Optimal. Leaf size=126 \[ -\frac {3 (5 b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 \sqrt {b} c^{7/2}}+\frac {3 \sqrt {x} (5 b B-A c)}{4 b c^3}-\frac {x^{3/2} (5 b B-A c)}{4 b c^2 (b+c x)}-\frac {x^{5/2} (b B-A c)}{2 b c (b+c x)^2} \]
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Rubi [A] time = 0.06, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {781, 78, 47, 50, 63, 205} \begin {gather*} -\frac {x^{3/2} (5 b B-A c)}{4 b c^2 (b+c x)}+\frac {3 \sqrt {x} (5 b B-A c)}{4 b c^3}-\frac {3 (5 b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 \sqrt {b} c^{7/2}}-\frac {x^{5/2} (b B-A c)}{2 b c (b+c x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 78
Rule 205
Rule 781
Rubi steps
\begin {align*} \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx &=\int \frac {x^{3/2} (A+B x)}{(b+c x)^3} \, dx\\ &=-\frac {(b B-A c) x^{5/2}}{2 b c (b+c x)^2}-\frac {\left (-\frac {5 b B}{2}+\frac {A c}{2}\right ) \int \frac {x^{3/2}}{(b+c x)^2} \, dx}{2 b c}\\ &=-\frac {(b B-A c) x^{5/2}}{2 b c (b+c x)^2}-\frac {(5 b B-A c) x^{3/2}}{4 b c^2 (b+c x)}+\frac {(3 (5 b B-A c)) \int \frac {\sqrt {x}}{b+c x} \, dx}{8 b c^2}\\ &=\frac {3 (5 b B-A c) \sqrt {x}}{4 b c^3}-\frac {(b B-A c) x^{5/2}}{2 b c (b+c x)^2}-\frac {(5 b B-A c) x^{3/2}}{4 b c^2 (b+c x)}-\frac {(3 (5 b B-A c)) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{8 c^3}\\ &=\frac {3 (5 b B-A c) \sqrt {x}}{4 b c^3}-\frac {(b B-A c) x^{5/2}}{2 b c (b+c x)^2}-\frac {(5 b B-A c) x^{3/2}}{4 b c^2 (b+c x)}-\frac {(3 (5 b B-A c)) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{4 c^3}\\ &=\frac {3 (5 b B-A c) \sqrt {x}}{4 b c^3}-\frac {(b B-A c) x^{5/2}}{2 b c (b+c x)^2}-\frac {(5 b B-A c) x^{3/2}}{4 b c^2 (b+c x)}-\frac {3 (5 b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 \sqrt {b} c^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 61, normalized size = 0.48 \begin {gather*} \frac {x^{5/2} \left (\frac {5 b^2 (A c-b B)}{(b+c x)^2}+(5 b B-A c) \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};-\frac {c x}{b}\right )\right )}{10 b^3 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 94, normalized size = 0.75 \begin {gather*} \frac {\sqrt {x} \left (-3 A b c-5 A c^2 x+15 b^2 B+25 b B c x+8 B c^2 x^2\right )}{4 c^3 (b+c x)^2}-\frac {3 (5 b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 \sqrt {b} c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 319, normalized size = 2.53 \begin {gather*} \left [\frac {3 \, {\left (5 \, B b^{3} - A b^{2} c + {\left (5 \, B b c^{2} - A c^{3}\right )} x^{2} + 2 \, {\left (5 \, B b^{2} c - A b c^{2}\right )} x\right )} \sqrt {-b c} \log \left (\frac {c x - b - 2 \, \sqrt {-b c} \sqrt {x}}{c x + b}\right ) + 2 \, {\left (8 \, B b c^{3} x^{2} + 15 \, B b^{3} c - 3 \, A b^{2} c^{2} + 5 \, {\left (5 \, B b^{2} c^{2} - A b c^{3}\right )} x\right )} \sqrt {x}}{8 \, {\left (b c^{6} x^{2} + 2 \, b^{2} c^{5} x + b^{3} c^{4}\right )}}, \frac {3 \, {\left (5 \, B b^{3} - A b^{2} c + {\left (5 \, B b c^{2} - A c^{3}\right )} x^{2} + 2 \, {\left (5 \, B b^{2} c - A b c^{2}\right )} x\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c}}{c \sqrt {x}}\right ) + {\left (8 \, B b c^{3} x^{2} + 15 \, B b^{3} c - 3 \, A b^{2} c^{2} + 5 \, {\left (5 \, B b^{2} c^{2} - A b c^{3}\right )} x\right )} \sqrt {x}}{4 \, {\left (b c^{6} x^{2} + 2 \, b^{2} c^{5} x + b^{3} c^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 87, normalized size = 0.69 \begin {gather*} \frac {2 \, B \sqrt {x}}{c^{3}} - \frac {3 \, {\left (5 \, B b - A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \, \sqrt {b c} c^{3}} + \frac {9 \, B b c x^{\frac {3}{2}} - 5 \, A c^{2} x^{\frac {3}{2}} + 7 \, B b^{2} \sqrt {x} - 3 \, A b c \sqrt {x}}{4 \, {\left (c x + b\right )}^{2} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 125, normalized size = 0.99 \begin {gather*} -\frac {5 A \,x^{\frac {3}{2}}}{4 \left (c x +b \right )^{2} c}+\frac {9 B b \,x^{\frac {3}{2}}}{4 \left (c x +b \right )^{2} c^{2}}-\frac {3 A b \sqrt {x}}{4 \left (c x +b \right )^{2} c^{2}}+\frac {7 B \,b^{2} \sqrt {x}}{4 \left (c x +b \right )^{2} c^{3}}+\frac {3 A \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \sqrt {b c}\, c^{2}}-\frac {15 B b \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \sqrt {b c}\, c^{3}}+\frac {2 B \sqrt {x}}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.46, size = 99, normalized size = 0.79 \begin {gather*} \frac {{\left (9 \, B b c - 5 \, A c^{2}\right )} x^{\frac {3}{2}} + {\left (7 \, B b^{2} - 3 \, A b c\right )} \sqrt {x}}{4 \, {\left (c^{5} x^{2} + 2 \, b c^{4} x + b^{2} c^{3}\right )}} + \frac {2 \, B \sqrt {x}}{c^{3}} - \frac {3 \, {\left (5 \, B b - A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \, \sqrt {b c} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 96, normalized size = 0.76 \begin {gather*} \frac {\sqrt {x}\,\left (\frac {7\,B\,b^2}{4}-\frac {3\,A\,b\,c}{4}\right )-x^{3/2}\,\left (\frac {5\,A\,c^2}{4}-\frac {9\,B\,b\,c}{4}\right )}{b^2\,c^3+2\,b\,c^4\,x+c^5\,x^2}+\frac {2\,B\,\sqrt {x}}{c^3}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (A\,c-5\,B\,b\right )}{4\,\sqrt {b}\,c^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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