Optimal. Leaf size=117 \[ \frac {16 b^2 \sqrt {a+b x} (6 A b-7 a B)}{105 a^4 \sqrt {x}}-\frac {8 b \sqrt {a+b x} (6 A b-7 a B)}{105 a^3 x^{3/2}}+\frac {2 \sqrt {a+b x} (6 A b-7 a B)}{35 a^2 x^{5/2}}-\frac {2 A \sqrt {a+b x}}{7 a x^{7/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \[ \frac {16 b^2 \sqrt {a+b x} (6 A b-7 a B)}{105 a^4 \sqrt {x}}-\frac {8 b \sqrt {a+b x} (6 A b-7 a B)}{105 a^3 x^{3/2}}+\frac {2 \sqrt {a+b x} (6 A b-7 a B)}{35 a^2 x^{5/2}}-\frac {2 A \sqrt {a+b x}}{7 a x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{9/2} \sqrt {a+b x}} \, dx &=-\frac {2 A \sqrt {a+b x}}{7 a x^{7/2}}+\frac {\left (2 \left (-3 A b+\frac {7 a B}{2}\right )\right ) \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{7 a}\\ &=-\frac {2 A \sqrt {a+b x}}{7 a x^{7/2}}+\frac {2 (6 A b-7 a B) \sqrt {a+b x}}{35 a^2 x^{5/2}}+\frac {(4 b (6 A b-7 a B)) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{35 a^2}\\ &=-\frac {2 A \sqrt {a+b x}}{7 a x^{7/2}}+\frac {2 (6 A b-7 a B) \sqrt {a+b x}}{35 a^2 x^{5/2}}-\frac {8 b (6 A b-7 a B) \sqrt {a+b x}}{105 a^3 x^{3/2}}-\frac {\left (8 b^2 (6 A b-7 a B)\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{105 a^3}\\ &=-\frac {2 A \sqrt {a+b x}}{7 a x^{7/2}}+\frac {2 (6 A b-7 a B) \sqrt {a+b x}}{35 a^2 x^{5/2}}-\frac {8 b (6 A b-7 a B) \sqrt {a+b x}}{105 a^3 x^{3/2}}+\frac {16 b^2 (6 A b-7 a B) \sqrt {a+b x}}{105 a^4 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 76, normalized size = 0.65 \[ -\frac {2 \sqrt {a+b x} \left (3 a^3 (5 A+7 B x)-2 a^2 b x (9 A+14 B x)+8 a b^2 x^2 (3 A+7 B x)-48 A b^3 x^3\right )}{105 a^4 x^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 78, normalized size = 0.67 \[ -\frac {2 \, {\left (15 \, A a^{3} + 8 \, {\left (7 \, B a b^{2} - 6 \, A b^{3}\right )} x^{3} - 4 \, {\left (7 \, B a^{2} b - 6 \, A a b^{2}\right )} x^{2} + 3 \, {\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} x\right )} \sqrt {b x + a}}{105 \, a^{4} x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.57, size = 137, normalized size = 1.17 \[ -\frac {2 \, {\left ({\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (7 \, B a b^{6} - 6 \, A b^{7}\right )} {\left (b x + a\right )}}{a^{4}} - \frac {7 \, {\left (7 \, B a^{2} b^{6} - 6 \, A a b^{7}\right )}}{a^{4}}\right )} + \frac {35 \, {\left (7 \, B a^{3} b^{6} - 6 \, A a^{2} b^{7}\right )}}{a^{4}}\right )} - \frac {105 \, {\left (B a^{4} b^{6} - A a^{3} b^{7}\right )}}{a^{4}}\right )} \sqrt {b x + a} b}{105 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {7}{2}} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 77, normalized size = 0.66 \[ -\frac {2 \sqrt {b x +a}\, \left (-48 A \,b^{3} x^{3}+56 B a \,b^{2} x^{3}+24 A a \,b^{2} x^{2}-28 B \,a^{2} b \,x^{2}-18 A \,a^{2} b x +21 B \,a^{3} x +15 A \,a^{3}\right )}{105 a^{4} x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 152, normalized size = 1.30 \[ -\frac {16 \, \sqrt {b x^{2} + a x} B b^{2}}{15 \, a^{3} x} + \frac {32 \, \sqrt {b x^{2} + a x} A b^{3}}{35 \, a^{4} x} + \frac {8 \, \sqrt {b x^{2} + a x} B b}{15 \, a^{2} x^{2}} - \frac {16 \, \sqrt {b x^{2} + a x} A b^{2}}{35 \, a^{3} x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} B}{5 \, a x^{3}} + \frac {12 \, \sqrt {b x^{2} + a x} A b}{35 \, a^{2} x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{7 \, a x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 79, normalized size = 0.68 \[ -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{7\,a}+\frac {x\,\left (42\,B\,a^3-36\,A\,a^2\,b\right )}{105\,a^4}-\frac {x^3\,\left (96\,A\,b^3-112\,B\,a\,b^2\right )}{105\,a^4}+\frac {8\,b\,x^2\,\left (6\,A\,b-7\,B\,a\right )}{105\,a^3}\right )}{x^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 59.76, size = 796, normalized size = 6.80 \[ - \frac {10 A a^{6} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} - \frac {18 A a^{5} b^{\frac {21}{2}} x \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} - \frac {10 A a^{4} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {10 A a^{3} b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {60 A a^{2} b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {80 A a b^{\frac {29}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {32 A b^{\frac {31}{2}} x^{6} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} - \frac {6 B a^{4} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {4 B a^{3} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {6 B a^{2} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {24 B a b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {16 B b^{\frac {17}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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