Optimal. Leaf size=178 \[ -\frac {32 b^2 \sqrt {b x+c x^2} (8 b B-7 A c)}{35 c^5 \sqrt {x}}+\frac {16 b \sqrt {x} \sqrt {b x+c x^2} (8 b B-7 A c)}{35 c^4}-\frac {12 x^{3/2} \sqrt {b x+c x^2} (8 b B-7 A c)}{35 c^3}+\frac {2 x^{5/2} \sqrt {b x+c x^2} (8 b B-7 A c)}{7 b c^2}-\frac {2 x^{9/2} (b B-A c)}{b c \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.15, antiderivative size = 178, 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} \begin {gather*} -\frac {32 b^2 \sqrt {b x+c x^2} (8 b B-7 A c)}{35 c^5 \sqrt {x}}+\frac {2 x^{5/2} \sqrt {b x+c x^2} (8 b B-7 A c)}{7 b c^2}-\frac {12 x^{3/2} \sqrt {b x+c x^2} (8 b B-7 A c)}{35 c^3}+\frac {16 b \sqrt {x} \sqrt {b x+c x^2} (8 b B-7 A c)}{35 c^4}-\frac {2 x^{9/2} (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^{9/2} (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (b B-A c) x^{9/2}}{b c \sqrt {b x+c x^2}}-\left (\frac {7 A}{b}-\frac {8 B}{c}\right ) \int \frac {x^{7/2}}{\sqrt {b x+c x^2}} \, dx\\ &=-\frac {2 (b B-A c) x^{9/2}}{b c \sqrt {b x+c x^2}}+\frac {2 (8 b B-7 A c) x^{5/2} \sqrt {b x+c x^2}}{7 b c^2}-\frac {(6 (8 b B-7 A c)) \int \frac {x^{5/2}}{\sqrt {b x+c x^2}} \, dx}{7 c^2}\\ &=-\frac {2 (b B-A c) x^{9/2}}{b c \sqrt {b x+c x^2}}-\frac {12 (8 b B-7 A c) x^{3/2} \sqrt {b x+c x^2}}{35 c^3}+\frac {2 (8 b B-7 A c) x^{5/2} \sqrt {b x+c x^2}}{7 b c^2}+\frac {(24 b (8 b B-7 A c)) \int \frac {x^{3/2}}{\sqrt {b x+c x^2}} \, dx}{35 c^3}\\ &=-\frac {2 (b B-A c) x^{9/2}}{b c \sqrt {b x+c x^2}}+\frac {16 b (8 b B-7 A c) \sqrt {x} \sqrt {b x+c x^2}}{35 c^4}-\frac {12 (8 b B-7 A c) x^{3/2} \sqrt {b x+c x^2}}{35 c^3}+\frac {2 (8 b B-7 A c) x^{5/2} \sqrt {b x+c x^2}}{7 b c^2}-\frac {\left (16 b^2 (8 b B-7 A c)\right ) \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx}{35 c^4}\\ &=-\frac {2 (b B-A c) x^{9/2}}{b c \sqrt {b x+c x^2}}-\frac {32 b^2 (8 b B-7 A c) \sqrt {b x+c x^2}}{35 c^5 \sqrt {x}}+\frac {16 b (8 b B-7 A c) \sqrt {x} \sqrt {b x+c x^2}}{35 c^4}-\frac {12 (8 b B-7 A c) x^{3/2} \sqrt {b x+c x^2}}{35 c^3}+\frac {2 (8 b B-7 A c) x^{5/2} \sqrt {b x+c x^2}}{7 b c^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 93, normalized size = 0.52 \begin {gather*} \frac {2 \sqrt {x} \left (16 b^3 c (7 A-4 B x)+8 b^2 c^2 x (7 A+2 B x)-2 b c^3 x^2 (7 A+4 B x)+c^4 x^3 (7 A+5 B x)-128 b^4 B\right )}{35 c^5 \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.81, size = 114, normalized size = 0.64 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (112 A b^3 c+56 A b^2 c^2 x-14 A b c^3 x^2+7 A c^4 x^3-128 b^4 B-64 b^3 B c x+16 b^2 B c^2 x^2-8 b B c^3 x^3+5 B c^4 x^4\right )}{35 c^5 \sqrt {x} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 