Optimal. Leaf size=113 \[ \frac {16 \sqrt {a+b x} (2 A b-a B)}{3 a^4 \sqrt {x}}-\frac {8 (2 A b-a B)}{3 a^3 \sqrt {x} \sqrt {a+b x}}-\frac {2 (2 A b-a B)}{3 a^2 \sqrt {x} (a+b x)^{3/2}}-\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 113, 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} \begin {gather*} \frac {16 \sqrt {a+b x} (2 A b-a B)}{3 a^4 \sqrt {x}}-\frac {8 (2 A b-a B)}{3 a^3 \sqrt {x} \sqrt {a+b x}}-\frac {2 (2 A b-a B)}{3 a^2 \sqrt {x} (a+b x)^{3/2}}-\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}} \end {gather*}
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)^{5/2}} \, dx &=-\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}}+\frac {\left (2 \left (-3 A b+\frac {3 a B}{2}\right )\right ) \int \frac {1}{x^{3/2} (a+b x)^{5/2}} \, dx}{3 a}\\ &=-\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}}-\frac {2 (2 A b-a B)}{3 a^2 \sqrt {x} (a+b x)^{3/2}}-\frac {(4 (2 A b-a B)) \int \frac {1}{x^{3/2} (a+b x)^{3/2}} \, dx}{3 a^2}\\ &=-\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}}-\frac {2 (2 A b-a B)}{3 a^2 \sqrt {x} (a+b x)^{3/2}}-\frac {8 (2 A b-a B)}{3 a^3 \sqrt {x} \sqrt {a+b x}}-\frac {(8 (2 A b-a B)) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{3 a^3}\\ &=-\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}}-\frac {2 (2 A b-a B)}{3 a^2 \sqrt {x} (a+b x)^{3/2}}-\frac {8 (2 A b-a B)}{3 a^3 \sqrt {x} \sqrt {a+b x}}+\frac {16 (2 A b-a B) \sqrt {a+b x}}{3 a^4 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 70, normalized size = 0.62 \begin {gather*} -\frac {2 \left (a^3 (A+3 B x)-6 a^2 b x (A-2 B x)+8 a b^2 x^2 (B x-3 A)-16 A b^3 x^3\right )}{3 a^4 x^{3/2} (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 81, normalized size = 0.72 \begin {gather*} -\frac {2 \left (a^3 A+3 a^3 B x-6 a^2 A b x+12 a^2 b B x^2-24 a A b^2 x^2+8 a b^2 B x^3-16 A b^3 x^3\right )}{3 a^4 x^{3/2} (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 100, normalized size = 0.88 \begin {gather*} -\frac {2 \, {\left (A a^{3} + 8 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} x^{3} + 12 \, {\left (B a^{2} b - 2 \, A a b^{2}\right )} x^{2} + 3 \, {\left (B a^{3} - 2 \, A a^{2} b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.18, size = 303, normalized size = 2.68 \begin {gather*} -\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (3 \, B a^{4} b^{3} {\left | b \right |} - 8 \, A a^{3} b^{4} {\left | b \right |}\right )} {\left (b x + a\right )}}{a^{7} b^{2}} - \frac {3 \, {\left (B a^{5} b^{3} {\left | b \right |} - 3 \, A a^{4} b^{4} {\left | b \right |}\right )}}{a^{7} b^{2}}\right )}}{3 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {3}{2}}} - \frac {4 \, {\left (3 \, B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {5}{2}} + 12 \, B a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {7}{2}} - 6 \, A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {7}{2}} + 5 \, B a^{3} b^{\frac {9}{2}} - 18 \, A a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {9}{2}} - 8 \, A a^{2} b^{\frac {11}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} a^{3} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 76, normalized size = 0.67 \begin {gather*} -\frac {2 \left (-16 A \,b^{3} x^{3}+8 B a \,b^{2} x^{3}-24 A a \,b^{2} x^{2}+12 B \,a^{2} b \,x^{2}-6 A \,a^{2} b x +3 B \,a^{3} x +A \,a^{3}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} a^{4} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 130, normalized size = 1.15 \begin {gather*} \frac {2 \, B x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a} - \frac {16 \, B b x}{3 \, \sqrt {b x^{2} + a x} a^{3}} - \frac {4 \, A b x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2}} + \frac {32 \, A b^{2} x}{3 \, \sqrt {b x^{2} + a x} a^{4}} - \frac {8 \, B}{3 \, \sqrt {b x^{2} + a x} a^{2}} - \frac {2 \, A}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a} + \frac {16 \, A b}{3 \, \sqrt {b x^{2} + a x} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 112, normalized size = 0.99 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{3\,a\,b^2}-\frac {8\,x^2\,\left (2\,A\,b-B\,a\right )}{a^3\,b}-\frac {x^3\,\left (32\,A\,b^3-16\,B\,a\,b^2\right )}{3\,a^4\,b^2}+\frac {x\,\left (6\,B\,a^3-12\,A\,a^2\,b\right )}{3\,a^4\,b^2}\right )}{x^{7/2}+\frac {2\,a\,x^{5/2}}{b}+\frac {a^2\,x^{3/2}}{b^2}} \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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