Optimal. Leaf size=143 \[ -\frac {16 b \sqrt {x} (2 b B-A c)}{3 c^4 \sqrt {b x+c x^2}}-\frac {8 x^{3/2} (2 b B-A c)}{3 c^3 \sqrt {b x+c x^2}}+\frac {2 x^{5/2} (2 b B-A c)}{3 b c^2 \sqrt {b x+c x^2}}-\frac {2 x^{9/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {788, 656, 648} \begin {gather*} \frac {2 x^{5/2} (2 b B-A c)}{3 b c^2 \sqrt {b x+c x^2}}-\frac {8 x^{3/2} (2 b B-A c)}{3 c^3 \sqrt {b x+c x^2}}-\frac {16 b \sqrt {x} (2 b B-A c)}{3 c^4 \sqrt {b x+c x^2}}-\frac {2 x^{9/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 788
Rubi steps
\begin {align*} \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b B-A c) x^{9/2}}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac {\left (2 \left (\frac {9}{2} (-b B+A c)-\frac {3}{2} (-b B+2 A c)\right )\right ) \int \frac {x^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b c}\\ &=-\frac {2 (b B-A c) x^{9/2}}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 (2 b B-A c) x^{5/2}}{3 b c^2 \sqrt {b x+c x^2}}-\frac {(4 (2 b B-A c)) \int \frac {x^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=-\frac {2 (b B-A c) x^{9/2}}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac {8 (2 b B-A c) x^{3/2}}{3 c^3 \sqrt {b x+c x^2}}+\frac {2 (2 b B-A c) x^{5/2}}{3 b c^2 \sqrt {b x+c x^2}}+\frac {(8 b (2 b B-A c)) \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 c^3}\\ &=-\frac {2 (b B-A c) x^{9/2}}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac {16 b (2 b B-A c) \sqrt {x}}{3 c^4 \sqrt {b x+c x^2}}-\frac {8 (2 b B-A c) x^{3/2}}{3 c^3 \sqrt {b x+c x^2}}+\frac {2 (2 b B-A c) x^{5/2}}{3 b c^2 \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 70, normalized size = 0.49 \begin {gather*} \frac {2 x^{3/2} \left (8 b^2 c (A-3 B x)-6 b c^2 x (B x-2 A)+c^3 x^2 (3 A+B x)-16 b^3 B\right )}{3 c^4 (x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.85, size = 82, normalized size = 0.57 \begin {gather*} \frac {2 x^{3/2} \left (8 A b^2 c+12 A b c^2 x+3 A c^3 x^2-16 b^3 B-24 b^2 B c x-6 b B c^2 x^2+B c^3 x^3\right )}{3 c^4 \left (b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 102, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (B c^{3} x^{3} - 16 \, B b^{3} + 8 \, A b^{2} c - 3 \, {\left (2 \, B b c^{2} - A c^{3}\right )} x^{2} - 12 \, {\left (2 \, B b^{2} c - A b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{3 \, {\left (c^{6} x^{3} + 2 \, b c^{5} x^{2} + b^{2} c^{4} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 112, normalized size = 0.78 \begin {gather*} \frac {16 \, {\left (2 \, B b^{2} - A b c\right )}}{3 \, \sqrt {b} c^{4}} - \frac {2 \, {\left (9 \, {\left (c x + b\right )} B b^{2} - B b^{3} - 6 \, {\left (c x + b\right )} A b c + A b^{2} c\right )}}{3 \, {\left (c x + b\right )}^{\frac {3}{2}} c^{4}} + \frac {2 \, {\left ({\left (c x + b\right )}^{\frac {3}{2}} B c^{8} - 9 \, \sqrt {c x + b} B b c^{8} + 3 \, \sqrt {c x + b} A c^{9}\right )}}{3 \, c^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 82, normalized size = 0.57 \begin {gather*} \frac {2 \left (c x +b \right ) \left (B \,c^{3} x^{3}+3 A \,c^{3} x^{2}-6 B b \,c^{2} x^{2}+12 A b \,c^{2} x -24 B \,b^{2} c x +8 A \,b^{2} c -16 b^{3} B \right ) x^{\frac {5}{2}}}{3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {2 \, {\left (B c x + B b\right )} \sqrt {c x + b} x^{3}}{3 \, {\left (c^{4} x^{3} + 3 \, b c^{3} x^{2} + 3 \, b^{2} c^{2} x + b^{3} c\right )}} + \int \frac {{\left (A b c x^{3} - {\left (2 \, B b^{2} + {\left (2 \, B b c - A c^{2}\right )} x\right )} x^{3}\right )} \sqrt {c x + b}}{c^{5} x^{5} + 4 \, b c^{4} x^{4} + 6 \, b^{2} c^{3} x^{3} + 4 \, b^{3} c^{2} x^{2} + b^{4} c x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{9/2}\,\left (A+B\,x\right )}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \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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