Optimal. Leaf size=306 \[ \frac{b^4 x^{11} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{11 (a+b x)}+\frac{a b^3 x^{10} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{2 (a+b x)}+\frac{10 a^2 b^2 x^9 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{9 (a+b x)}+\frac{5 a^3 b x^8 \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{8 (a+b x)}+\frac{a^4 x^7 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{7 (a+b x)}+\frac{a^5 A x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac{b^5 B x^{12} \sqrt{a^2+2 a b x+b^2 x^2}}{12 (a+b x)} \]
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Rubi [A] time = 0.140678, antiderivative size = 306, 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 = {770, 76} \[ \frac{b^4 x^{11} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{11 (a+b x)}+\frac{a b^3 x^{10} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{2 (a+b x)}+\frac{10 a^2 b^2 x^9 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{9 (a+b x)}+\frac{5 a^3 b x^8 \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{8 (a+b x)}+\frac{a^4 x^7 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{7 (a+b x)}+\frac{a^5 A x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac{b^5 B x^{12} \sqrt{a^2+2 a b x+b^2 x^2}}{12 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 76
Rubi steps
\begin{align*} \int x^5 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int x^5 \left (a b+b^2 x\right )^5 (A+B x) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a^5 A b^5 x^5+a^4 b^5 (5 A b+a B) x^6+5 a^3 b^6 (2 A b+a B) x^7+10 a^2 b^7 (A b+a B) x^8+5 a b^8 (A b+2 a B) x^9+b^9 (A b+5 a B) x^{10}+b^{10} B x^{11}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{a^5 A x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac{a^4 (5 A b+a B) x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{5 a^3 b (2 A b+a B) x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (a+b x)}+\frac{10 a^2 b^2 (A b+a B) x^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{a b^3 (A b+2 a B) x^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b^4 (A b+5 a B) x^{11} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac{b^5 B x^{12} \sqrt{a^2+2 a b x+b^2 x^2}}{12 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0450954, size = 125, normalized size = 0.41 \[ \frac{x^6 \sqrt{(a+b x)^2} \left (770 a^3 b^2 x^2 (9 A+8 B x)+616 a^2 b^3 x^3 (10 A+9 B x)+495 a^4 b x (8 A+7 B x)+132 a^5 (7 A+6 B x)+252 a b^4 x^4 (11 A+10 B x)+42 b^5 x^5 (12 A+11 B x)\right )}{5544 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 140, normalized size = 0.5 \begin{align*}{\frac{{x}^{6} \left ( 462\,B{b}^{5}{x}^{6}+504\,{x}^{5}A{b}^{5}+2520\,{x}^{5}Ba{b}^{4}+2772\,{x}^{4}Aa{b}^{4}+5544\,{x}^{4}B{a}^{2}{b}^{3}+6160\,{x}^{3}A{a}^{2}{b}^{3}+6160\,{x}^{3}B{a}^{3}{b}^{2}+6930\,{x}^{2}A{a}^{3}{b}^{2}+3465\,{x}^{2}B{a}^{4}b+3960\,xA{a}^{4}b+792\,xB{a}^{5}+924\,A{a}^{5} \right ) }{5544\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42242, size = 269, normalized size = 0.88 \begin{align*} \frac{1}{12} \, B b^{5} x^{12} + \frac{1}{6} \, A a^{5} x^{6} + \frac{1}{11} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{11} + \frac{1}{2} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{10} + \frac{10}{9} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac{5}{8} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{8} + \frac{1}{7} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32363, size = 297, normalized size = 0.97 \begin{align*} \frac{1}{12} \, B b^{5} x^{12} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{11} \, B a b^{4} x^{11} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{11} \, A b^{5} x^{11} \mathrm{sgn}\left (b x + a\right ) + B a^{2} b^{3} x^{10} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, A a b^{4} x^{10} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{9} \, B a^{3} b^{2} x^{9} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{9} \, A a^{2} b^{3} x^{9} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{8} \, B a^{4} b x^{8} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{4} \, A a^{3} b^{2} x^{8} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{7} \, B a^{5} x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{7} \, A a^{4} b x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, A a^{5} x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{{\left (B a^{12} - 2 \, A a^{11} b\right )} \mathrm{sgn}\left (b x + a\right )}{5544 \, b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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