Optimal. Leaf size=107 \[ -\frac {2 a^4 A}{7 x^{7/2}}-\frac {2 a^3 (4 A b+a B)}{5 x^{5/2}}-\frac {4 a^2 b (3 A b+2 a B)}{3 x^{3/2}}-\frac {4 a b^2 (2 A b+3 a B)}{\sqrt {x}}+2 b^3 (A b+4 a B) \sqrt {x}+\frac {2}{3} b^4 B x^{3/2} \]
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Rubi [A]
time = 0.04, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 77}
\begin {gather*} -\frac {2 a^4 A}{7 x^{7/2}}-\frac {2 a^3 (a B+4 A b)}{5 x^{5/2}}-\frac {4 a^2 b (2 a B+3 A b)}{3 x^{3/2}}+2 b^3 \sqrt {x} (4 a B+A b)-\frac {4 a b^2 (3 a B+2 A b)}{\sqrt {x}}+\frac {2}{3} b^4 B x^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{9/2}} \, dx &=\int \frac {(a+b x)^4 (A+B x)}{x^{9/2}} \, dx\\ &=\int \left (\frac {a^4 A}{x^{9/2}}+\frac {a^3 (4 A b+a B)}{x^{7/2}}+\frac {2 a^2 b (3 A b+2 a B)}{x^{5/2}}+\frac {2 a b^2 (2 A b+3 a B)}{x^{3/2}}+\frac {b^3 (A b+4 a B)}{\sqrt {x}}+b^4 B \sqrt {x}\right ) \, dx\\ &=-\frac {2 a^4 A}{7 x^{7/2}}-\frac {2 a^3 (4 A b+a B)}{5 x^{5/2}}-\frac {4 a^2 b (3 A b+2 a B)}{3 x^{3/2}}-\frac {4 a b^2 (2 A b+3 a B)}{\sqrt {x}}+2 b^3 (A b+4 a B) \sqrt {x}+\frac {2}{3} b^4 B x^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 85, normalized size = 0.79 \begin {gather*} -\frac {2 \left (420 a b^3 x^3 (A-B x)-35 b^4 x^4 (3 A+B x)+210 a^2 b^2 x^2 (A+3 B x)+28 a^3 b x (3 A+5 B x)+3 a^4 (5 A+7 B x)\right )}{105 x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.61, size = 91, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {2 b^{4} B \,x^{\frac {3}{2}}}{3}+2 A \,b^{4} \sqrt {x}+8 B a \,b^{3} \sqrt {x}-\frac {2 a^{4} A}{7 x^{\frac {7}{2}}}-\frac {4 a \,b^{2} \left (2 A b +3 B a \right )}{\sqrt {x}}-\frac {4 a^{2} b \left (3 A b +2 B a \right )}{3 x^{\frac {3}{2}}}-\frac {2 a^{3} \left (4 A b +B a \right )}{5 x^{\frac {5}{2}}}\) | \(91\) |
default | \(\frac {2 b^{4} B \,x^{\frac {3}{2}}}{3}+2 A \,b^{4} \sqrt {x}+8 B a \,b^{3} \sqrt {x}-\frac {2 a^{4} A}{7 x^{\frac {7}{2}}}-\frac {4 a \,b^{2} \left (2 A b +3 B a \right )}{\sqrt {x}}-\frac {4 a^{2} b \left (3 A b +2 B a \right )}{3 x^{\frac {3}{2}}}-\frac {2 a^{3} \left (4 A b +B a \right )}{5 x^{\frac {5}{2}}}\) | \(91\) |
gosper | \(-\frac {2 \left (-35 B \,b^{4} x^{5}-105 A \,b^{4} x^{4}-420 x^{4} B a \,b^{3}+420 a A \,b^{3} x^{3}+630 x^{3} B \,a^{2} b^{2}+210 a^{2} A \,b^{2} x^{2}+140 x^{2} B \,a^{3} b +84 A \,a^{3} b x +21 B \,a^{4} x +15 a^{4} A \right )}{105 x^{\frac {7}{2}}}\) | \(100\) |
trager | \(-\frac {2 \left (-35 B \,b^{4} x^{5}-105 A \,b^{4} x^{4}-420 x^{4} B a \,b^{3}+420 a A \,b^{3} x^{3}+630 x^{3} B \,a^{2} b^{2}+210 a^{2} A \,b^{2} x^{2}+140 x^{2} B \,a^{3} b +84 A \,a^{3} b x +21 B \,a^{4} x +15 a^{4} A \right )}{105 x^{\frac {7}{2}}}\) | \(100\) |
risch | \(-\frac {2 \left (-35 B \,b^{4} x^{5}-105 A \,b^{4} x^{4}-420 x^{4} B a \,b^{3}+420 a A \,b^{3} x^{3}+630 x^{3} B \,a^{2} b^{2}+210 a^{2} A \,b^{2} x^{2}+140 x^{2} B \,a^{3} b +84 A \,a^{3} b x +21 B \,a^{4} x +15 a^{4} A \right )}{105 x^{\frac {7}{2}}}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 100, normalized size = 0.93 \begin {gather*} \frac {2}{3} \, B b^{4} x^{\frac {3}{2}} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \sqrt {x} - \frac {2 \, {\left (15 \, A a^{4} + 210 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 70 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 21 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )}}{105 \, x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.33, size = 99, normalized size = 0.93 \begin {gather*} \frac {2 \, {\left (35 \, B b^{4} x^{5} - 15 \, A a^{4} + 105 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} - 210 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} - 70 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} - 21 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )}}{105 \, x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.44, size = 139, normalized size = 1.30 \begin {gather*} - \frac {2 A a^{4}}{7 x^{\frac {7}{2}}} - \frac {8 A a^{3} b}{5 x^{\frac {5}{2}}} - \frac {4 A a^{2} b^{2}}{x^{\frac {3}{2}}} - \frac {8 A a b^{3}}{\sqrt {x}} + 2 A b^{4} \sqrt {x} - \frac {2 B a^{4}}{5 x^{\frac {5}{2}}} - \frac {8 B a^{3} b}{3 x^{\frac {3}{2}}} - \frac {12 B a^{2} b^{2}}{\sqrt {x}} + 8 B a b^{3} \sqrt {x} + \frac {2 B b^{4} x^{\frac {3}{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.94, size = 100, normalized size = 0.93 \begin {gather*} \frac {2}{3} \, B b^{4} x^{\frac {3}{2}} + 8 \, B a b^{3} \sqrt {x} + 2 \, A b^{4} \sqrt {x} - \frac {2 \, {\left (630 \, B a^{2} b^{2} x^{3} + 420 \, A a b^{3} x^{3} + 140 \, B a^{3} b x^{2} + 210 \, A a^{2} b^{2} x^{2} + 21 \, B a^{4} x + 84 \, A a^{3} b x + 15 \, A a^{4}\right )}}{105 \, x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 98, normalized size = 0.92 \begin {gather*} \sqrt {x}\,\left (2\,A\,b^4+8\,B\,a\,b^3\right )-\frac {x\,\left (\frac {2\,B\,a^4}{5}+\frac {8\,A\,b\,a^3}{5}\right )+\frac {2\,A\,a^4}{7}+x^2\,\left (\frac {8\,B\,a^3\,b}{3}+4\,A\,a^2\,b^2\right )+x^3\,\left (12\,B\,a^2\,b^2+8\,A\,a\,b^3\right )}{x^{7/2}}+\frac {2\,B\,b^4\,x^{3/2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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