Optimal. Leaf size=59 \[ -\frac {2 A b^2}{\sqrt {x}}+\frac {2}{3} c x^{3/2} (A c+2 b B)+2 b \sqrt {x} (2 A c+b B)+\frac {2}{5} B c^2 x^{5/2} \]
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Rubi [A] time = 0.03, antiderivative size = 59, 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*} -\frac {2 A b^2}{\sqrt {x}}+\frac {2}{3} c x^{3/2} (A c+2 b B)+2 b \sqrt {x} (2 A c+b B)+\frac {2}{5} B c^2 x^{5/2} \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^{7/2}} \, dx &=\int \left (\frac {A b^2}{x^{3/2}}+\frac {b (b B+2 A c)}{\sqrt {x}}+c (2 b B+A c) \sqrt {x}+B c^2 x^{3/2}\right ) \, dx\\ &=-\frac {2 A b^2}{\sqrt {x}}+2 b (b B+2 A c) \sqrt {x}+\frac {2}{3} c (2 b B+A c) x^{3/2}+\frac {2}{5} B c^2 x^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 54, normalized size = 0.92 \begin {gather*} \frac {10 A \left (-3 b^2+6 b c x+c^2 x^2\right )+2 B x \left (15 b^2+10 b c x+3 c^2 x^2\right )}{15 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 55, normalized size = 0.93 \begin {gather*} \frac {2 \left (-15 A b^2+30 A b c x+5 A c^2 x^2+15 b^2 B x+10 b B c x^2+3 B c^2 x^3\right )}{15 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 51, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left (3 \, B c^{2} x^{3} - 15 \, A b^{2} + 5 \, {\left (2 \, B b c + A c^{2}\right )} x^{2} + 15 \, {\left (B b^{2} + 2 \, A b c\right )} x\right )}}{15 \, \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 53, normalized size = 0.90 \begin {gather*} \frac {2}{5} \, B c^{2} x^{\frac {5}{2}} + \frac {4}{3} \, B b c x^{\frac {3}{2}} + \frac {2}{3} \, A c^{2} x^{\frac {3}{2}} + 2 \, B b^{2} \sqrt {x} + 4 \, A b c \sqrt {x} - \frac {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 = 0.88 \begin {gather*} -\frac {2 \left (-3 B \,c^{2} x^{3}-5 A \,c^{2} x^{2}-10 B b c \,x^{2}-30 A b c x -15 B \,b^{2} x +15 A \,b^{2}\right )}{15 \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 51, normalized size = 0.86 \begin {gather*} \frac {2}{5} \, B c^{2} x^{\frac {5}{2}} - \frac {2 \, A b^{2}}{\sqrt {x}} + \frac {2}{3} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {3}{2}} + 2 \, {\left (B b^{2} + 2 \, A b c\right )} \sqrt {x} \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 = 0.86 \begin {gather*} \sqrt {x}\,\left (2\,B\,b^2+4\,A\,c\,b\right )+x^{3/2}\,\left (\frac {2\,A\,c^2}{3}+\frac {4\,B\,b\,c}{3}\right )-\frac {2\,A\,b^2}{\sqrt {x}}+\frac {2\,B\,c^2\,x^{5/2}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.90, size = 75, normalized size = 1.27 \begin {gather*} - \frac {2 A b^{2}}{\sqrt {x}} + 4 A b c \sqrt {x} + \frac {2 A c^{2} x^{\frac {3}{2}}}{3} + 2 B b^{2} \sqrt {x} + \frac {4 B b c x^{\frac {3}{2}}}{3} + \frac {2 B c^{2} x^{\frac {5}{2}}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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