a x^2 \sqrt {a x+b x^3}+\frac {2}{11} x \left (a x+b x^3\right )^{3/2}-\frac {\left (4 a^3\right ) \int \frac {1}{\sqrt {a x+b x^3}} \, dx}{77 b}\\ &=\frac {8 a^2 \sqrt {a x+b x^3}}{77 b}+\frac {12}{77} a x^2 \sqrt {a x+b x^3}+\frac {2}{11} x \left (a x+b x^3\right )^{3/2}-\frac {\left (4 a^3 \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x^2}} \, dx}{77 b \sqrt {a x+b x^3}}\\ &=\frac {8 a^2 \sqrt {a x+b x^3}}{77 b}+\frac {12}{77} a x^2 \sqrt {a x+b x^3}+\frac {2}{11} x \left (a x+b x^3\right )^{3/2}-\frac {\left (8 a^3 \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{77 b \sqrt {a x+b x^3}}\\ &=\frac {8 a^2 \sqrt {a x+b x^3}}{77 b}+\frac {12}{77} a x^2 \sqrt {a x+b x^3}+\frac {2}{11} x \left (a x+b x^3\right )^{3/2}-\frac {4 a^{11/4} \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 )}{77 b^{5/4} \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 = 10.03, size = 83, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {x \left (a+b x^2\right )} \left (\left (a+b x^2\right )^2 \sqrt {1+\frac {b x^2}{a}}-a^2 \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^2}{a}\right )\right )}{11 b \sqrt {1+\frac {b x^2}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 166, normalized size = 1.05
method | result | size |
risch | \(\frac {2 \left (7 b^{2} x^{4}+13 a b \,x^{2}+4 a^{2}\right ) x \left (b \,x^{2}+a \right )}{77 b \sqrt {x \left (b \,x^{2}+a \right )}}-\frac {4 a^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{77 b^{2} \sqrt {b \,x^{3}+a x}}\) | \(158\) |
default | \(\frac {2 b \,x^{4} \sqrt {b \,x^{3}+a x}}{11}+\frac {26 a \,x^{2} \sqrt {b \,x^{3}+a x}}{77}+\frac {8 a^{2} \sqrt {b \,x^{3}+a x}}{77 b}-\frac {4 a^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{77 b^{2} \sqrt {b \,x^{3}+a x}}\) | \(166\) |
elliptic | \(\frac {2 b \,x^{4} \sqrt {b \,x^{3}+a x}}{11}+\frac {26 a \,x^{2} \sqrt {b \,x^{3}+a x}}{77}+\frac {8 a^{2} \sqrt {b \,x^{3}+a x}}{77 b}-\frac {4 a^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{77 b^{2} \sqrt {b \,x^{3}+a x}}\) | \(166\) |
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 = 0.54, size = 60, normalized size = 0.38 \begin {gather*} -\frac {2 \, {\left (4 \, a^{3} \sqrt {b} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - {\left (7 \, b^{3} x^{4} + 13 \, a b^{2} x^{2} + 4 \, a^{2} b\right )} \sqrt {b x^{3} + a x}\right )}}{77 \, b^{2}} \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 \left (a x + b x^{3}\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 [B]
time = 5.00, size = 40, normalized size = 0.25 \begin {gather*} \frac {2\,x\,{\left (b\,x^3+a\,x\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {5}{4};\ \frac {9}{4};\ -\frac {b\,x^2}{a}\right )}{5\,{\left (\frac {b\,x^2}{a}+1\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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