Optimal. Leaf size=109 \[ \frac{4}{5} a^2 b x^{5/2} (2 a B+3 A b)+\frac{2}{3} a^3 x^{3/2} (a B+4 A b)+2 a^4 A \sqrt{x}+\frac{2}{9} b^3 x^{9/2} (4 a B+A b)+\frac{4}{7} a b^2 x^{7/2} (3 a B+2 A b)+\frac{2}{11} b^4 B x^{11/2} \]
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Rubi [A] time = 0.0554192, antiderivative size = 109, 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} \[ \frac{4}{5} a^2 b x^{5/2} (2 a B+3 A b)+\frac{2}{3} a^3 x^{3/2} (a B+4 A b)+2 a^4 A \sqrt{x}+\frac{2}{9} b^3 x^{9/2} (4 a B+A b)+\frac{4}{7} a b^2 x^{7/2} (3 a B+2 A b)+\frac{2}{11} b^4 B x^{11/2} \]
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}{\sqrt{x}} \, dx &=\int \frac{(a+b x)^4 (A+B x)}{\sqrt{x}} \, dx\\ &=\int \left (\frac{a^4 A}{\sqrt{x}}+a^3 (4 A b+a B) \sqrt{x}+2 a^2 b (3 A b+2 a B) x^{3/2}+2 a b^2 (2 A b+3 a B) x^{5/2}+b^3 (A b+4 a B) x^{7/2}+b^4 B x^{9/2}\right ) \, dx\\ &=2 a^4 A \sqrt{x}+\frac{2}{3} a^3 (4 A b+a B) x^{3/2}+\frac{4}{5} a^2 b (3 A b+2 a B) x^{5/2}+\frac{4}{7} a b^2 (2 A b+3 a B) x^{7/2}+\frac{2}{9} b^3 (A b+4 a B) x^{9/2}+\frac{2}{11} b^4 B x^{11/2}\\ \end{align*}
Mathematica [A] time = 0.0325057, size = 89, normalized size = 0.82 \[ \frac{2 \sqrt{x} \left (594 a^2 b^2 x^2 (7 A+5 B x)+924 a^3 b x (5 A+3 B x)+1155 a^4 (3 A+B x)+220 a b^3 x^3 (9 A+7 B x)+35 b^4 x^4 (11 A+9 B x)\right )}{3465} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 100, normalized size = 0.9 \begin{align*}{\frac{630\,{b}^{4}B{x}^{5}+770\,A{b}^{4}{x}^{4}+3080\,B{x}^{4}a{b}^{3}+3960\,aA{b}^{3}{x}^{3}+5940\,B{x}^{3}{a}^{2}{b}^{2}+8316\,{a}^{2}A{b}^{2}{x}^{2}+5544\,B{x}^{2}{a}^{3}b+9240\,{a}^{3}Abx+2310\,{a}^{4}Bx+6930\,A{a}^{4}}{3465}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05721, size = 134, normalized size = 1.23 \begin{align*} \frac{2}{11} \, B b^{4} x^{\frac{11}{2}} + 2 \, A a^{4} \sqrt{x} + \frac{2}{9} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac{9}{2}} + \frac{4}{7} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{\frac{7}{2}} + \frac{4}{5} \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47111, size = 239, normalized size = 2.19 \begin{align*} \frac{2}{3465} \,{\left (315 \, B b^{4} x^{5} + 3465 \, A a^{4} + 385 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 990 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 1386 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 1155 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.69109, size = 146, normalized size = 1.34 \begin{align*} 2 A a^{4} \sqrt{x} + \frac{8 A a^{3} b x^{\frac{3}{2}}}{3} + \frac{12 A a^{2} b^{2} x^{\frac{5}{2}}}{5} + \frac{8 A a b^{3} x^{\frac{7}{2}}}{7} + \frac{2 A b^{4} x^{\frac{9}{2}}}{9} + \frac{2 B a^{4} x^{\frac{3}{2}}}{3} + \frac{8 B a^{3} b x^{\frac{5}{2}}}{5} + \frac{12 B a^{2} b^{2} x^{\frac{7}{2}}}{7} + \frac{8 B a b^{3} x^{\frac{9}{2}}}{9} + \frac{2 B b^{4} x^{\frac{11}{2}}}{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24244, size = 136, normalized size = 1.25 \begin{align*} \frac{2}{11} \, B b^{4} x^{\frac{11}{2}} + \frac{8}{9} \, B a b^{3} x^{\frac{9}{2}} + \frac{2}{9} \, A b^{4} x^{\frac{9}{2}} + \frac{12}{7} \, B a^{2} b^{2} x^{\frac{7}{2}} + \frac{8}{7} \, A a b^{3} x^{\frac{7}{2}} + \frac{8}{5} \, B a^{3} b x^{\frac{5}{2}} + \frac{12}{5} \, A a^{2} b^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B a^{4} x^{\frac{3}{2}} + \frac{8}{3} \, A a^{3} b x^{\frac{3}{2}} + 2 \, A a^{4} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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