Optimal. Leaf size=81 \[ -\frac {2 a^3 A}{5 x^{5/2}}-\frac {2 a^2 (3 A b+a B)}{3 x^{3/2}}-\frac {6 a b (A b+a B)}{\sqrt {x}}+2 b^2 (A b+3 a B) \sqrt {x}+\frac {2}{3} b^3 B x^{3/2} \]
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Rubi [A]
time = 0.03, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77}
\begin {gather*} -\frac {2 a^3 A}{5 x^{5/2}}-\frac {2 a^2 (a B+3 A b)}{3 x^{3/2}}+2 b^2 \sqrt {x} (3 a B+A b)-\frac {6 a b (a B+A b)}{\sqrt {x}}+\frac {2}{3} b^3 B x^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(a+b x)^3 (A+B x)}{x^{7/2}} \, dx &=\int \left (\frac {a^3 A}{x^{7/2}}+\frac {a^2 (3 A b+a B)}{x^{5/2}}+\frac {3 a b (A b+a B)}{x^{3/2}}+\frac {b^2 (A b+3 a B)}{\sqrt {x}}+b^3 B \sqrt {x}\right ) \, dx\\ &=-\frac {2 a^3 A}{5 x^{5/2}}-\frac {2 a^2 (3 A b+a B)}{3 x^{3/2}}-\frac {6 a b (A b+a B)}{\sqrt {x}}+2 b^2 (A b+3 a B) \sqrt {x}+\frac {2}{3} b^3 B x^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 65, normalized size = 0.80 \begin {gather*} -\frac {2 \left (45 a b^2 x^2 (A-B x)-5 b^3 x^3 (3 A+B x)+15 a^2 b x (A+3 B x)+a^3 (3 A+5 B x)\right )}{15 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 69, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {2 b^{3} B \,x^{\frac {3}{2}}}{3}+2 b^{3} A \sqrt {x}+6 a \,b^{2} B \sqrt {x}-\frac {6 a b \left (A b +B a \right )}{\sqrt {x}}-\frac {2 a^{3} A}{5 x^{\frac {5}{2}}}-\frac {2 a^{2} \left (3 A b +B a \right )}{3 x^{\frac {3}{2}}}\) | \(69\) |
default | \(\frac {2 b^{3} B \,x^{\frac {3}{2}}}{3}+2 b^{3} A \sqrt {x}+6 a \,b^{2} B \sqrt {x}-\frac {6 a b \left (A b +B a \right )}{\sqrt {x}}-\frac {2 a^{3} A}{5 x^{\frac {5}{2}}}-\frac {2 a^{2} \left (3 A b +B a \right )}{3 x^{\frac {3}{2}}}\) | \(69\) |
gosper | \(-\frac {2 \left (-5 b^{3} B \,x^{4}-15 A \,b^{3} x^{3}-45 B a \,b^{2} x^{3}+45 a A \,b^{2} x^{2}+45 B \,a^{2} b \,x^{2}+15 a^{2} A b x +5 a^{3} B x +3 a^{3} A \right )}{15 x^{\frac {5}{2}}}\) | \(76\) |
trager | \(-\frac {2 \left (-5 b^{3} B \,x^{4}-15 A \,b^{3} x^{3}-45 B a \,b^{2} x^{3}+45 a A \,b^{2} x^{2}+45 B \,a^{2} b \,x^{2}+15 a^{2} A b x +5 a^{3} B x +3 a^{3} A \right )}{15 x^{\frac {5}{2}}}\) | \(76\) |
risch | \(-\frac {2 \left (-5 b^{3} B \,x^{4}-15 A \,b^{3} x^{3}-45 B a \,b^{2} x^{3}+45 a A \,b^{2} x^{2}+45 B \,a^{2} b \,x^{2}+15 a^{2} A b x +5 a^{3} B x +3 a^{3} A \right )}{15 x^{\frac {5}{2}}}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 74, normalized size = 0.91 \begin {gather*} \frac {2}{3} \, B b^{3} x^{\frac {3}{2}} + 2 \, {\left (3 \, B a b^{2} + A b^{3}\right )} \sqrt {x} - \frac {2 \, {\left (3 \, A a^{3} + 45 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.99, size = 73, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (5 \, B b^{3} x^{4} - 3 \, A a^{3} + 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} - 45 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} - 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.39, size = 105, normalized size = 1.30 \begin {gather*} - \frac {2 A a^{3}}{5 x^{\frac {5}{2}}} - \frac {2 A a^{2} b}{x^{\frac {3}{2}}} - \frac {6 A a b^{2}}{\sqrt {x}} + 2 A b^{3} \sqrt {x} - \frac {2 B a^{3}}{3 x^{\frac {3}{2}}} - \frac {6 B a^{2} b}{\sqrt {x}} + 6 B a b^{2} \sqrt {x} + \frac {2 B b^{3} x^{\frac {3}{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.84, size = 76, normalized size = 0.94 \begin {gather*} \frac {2}{3} \, B b^{3} x^{\frac {3}{2}} + 6 \, B a b^{2} \sqrt {x} + 2 \, A b^{3} \sqrt {x} - \frac {2 \, {\left (45 \, B a^{2} b x^{2} + 45 \, A a b^{2} x^{2} + 5 \, B a^{3} x + 15 \, A a^{2} b x + 3 \, A a^{3}\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 75, normalized size = 0.93 \begin {gather*} -\frac {10\,B\,a^3\,x+6\,A\,a^3+90\,B\,a^2\,b\,x^2+30\,A\,a^2\,b\,x-90\,B\,a\,b^2\,x^3+90\,A\,a\,b^2\,x^2-10\,B\,b^3\,x^4-30\,A\,b^3\,x^3}{15\,x^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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