Optimal. Leaf size=107 \[ -\frac {2 a^4 A}{\sqrt {x}}+2 a^3 \sqrt {x} (a B+4 A b)+\frac {4}{3} a^2 b x^{3/2} (2 a B+3 A b)+\frac {2}{7} b^3 x^{7/2} (4 a B+A b)+\frac {4}{5} a b^2 x^{5/2} (3 a B+2 A b)+\frac {2}{9} b^4 B x^{9/2} \]
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Rubi [A] time = 0.05, 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, 76} \begin {gather*} \frac {4}{3} a^2 b x^{3/2} (2 a B+3 A b)+2 a^3 \sqrt {x} (a B+4 A b)-\frac {2 a^4 A}{\sqrt {x}}+\frac {2}{7} b^3 x^{7/2} (4 a B+A b)+\frac {4}{5} a b^2 x^{5/2} (3 a B+2 A b)+\frac {2}{9} b^4 B x^{9/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{3/2}} \, dx &=\int \frac {(a+b x)^4 (A+B x)}{x^{3/2}} \, dx\\ &=\int \left (\frac {a^4 A}{x^{3/2}}+\frac {a^3 (4 A b+a B)}{\sqrt {x}}+2 a^2 b (3 A b+2 a B) \sqrt {x}+2 a b^2 (2 A b+3 a B) x^{3/2}+b^3 (A b+4 a B) x^{5/2}+b^4 B x^{7/2}\right ) \, dx\\ &=-\frac {2 a^4 A}{\sqrt {x}}+2 a^3 (4 A b+a B) \sqrt {x}+\frac {4}{3} a^2 b (3 A b+2 a B) x^{3/2}+\frac {4}{5} a b^2 (2 A b+3 a B) x^{5/2}+\frac {2}{7} b^3 (A b+4 a B) x^{7/2}+\frac {2}{9} b^4 B x^{9/2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 87, normalized size = 0.81 \begin {gather*} \frac {-630 a^4 (A-B x)+840 a^3 b x (3 A+B x)+252 a^2 b^2 x^2 (5 A+3 B x)+72 a b^3 x^3 (7 A+5 B x)+10 b^4 x^4 (9 A+7 B x)}{315 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 103, normalized size = 0.96 \begin {gather*} \frac {2 \left (-315 a^4 A+315 a^4 B x+1260 a^3 A b x+420 a^3 b B x^2+630 a^2 A b^2 x^2+378 a^2 b^2 B x^3+252 a A b^3 x^3+180 a b^3 B x^4+45 A b^4 x^4+35 b^4 B x^5\right )}{315 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 99, normalized size = 0.93 \begin {gather*} \frac {2 \, {\left (35 \, B b^{4} x^{5} - 315 \, A a^{4} + 45 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 126 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 210 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 315 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )}}{315 \, \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 101, normalized size = 0.94 \begin {gather*} \frac {2}{9} \, B b^{4} x^{\frac {9}{2}} + \frac {8}{7} \, B a b^{3} x^{\frac {7}{2}} + \frac {2}{7} \, A b^{4} x^{\frac {7}{2}} + \frac {12}{5} \, B a^{2} b^{2} x^{\frac {5}{2}} + \frac {8}{5} \, A a b^{3} x^{\frac {5}{2}} + \frac {8}{3} \, B a^{3} b x^{\frac {3}{2}} + 4 \, A a^{2} b^{2} x^{\frac {3}{2}} + 2 \, B a^{4} \sqrt {x} + 8 \, A a^{3} b \sqrt {x} - \frac {2 \, A a^{4}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 100, normalized size = 0.93 \begin {gather*} -\frac {2 \left (-35 b^{4} B \,x^{5}-45 A \,b^{4} x^{4}-180 x^{4} B a \,b^{3}-252 A a \,b^{3} x^{3}-378 B \,a^{2} b^{2} x^{3}-630 A \,a^{2} b^{2} x^{2}-420 B \,a^{3} b \,x^{2}-1260 A \,a^{3} b x -315 B \,a^{4} x +315 A \,a^{4}\right )}{315 \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 99, normalized size = 0.93 \begin {gather*} \frac {2}{9} \, B b^{4} x^{\frac {9}{2}} - \frac {2 \, A a^{4}}{\sqrt {x}} + \frac {2}{7} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac {7}{2}} + \frac {4}{5} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{\frac {5}{2}} + \frac {4}{3} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{\frac {3}{2}} + 2 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 91, normalized size = 0.85 \begin {gather*} \sqrt {x}\,\left (2\,B\,a^4+8\,A\,b\,a^3\right )+x^{7/2}\,\left (\frac {2\,A\,b^4}{7}+\frac {8\,B\,a\,b^3}{7}\right )-\frac {2\,A\,a^4}{\sqrt {x}}+\frac {2\,B\,b^4\,x^{9/2}}{9}+\frac {4\,a^2\,b\,x^{3/2}\,\left (3\,A\,b+2\,B\,a\right )}{3}+\frac {4\,a\,b^2\,x^{5/2}\,\left (2\,A\,b+3\,B\,a\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.80, size = 141, normalized size = 1.32 \begin {gather*} - \frac {2 A a^{4}}{\sqrt {x}} + 8 A a^{3} b \sqrt {x} + 4 A a^{2} b^{2} x^{\frac {3}{2}} + \frac {8 A a b^{3} x^{\frac {5}{2}}}{5} + \frac {2 A b^{4} x^{\frac {7}{2}}}{7} + 2 B a^{4} \sqrt {x} + \frac {8 B a^{3} b x^{\frac {3}{2}}}{3} + \frac {12 B a^{2} b^{2} x^{\frac {5}{2}}}{5} + \frac {8 B a b^{3} x^{\frac {7}{2}}}{7} + \frac {2 B b^{4} x^{\frac {9}{2}}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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