Optimal. Leaf size=61 \[ 2 A b^2 \sqrt {x}+\frac {2}{5} c x^{5/2} (A c+2 b B)+\frac {2}{3} b x^{3/2} (2 A c+b B)+\frac {2}{7} B c^2 x^{7/2} \]
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Rubi [A] time = 0.03, antiderivative size = 61, 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 = {765} \begin {gather*} 2 A b^2 \sqrt {x}+\frac {2}{5} c x^{5/2} (A c+2 b B)+\frac {2}{3} b x^{3/2} (2 A c+b B)+\frac {2}{7} B c^2 x^{7/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^2}{x^{5/2}} \, dx &=\int \left (\frac {A b^2}{\sqrt {x}}+b (b B+2 A c) \sqrt {x}+c (2 b B+A c) x^{3/2}+B c^2 x^{5/2}\right ) \, dx\\ &=2 A b^2 \sqrt {x}+\frac {2}{3} b (b B+2 A c) x^{3/2}+\frac {2}{5} c (2 b B+A c) x^{5/2}+\frac {2}{7} B c^2 x^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 54, normalized size = 0.89 \begin {gather*} \frac {2}{105} \sqrt {x} \left (7 A \left (15 b^2+10 b c x+3 c^2 x^2\right )+B x \left (35 b^2+42 b c x+15 c^2 x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 69, normalized size = 1.13 \begin {gather*} \frac {2}{105} \left (105 A b^2 \sqrt {x}+70 A b c x^{3/2}+21 A c^2 x^{5/2}+35 b^2 B x^{3/2}+42 b B c x^{5/2}+15 B c^2 x^{7/2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 51, normalized size = 0.84 \begin {gather*} \frac {2}{105} \, {\left (15 \, B c^{2} x^{3} + 105 \, A b^{2} + 21 \, {\left (2 \, B b c + A c^{2}\right )} x^{2} + 35 \, {\left (B b^{2} + 2 \, A b c\right )} x\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 53, normalized size = 0.87 \begin {gather*} \frac {2}{7} \, B c^{2} x^{\frac {7}{2}} + \frac {4}{5} \, B b c x^{\frac {5}{2}} + \frac {2}{5} \, A c^{2} x^{\frac {5}{2}} + \frac {2}{3} \, B b^{2} x^{\frac {3}{2}} + \frac {4}{3} \, A b c x^{\frac {3}{2}} + 2 \, A b^{2} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 52, normalized size = 0.85 \begin {gather*} \frac {2 \left (15 B \,c^{2} x^{3}+21 A \,c^{2} x^{2}+42 B b c \,x^{2}+70 A b c x +35 B \,b^{2} x +105 A \,b^{2}\right ) \sqrt {x}}{105} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 51, normalized size = 0.84 \begin {gather*} \frac {2}{7} \, B c^{2} x^{\frac {7}{2}} + 2 \, A b^{2} \sqrt {x} + \frac {2}{5} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (B b^{2} + 2 \, A b c\right )} x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 51, normalized size = 0.84 \begin {gather*} x^{3/2}\,\left (\frac {2\,B\,b^2}{3}+\frac {4\,A\,c\,b}{3}\right )+x^{5/2}\,\left (\frac {2\,A\,c^2}{5}+\frac {4\,B\,b\,c}{5}\right )+2\,A\,b^2\,\sqrt {x}+\frac {2\,B\,c^2\,x^{7/2}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.84, size = 78, normalized size = 1.28 \begin {gather*} 2 A b^{2} \sqrt {x} + \frac {4 A b c x^{\frac {3}{2}}}{3} + \frac {2 A c^{2} x^{\frac {5}{2}}}{5} + \frac {2 B b^{2} x^{\frac {3}{2}}}{3} + \frac {4 B b c x^{\frac {5}{2}}}{5} + \frac {2 B c^{2} x^{\frac {7}{2}}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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