Optimal. Leaf size=83 \[ 2 a^3 A \sqrt {x}+\frac {2}{5} a^2 x^{5/2} (a B+3 A b)+\frac {2}{13} b^2 x^{13/2} (3 a B+A b)+\frac {2}{3} a b x^{9/2} (a B+A b)+\frac {2}{17} b^3 B x^{17/2} \]
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Rubi [A] time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {448} \[ \frac {2}{5} a^2 x^{5/2} (a B+3 A b)+2 a^3 A \sqrt {x}+\frac {2}{13} b^2 x^{13/2} (3 a B+A b)+\frac {2}{3} a b x^{9/2} (a B+A b)+\frac {2}{17} b^3 B x^{17/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{\sqrt {x}} \, dx &=\int \left (\frac {a^3 A}{\sqrt {x}}+a^2 (3 A b+a B) x^{3/2}+3 a b (A b+a B) x^{7/2}+b^2 (A b+3 a B) x^{11/2}+b^3 B x^{15/2}\right ) \, dx\\ &=2 a^3 A \sqrt {x}+\frac {2}{5} a^2 (3 A b+a B) x^{5/2}+\frac {2}{3} a b (A b+a B) x^{9/2}+\frac {2}{13} b^2 (A b+3 a B) x^{13/2}+\frac {2}{17} b^3 B x^{17/2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 71, normalized size = 0.86 \[ \frac {2 \sqrt {x} \left (3315 a^3 A+663 a^2 x^2 (a B+3 A b)+255 b^2 x^6 (3 a B+A b)+1105 a b x^4 (a B+A b)+195 b^3 B x^8\right )}{3315} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 75, normalized size = 0.90 \[ \frac {2}{3315} \, {\left (195 \, B b^{3} x^{8} + 255 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 1105 \, {\left (B a^{2} b + A a b^{2}\right )} x^{4} + 3315 \, A a^{3} + 663 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 77, normalized size = 0.93 \[ \frac {2}{17} \, B b^{3} x^{\frac {17}{2}} + \frac {6}{13} \, B a b^{2} x^{\frac {13}{2}} + \frac {2}{13} \, A b^{3} x^{\frac {13}{2}} + \frac {2}{3} \, B a^{2} b x^{\frac {9}{2}} + \frac {2}{3} \, A a b^{2} x^{\frac {9}{2}} + \frac {2}{5} \, B a^{3} x^{\frac {5}{2}} + \frac {6}{5} \, A a^{2} b x^{\frac {5}{2}} + 2 \, A a^{3} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 80, normalized size = 0.96 \[ \frac {2 \left (195 B \,b^{3} x^{8}+255 x^{6} A \,b^{3}+765 B a \,b^{2} x^{6}+1105 x^{4} A a \,b^{2}+1105 x^{4} B \,a^{2} b +1989 A \,a^{2} b \,x^{2}+663 B \,a^{3} x^{2}+3315 A \,a^{3}\right ) \sqrt {x}}{3315} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 73, normalized size = 0.88 \[ \frac {2}{17} \, B b^{3} x^{\frac {17}{2}} + \frac {2}{13} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {13}{2}} + \frac {2}{3} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {9}{2}} + 2 \, A a^{3} \sqrt {x} + \frac {2}{5} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 69, normalized size = 0.83 \[ x^{5/2}\,\left (\frac {2\,B\,a^3}{5}+\frac {6\,A\,b\,a^2}{5}\right )+x^{13/2}\,\left (\frac {2\,A\,b^3}{13}+\frac {6\,B\,a\,b^2}{13}\right )+2\,A\,a^3\,\sqrt {x}+\frac {2\,B\,b^3\,x^{17/2}}{17}+\frac {2\,a\,b\,x^{9/2}\,\left (A\,b+B\,a\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.82, size = 112, normalized size = 1.35 \[ 2 A a^{3} \sqrt {x} + \frac {6 A a^{2} b x^{\frac {5}{2}}}{5} + \frac {2 A a b^{2} x^{\frac {9}{2}}}{3} + \frac {2 A b^{3} x^{\frac {13}{2}}}{13} + \frac {2 B a^{3} x^{\frac {5}{2}}}{5} + \frac {2 B a^{2} b x^{\frac {9}{2}}}{3} + \frac {6 B a b^{2} x^{\frac {13}{2}}}{13} + \frac {2 B b^{3} x^{\frac {17}{2}}}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
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