Optimal. Leaf size=141 \[ \frac {16 b \sqrt {b x+c x^2} (6 b B-5 A c)}{15 c^4 \sqrt {x}}-\frac {8 \sqrt {x} \sqrt {b x+c x^2} (6 b B-5 A c)}{15 c^3}+\frac {2 x^{3/2} \sqrt {b x+c x^2} (6 b B-5 A c)}{5 b c^2}-\frac {2 x^{7/2} (b B-A c)}{b c \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.11, antiderivative size = 141, 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^{3/2} \sqrt {b x+c x^2} (6 b B-5 A c)}{5 b c^2}-\frac {8 \sqrt {x} \sqrt {b x+c x^2} (6 b B-5 A c)}{15 c^3}+\frac {16 b \sqrt {b x+c x^2} (6 b B-5 A c)}{15 c^4 \sqrt {x}}-\frac {2 x^{7/2} (b B-A c)}{b c \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 788
Rubi steps
\begin {align*} \int \frac {x^{7/2} (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (b B-A c) x^{7/2}}{b c \sqrt {b x+c x^2}}-\left (\frac {5 A}{b}-\frac {6 B}{c}\right ) \int \frac {x^{5/2}}{\sqrt {b x+c x^2}} \, dx\\ &=-\frac {2 (b B-A c) x^{7/2}}{b c \sqrt {b x+c x^2}}+\frac {2 (6 b B-5 A c) x^{3/2} \sqrt {b x+c x^2}}{5 b c^2}-\frac {(4 (6 b B-5 A c)) \int \frac {x^{3/2}}{\sqrt {b x+c x^2}} \, dx}{5 c^2}\\ &=-\frac {2 (b B-A c) x^{7/2}}{b c \sqrt {b x+c x^2}}-\frac {8 (6 b B-5 A c) \sqrt {x} \sqrt {b x+c x^2}}{15 c^3}+\frac {2 (6 b B-5 A c) x^{3/2} \sqrt {b x+c x^2}}{5 b c^2}+\frac {(8 b (6 b B-5 A c)) \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx}{15 c^3}\\ &=-\frac {2 (b B-A c) x^{7/2}}{b c \sqrt {b x+c x^2}}+\frac {16 b (6 b B-5 A c) \sqrt {b x+c x^2}}{15 c^4 \sqrt {x}}-\frac {8 (6 b B-5 A c) \sqrt {x} \sqrt {b x+c x^2}}{15 c^3}+\frac {2 (6 b B-5 A c) x^{3/2} \sqrt {b x+c x^2}}{5 b c^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 74, normalized size = 0.52 \begin {gather*} \frac {2 \sqrt {x} \left (-8 b^2 c (5 A-3 B x)-2 b c^2 x (10 A+3 B x)+c^3 x^2 (5 A+3 B x)+48 b^3 B\right )}{15 c^4 \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.72, size = 90, normalized size = 0.64 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-40 A b^2 c-20 A b c^2 x+5 A c^3 x^2+48 b^3 B+24 b^2 B c x-6 b B c^2 x^2+3 B c^3 x^3\right )}{15 c^4 \sqrt {x} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 92, normalized size = 0.65 \begin {gather*} \frac {2 \, {\left (3 \, B c^{3} x^{3} + 48 \, B b^{3} - 40 \, A b^{2} c - {\left (6 \, B b c^{2} - 5 \, A c^{3}\right )} x^{2} + 4 \, {\left (6 \, B b^{2} c - 5 \, A b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{15 \, {\left (c^{5} x^{2} + b c^{4} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 124, normalized size = 0.88 \begin {gather*} \frac {2 \, {\left (B b^{3} - A b^{2} c\right )}}{\sqrt {c x + b} c^{4}} - \frac {16 \, {\left (6 \, B b^{3} - 5 \, A b^{2} c\right )}}{15 \, \sqrt {b} c^{4}} + \frac {2 \, {\left (3 \, {\left (c x + b\right )}^{\frac {5}{2}} B c^{16} - 15 \, {\left (c x + b\right )}^{\frac {3}{2}} B b c^{16} + 45 \, \sqrt {c x + b} B b^{2} c^{16} + 5 \, {\left (c x + b\right )}^{\frac {3}{2}} A c^{17} - 30 \, \sqrt {c x + b} A b c^{17}\right )}}{15 \, c^{20}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 83, normalized size = 0.59 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-3 B \,c^{3} x^{3}-5 A \,c^{3} x^{2}+6 B b \,c^{2} x^{2}+20 A b \,c^{2} x -24 B \,b^{2} c x +40 A \,b^{2} c -48 b^{3} B \right ) x^{\frac {3}{2}}}{15 \left (c \,x^{2}+b x \right )^{\frac {3}{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 ({\left (3 \, B c^{3} x^{2} + B b c^{2} x - 2 \, B b^{2} c\right )} x^{3} - {\left (4 \, B b^{3} + {\left (4 \, B b c^{2} - 5 \, A c^{3}\right )} x^{2} + {\left (8 \, B b^{2} c - 5 \, A b c^{2}\right )} x\right )} x^{2}\right )} \sqrt {c x + b}}{15 \, {\left (c^{5} x^{3} + 2 \, b c^{4} x^{2} + b^{2} c^{3} x\right )}} - \int -\frac {4 \, {\left (2 \, B b^{4} + {\left (7 \, B b^{2} c^{2} - 5 \, A b c^{3}\right )} x^{2} + {\left (9 \, B b^{3} c - 5 \, A b^{2} c^{2}\right )} x\right )} \sqrt {c x + b} x^{2}}{15 \, {\left (c^{6} x^{5} + 3 \, b c^{5} x^{4} + 3 \, b^{2} c^{4} x^{3} + b^{3} c^{3} x^{2}\right )}}\,{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^{7/2}\,\left (A+B\,x\right )}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,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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