Optimal. Leaf size=107 \[ -\frac{2 a^3 (a B+4 A b)}{5 x^{5/2}}-\frac{4 a^2 b (2 a B+3 A b)}{3 x^{3/2}}-\frac{2 a^4 A}{7 x^{7/2}}-\frac{4 a b^2 (3 a B+2 A b)}{\sqrt{x}}+2 b^3 \sqrt{x} (4 a B+A b)+\frac{2}{3} b^4 B x^{3/2} \]
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Rubi [A] time = 0.0521916, antiderivative size = 107, normalized size of antiderivative = 1., 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} \[ -\frac{2 a^3 (a B+4 A b)}{5 x^{5/2}}-\frac{4 a^2 b (2 a B+3 A b)}{3 x^{3/2}}-\frac{2 a^4 A}{7 x^{7/2}}-\frac{4 a b^2 (3 a B+2 A b)}{\sqrt{x}}+2 b^3 \sqrt{x} (4 a B+A b)+\frac{2}{3} b^4 B x^{3/2} \]
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^{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*}
Mathematica [A] time = 0.0273657, size = 85, normalized size = 0.79 \[ -\frac{2 \left (210 a^2 b^2 x^2 (A+3 B x)+28 a^3 b x (3 A+5 B x)+3 a^4 (5 A+7 B x)+420 a b^3 x^3 (A-B x)-35 b^4 x^4 (3 A+B x)\right )}{105 x^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 100, normalized size = 0.9 \begin{align*} -{\frac{-70\,{b}^{4}B{x}^{5}-210\,A{b}^{4}{x}^{4}-840\,B{x}^{4}a{b}^{3}+840\,aA{b}^{3}{x}^{3}+1260\,B{x}^{3}{a}^{2}{b}^{2}+420\,{a}^{2}A{b}^{2}{x}^{2}+280\,B{x}^{2}{a}^{3}b+168\,{a}^{3}Abx+42\,{a}^{4}Bx+30\,A{a}^{4}}{105}{x}^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05015, size = 135, normalized size = 1.26 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57267, size = 228, normalized size = 2.13 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.88497, size = 139, normalized size = 1.3 \begin{align*} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14114, size = 135, normalized size = 1.26 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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