Optimal. Leaf size=63 \[ \frac {2}{3} A b^2 x^{3/2}+\frac {2}{11} c x^{11/2} (A c+2 b B)+\frac {2}{7} b x^{7/2} (2 A c+b B)+\frac {2}{15} B c^2 x^{15/2} \]
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Rubi [A] time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1584, 448} \[ \frac {2}{3} A b^2 x^{3/2}+\frac {2}{11} c x^{11/2} (A c+2 b B)+\frac {2}{7} b x^{7/2} (2 A c+b B)+\frac {2}{15} B c^2 x^{15/2} \]
Antiderivative was successfully verified.
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Rule 448
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{x^{7/2}} \, dx &=\int \sqrt {x} \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2 \sqrt {x}+b (b B+2 A c) x^{5/2}+c (2 b B+A c) x^{9/2}+B c^2 x^{13/2}\right ) \, dx\\ &=\frac {2}{3} A b^2 x^{3/2}+\frac {2}{7} b (b B+2 A c) x^{7/2}+\frac {2}{11} c (2 b B+A c) x^{11/2}+\frac {2}{15} B c^2 x^{15/2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 0.84 \[ \frac {2 x^{3/2} \left (385 A b^2+105 c x^4 (A c+2 b B)+165 b x^2 (2 A c+b B)+77 B c^2 x^6\right )}{1155} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 54, normalized size = 0.86 \[ \frac {2}{1155} \, {\left (77 \, B c^{2} x^{7} + 105 \, {\left (2 \, B b c + A c^{2}\right )} x^{5} + 385 \, A b^{2} x + 165 \, {\left (B b^{2} + 2 \, A b c\right )} x^{3}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 53, normalized size = 0.84 \[ \frac {2}{15} \, B c^{2} x^{\frac {15}{2}} + \frac {4}{11} \, B b c x^{\frac {11}{2}} + \frac {2}{11} \, A c^{2} x^{\frac {11}{2}} + \frac {2}{7} \, B b^{2} x^{\frac {7}{2}} + \frac {4}{7} \, A b c x^{\frac {7}{2}} + \frac {2}{3} \, A b^{2} x^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 56, normalized size = 0.89 \[ \frac {2 \left (77 B \,c^{2} x^{6}+105 A \,c^{2} x^{4}+210 B b c \,x^{4}+330 A b c \,x^{2}+165 B \,b^{2} x^{2}+385 b^{2} A \right ) x^{\frac {3}{2}}}{1155} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 51, normalized size = 0.81 \[ \frac {2}{15} \, B c^{2} x^{\frac {15}{2}} + \frac {2}{11} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {11}{2}} + \frac {2}{3} \, A b^{2} x^{\frac {3}{2}} + \frac {2}{7} \, {\left (B b^{2} + 2 \, A b c\right )} x^{\frac {7}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 51, normalized size = 0.81 \[ x^{7/2}\,\left (\frac {2\,B\,b^2}{7}+\frac {4\,A\,c\,b}{7}\right )+x^{11/2}\,\left (\frac {2\,A\,c^2}{11}+\frac {4\,B\,b\,c}{11}\right )+\frac {2\,A\,b^2\,x^{3/2}}{3}+\frac {2\,B\,c^2\,x^{15/2}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.46, size = 80, normalized size = 1.27 \[ \frac {2 A b^{2} x^{\frac {3}{2}}}{3} + \frac {4 A b c x^{\frac {7}{2}}}{7} + \frac {2 A c^{2} x^{\frac {11}{2}}}{11} + \frac {2 B b^{2} x^{\frac {7}{2}}}{7} + \frac {4 B b c x^{\frac {11}{2}}}{11} + \frac {2 B c^{2} x^{\frac {15}{2}}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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