116, normalized size = 0.65 \begin {gather*} \frac {2 \, {\left (5 \, B c^{4} x^{4} - 128 \, B b^{4} + 112 \, A b^{3} c - {\left (8 \, B b c^{3} - 7 \, A c^{4}\right )} x^{3} + 2 \, {\left (8 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{2} - 8 \, {\left (8 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{35 \, {\left (c^{6} x^{2} + b c^{5} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 156, normalized size = 0.88 \begin {gather*} -\frac {2 \, {\left (B b^{4} - A b^{3} c\right )}}{\sqrt {c x + b} c^{5}} + \frac {32 \, {\left (8 \, B b^{4} - 7 \, A b^{3} c\right )}}{35 \, \sqrt {b} c^{5}} + \frac {2 \, {\left (5 \, {\left (c x + b\right )}^{\frac {7}{2}} B c^{30} - 28 \, {\left (c x + b\right )}^{\frac {5}{2}} B b c^{30} + 70 \, {\left (c x + b\right )}^{\frac {3}{2}} B b^{2} c^{30} - 140 \, \sqrt {c x + b} B b^{3} c^{30} + 7 \, {\left (c x + b\right )}^{\frac {5}{2}} A c^{31} - 35 \, {\left (c x + b\right )}^{\frac {3}{2}} A b c^{31} + 105 \, \sqrt {c x + b} A b^{2} c^{31}\right )}}{35 \, c^{35}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 107, normalized size = 0.60 \begin {gather*} \frac {2 \left (c x +b \right ) \left (5 B \,x^{4} c^{4}+7 A \,c^{4} x^{3}-8 B b \,c^{3} x^{3}-14 A b \,c^{3} x^{2}+16 B \,b^{2} c^{2} x^{2}+56 A \,b^{2} c^{2} x -64 B \,b^{3} c x +112 A \,b^{3} c -128 b^{4} B \right ) x^{\frac {3}{2}}}{35 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{5}} \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 (15 \, B c^{5} x^{3} + 3 \, B b c^{4} x^{2} - 4 \, B b^{2} c^{3} x + 8 \, B b^{3} c^{2}\right )} x^{4} + {\left (16 \, B b^{4} c - 3 \, {\left (4 \, B b c^{4} - 7 \, A c^{5}\right )} x^{3} - {\left (8 \, B b^{2} c^{3} - 7 \, A b c^{4}\right )} x^{2} + 2 \, {\left (10 \, B b^{3} c^{2} - 7 \, A b^{2} c^{3}\right )} x\right )} x^{3} + 4 \, {\left (2 \, B b^{5} + {\left (9 \, B b^{2} c^{3} - 7 \, A b c^{4}\right )} x^{3} + 2 \, {\left (10 \, B b^{3} c^{2} - 7 \, A b^{2} c^{3}\right )} x^{2} + {\left (13 \, B b^{4} c - 7 \, A b^{3} c^{2}\right )} x\right )} x^{2}\right )} \sqrt {c x + b}}{105 \, {\left (c^{7} x^{4} + 2 \, b c^{6} x^{3} + b^{2} c^{5} x^{2}\right )}} + \int -\frac {4 \, {\left (4 \, B b^{5} - 2 \, A b^{4} c + {\left (9 \, B b^{3} c^{2} - 7 \, A b^{2} c^{3}\right )} x^{2} + {\left (13 \, B b^{4} c - 9 \, A b^{3} c^{2}\right )} x\right )} \sqrt {c x + b} x^{2}}{15 \, {\left (c^{7} x^{5} + 3 \, b c^{6} x^{4} + 3 \, b^{2} c^{5} x^{3} + b^{3} c^{4} 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^{9/2}\,\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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