Optimal. Leaf size=83 \[ \frac {4 \sqrt {a+b x} (4 A b-3 a B)}{3 a^3 \sqrt {x}}-\frac {2 (4 A b-3 a B)}{3 a^2 \sqrt {x} \sqrt {a+b x}}-\frac {2 A}{3 a x^{3/2} \sqrt {a+b x}} \]
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Rubi [A] time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \[ \frac {4 \sqrt {a+b x} (4 A b-3 a B)}{3 a^3 \sqrt {x}}-\frac {2 (4 A b-3 a B)}{3 a^2 \sqrt {x} \sqrt {a+b x}}-\frac {2 A}{3 a x^{3/2} \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{5/2} (a+b x)^{3/2}} \, dx &=-\frac {2 A}{3 a x^{3/2} \sqrt {a+b x}}+\frac {\left (2 \left (-2 A b+\frac {3 a B}{2}\right )\right ) \int \frac {1}{x^{3/2} (a+b x)^{3/2}} \, dx}{3 a}\\ &=-\frac {2 A}{3 a x^{3/2} \sqrt {a+b x}}-\frac {2 (4 A b-3 a B)}{3 a^2 \sqrt {x} \sqrt {a+b x}}-\frac {(2 (4 A b-3 a B)) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{3 a^2}\\ &=-\frac {2 A}{3 a x^{3/2} \sqrt {a+b x}}-\frac {2 (4 A b-3 a B)}{3 a^2 \sqrt {x} \sqrt {a+b x}}+\frac {4 (4 A b-3 a B) \sqrt {a+b x}}{3 a^3 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 54, normalized size = 0.65 \[ -\frac {2 \left (a^2 (A+3 B x)+2 a b x (3 B x-2 A)-8 A b^2 x^2\right )}{3 a^3 x^{3/2} \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 67, normalized size = 0.81 \[ -\frac {2 \, {\left (A a^{2} + 2 \, {\left (3 \, B a b - 4 \, A b^{2}\right )} x^{2} + {\left (3 \, B a^{2} - 4 \, A a b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.23, size = 215, normalized size = 2.59 \[ -\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (3 \, B a^{3} b^{3} {\left | b \right |} - 5 \, A a^{2} b^{4} {\left | b \right |}\right )} {\left (b x + a\right )}}{a^{5} b^{2}} - \frac {3 \, {\left (B a^{4} b^{3} {\left | b \right |} - 2 \, A a^{3} b^{4} {\left | b \right |}\right )}}{a^{5} b^{2}}\right )}}{3 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {3}{2}}} - \frac {4 \, {\left (B^{2} a^{2} b^{5} - 2 \, A B a b^{6} + A^{2} b^{7}\right )}}{{\left (B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {5}{2}} + B a^{2} b^{\frac {7}{2}} - A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {7}{2}} - A a b^{\frac {9}{2}}\right )} a^{2} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 52, normalized size = 0.63 \[ -\frac {2 \left (-8 A \,b^{2} x^{2}+6 B a b \,x^{2}-4 A a b x +3 B \,a^{2} x +A \,a^{2}\right )}{3 \sqrt {b x +a}\, a^{3} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 96, normalized size = 1.16 \[ -\frac {4 \, B b x}{\sqrt {b x^{2} + a x} a^{2}} + \frac {16 \, A b^{2} x}{3 \, \sqrt {b x^{2} + a x} a^{3}} - \frac {2 \, B}{\sqrt {b x^{2} + a x} a} + \frac {8 \, A b}{3 \, \sqrt {b x^{2} + a x} a^{2}} - \frac {2 \, A}{3 \, \sqrt {b x^{2} + a x} a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 77, normalized size = 0.93 \[ -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{3\,a\,b}+\frac {x\,\left (6\,B\,a^2-8\,A\,a\,b\right )}{3\,a^3\,b}-\frac {x^2\,\left (16\,A\,b^2-12\,B\,a\,b\right )}{3\,a^3\,b}\right )}{x^{5/2}+\frac {a\,x^{3/2}}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 68.06, size = 265, normalized size = 3.19 \[ A \left (- \frac {2 a^{3} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac {6 a^{2} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac {24 a b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac {16 b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}}\right ) + B \left (- \frac {2}{a \sqrt {b} x \sqrt {\frac {a}{b x} + 1}} - \frac {4 \sqrt {b}}{a^{2} \sqrt {\frac {a}{b x} + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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