Optimal. Leaf size=180 \[ \frac {32 b^2 \sqrt {x} (8 b B-5 A c)}{15 c^5 \sqrt {b x+c x^2}}+\frac {16 b x^{3/2} (8 b B-5 A c)}{15 c^4 \sqrt {b x+c x^2}}-\frac {4 x^{5/2} (8 b B-5 A c)}{15 c^3 \sqrt {b x+c x^2}}+\frac {2 x^{7/2} (8 b B-5 A c)}{15 b c^2 \sqrt {b x+c x^2}}-\frac {2 x^{11/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.16, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {788, 656, 648} \[ \frac {32 b^2 \sqrt {x} (8 b B-5 A c)}{15 c^5 \sqrt {b x+c x^2}}+\frac {2 x^{7/2} (8 b B-5 A c)}{15 b c^2 \sqrt {b x+c x^2}}-\frac {4 x^{5/2} (8 b B-5 A c)}{15 c^3 \sqrt {b x+c x^2}}+\frac {16 b x^{3/2} (8 b B-5 A c)}{15 c^4 \sqrt {b x+c x^2}}-\frac {2 x^{11/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 788
Rubi steps
\begin {align*} \int \frac {x^{11/2} (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b B-A c) x^{11/2}}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac {1}{3} \left (\frac {5 A}{b}-\frac {8 B}{c}\right ) \int \frac {x^{9/2}}{\left (b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (b B-A c) x^{11/2}}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 (8 b B-5 A c) x^{7/2}}{15 b c^2 \sqrt {b x+c x^2}}-\frac {(2 (8 b B-5 A c)) \int \frac {x^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{5 c^2}\\ &=-\frac {2 (b B-A c) x^{11/2}}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac {4 (8 b B-5 A c) x^{5/2}}{15 c^3 \sqrt {b x+c x^2}}+\frac {2 (8 b B-5 A c) x^{7/2}}{15 b c^2 \sqrt {b x+c x^2}}+\frac {(8 b (8 b B-5 A c)) \int \frac {x^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{15 c^3}\\ &=-\frac {2 (b B-A c) x^{11/2}}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {16 b (8 b B-5 A c) x^{3/2}}{15 c^4 \sqrt {b x+c x^2}}-\frac {4 (8 b B-5 A c) x^{5/2}}{15 c^3 \sqrt {b x+c x^2}}+\frac {2 (8 b B-5 A c) x^{7/2}}{15 b c^2 \sqrt {b x+c x^2}}-\frac {\left (16 b^2 (8 b B-5 A c)\right ) \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{15 c^4}\\ &=-\frac {2 (b B-A c) x^{11/2}}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {32 b^2 (8 b B-5 A c) \sqrt {x}}{15 c^5 \sqrt {b x+c x^2}}+\frac {16 b (8 b B-5 A c) x^{3/2}}{15 c^4 \sqrt {b x+c x^2}}-\frac {4 (8 b B-5 A c) x^{5/2}}{15 c^3 \sqrt {b x+c x^2}}+\frac {2 (8 b B-5 A c) x^{7/2}}{15 b c^2 \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 93, normalized size = 0.52 \[ \frac {2 x^{3/2} \left (b^3 (192 B c x-80 A c)+24 b^2 c^2 x (2 B x-5 A)-2 b c^3 x^2 (15 A+4 B x)+c^4 x^3 (5 A+3 B x)+128 b^4 B\right )}{15 c^5 (x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 127, normalized size = 0.71 \[ \frac {2 \, {\left (3 \, B c^{4} x^{4} + 128 \, B b^{4} - 80 \, A b^{3} c - {\left (8 \, B b c^{3} - 5 \, A c^{4}\right )} x^{3} + 6 \, {\left (8 \, B b^{2} c^{2} - 5 \, A b c^{3}\right )} x^{2} + 24 \, {\left (8 \, B b^{3} c - 5 \, A b^{2} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{15 \, {\left (c^{7} x^{3} + 2 \, b c^{6} x^{2} + b^{2} c^{5} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 147, normalized size = 0.82 \[ -\frac {32 \, {\left (8 \, B b^{3} - 5 \, A b^{2} c\right )}}{15 \, \sqrt {b} c^{5}} + \frac {2 \, {\left (12 \, {\left (c x + b\right )} B b^{3} - B b^{4} - 9 \, {\left (c x + b\right )} A b^{2} c + A b^{3} c\right )}}{3 \, {\left (c x + b\right )}^{\frac {3}{2}} c^{5}} + \frac {2 \, {\left (3 \, {\left (c x + b\right )}^{\frac {5}{2}} B c^{20} - 20 \, {\left (c x + b\right )}^{\frac {3}{2}} B b c^{20} + 90 \, \sqrt {c x + b} B b^{2} c^{20} + 5 \, {\left (c x + b\right )}^{\frac {3}{2}} A c^{21} - 45 \, \sqrt {c x + b} A b c^{21}\right )}}{15 \, c^{25}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 107, normalized size = 0.59 \[ -\frac {2 \left (c x +b \right ) \left (-3 B \,x^{4} c^{4}-5 A \,c^{4} x^{3}+8 B b \,c^{3} x^{3}+30 A b \,c^{3} x^{2}-48 B \,b^{2} c^{2} x^{2}+120 A \,b^{2} c^{2} x -192 B \,b^{3} c x +80 A \,b^{3} c -128 b^{4} B \right ) x^{\frac {5}{2}}}{15 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} c^{5}} \]
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 \[ \frac {2 \, {\left ({\left (3 \, B c^{3} x^{2} + B b c^{2} x - 2 \, B b^{2} c\right )} x^{4} - {\left (6 \, B b^{3} + {\left (6 \, B b c^{2} - 5 \, A c^{3}\right )} x^{2} + {\left (12 \, B b^{2} c - 5 \, A b c^{2}\right )} x\right )} x^{3}\right )} \sqrt {c x + b}}{15 \, {\left (c^{6} x^{4} + 3 \, b c^{5} x^{3} + 3 \, b^{2} c^{4} x^{2} + b^{3} c^{3} x\right )}} - \int -\frac {2 \, {\left (4 \, B b^{4} + {\left (9 \, B b^{2} c^{2} - 5 \, A b c^{3}\right )} x^{2} + {\left (13 \, B b^{3} c - 5 \, A b^{2} c^{2}\right )} x\right )} \sqrt {c x + b} x^{3}}{5 \, {\left (c^{7} x^{6} + 4 \, b c^{6} x^{5} + 6 \, b^{2} c^{5} x^{4} + 4 \, b^{3} c^{4} x^{3} + b^{4} c^{3} x^{2}\right )}}\,{d x} \]
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 \[ \int \frac {x^{11/2}\,\left (A+B\,x\right )}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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