Optimal. Leaf size=107 \[ \frac{4}{3} a^2 b x^{3/2} (2 a B+3 A b)+2 a^3 \sqrt{x} (a B+4 A b)-\frac{2 a^4 A}{\sqrt{x}}+\frac{2}{7} b^3 x^{7/2} (4 a B+A b)+\frac{4}{5} a b^2 x^{5/2} (3 a B+2 A b)+\frac{2}{9} b^4 B x^{9/2} \]
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Rubi [A] time = 0.0510342, 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} \[ \frac{4}{3} a^2 b x^{3/2} (2 a B+3 A b)+2 a^3 \sqrt{x} (a B+4 A b)-\frac{2 a^4 A}{\sqrt{x}}+\frac{2}{7} b^3 x^{7/2} (4 a B+A b)+\frac{4}{5} a b^2 x^{5/2} (3 a B+2 A b)+\frac{2}{9} b^4 B x^{9/2} \]
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*}
Mathematica [A] time = 0.0245927, size = 87, normalized size = 0.81 \[ \frac{252 a^2 b^2 x^2 (5 A+3 B x)+840 a^3 b x (3 A+B x)-630 a^4 (A-B x)+72 a b^3 x^3 (7 A+5 B x)+10 b^4 x^4 (9 A+7 B x)}{315 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 100, normalized size = 0.9 \begin{align*} -{\frac{-70\,{b}^{4}B{x}^{5}-90\,A{b}^{4}{x}^{4}-360\,B{x}^{4}a{b}^{3}-504\,aA{b}^{3}{x}^{3}-756\,B{x}^{3}{a}^{2}{b}^{2}-1260\,{a}^{2}A{b}^{2}{x}^{2}-840\,B{x}^{2}{a}^{3}b-2520\,{a}^{3}Abx-630\,{a}^{4}Bx+630\,A{a}^{4}}{315}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.999749, size = 134, normalized size = 1.25 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53138, size = 231, normalized size = 2.16 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.0922, size = 141, normalized size = 1.32 \begin{align*} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18998, size = 136, normalized size = 1.27 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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