Optimal. Leaf size=306 \[ \frac{b^4 x^{10} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{10 (a+b x)}+\frac{5 a b^3 x^9 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{9 (a+b x)}+\frac{5 a^2 b^2 x^8 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{4 (a+b x)}+\frac{5 a^3 b x^7 \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{7 (a+b x)}+\frac{a^4 x^6 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{6 (a+b x)}+\frac{a^5 A x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{b^5 B x^{11} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)} \]
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Rubi [A] time = 0.127083, 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} \[ \frac{b^4 x^{10} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{10 (a+b x)}+\frac{5 a b^3 x^9 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{9 (a+b x)}+\frac{5 a^2 b^2 x^8 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{4 (a+b x)}+\frac{5 a^3 b x^7 \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{7 (a+b x)}+\frac{a^4 x^6 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{6 (a+b x)}+\frac{a^5 A x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{b^5 B x^{11} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 76
Rubi steps
\begin{align*} \int x^4 (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^4 \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^4+a^4 b^5 (5 A b+a B) x^5+5 a^3 b^6 (2 A b+a B) x^6+10 a^2 b^7 (A b+a B) x^7+5 a b^8 (A b+2 a B) x^8+b^9 (A b+5 a B) x^9+b^{10} B x^{10}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{a^5 A x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{a^4 (5 A b+a B) x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac{5 a^3 b (2 A b+a B) x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{5 a^2 b^2 (A b+a B) x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{5 a b^3 (A b+2 a B) x^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{b^4 (A b+5 a B) x^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{10 (a+b x)}+\frac{b^5 B x^{11} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.04108, size = 125, normalized size = 0.41 \[ \frac{x^5 \sqrt{(a+b x)^2} \left (2475 a^3 b^2 x^2 (8 A+7 B x)+1925 a^2 b^3 x^3 (9 A+8 B x)+1650 a^4 b x (7 A+6 B x)+462 a^5 (6 A+5 B x)+770 a b^4 x^4 (10 A+9 B x)+126 b^5 x^5 (11 A+10 B x)\right )}{13860 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 140, normalized size = 0.5 \begin{align*}{\frac{{x}^{5} \left ( 1260\,B{b}^{5}{x}^{6}+1386\,{x}^{5}A{b}^{5}+6930\,{x}^{5}Ba{b}^{4}+7700\,{x}^{4}Aa{b}^{4}+15400\,{x}^{4}B{a}^{2}{b}^{3}+17325\,{x}^{3}A{a}^{2}{b}^{3}+17325\,{x}^{3}B{a}^{3}{b}^{2}+19800\,{x}^{2}A{a}^{3}{b}^{2}+9900\,{x}^{2}B{a}^{4}b+11550\,xA{a}^{4}b+2310\,xB{a}^{5}+2772\,A{a}^{5} \right ) }{13860\, \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 = 1.34025, size = 266, normalized size = 0.87 \begin{align*} \frac{1}{11} \, B b^{5} x^{11} + \frac{1}{5} \, A a^{5} x^{5} + \frac{1}{10} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + \frac{5}{9} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{9} + \frac{5}{4} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{8} + \frac{5}{7} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{6} \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^{4} \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 = 1.16247, size = 300, normalized size = 0.98 \begin{align*} \frac{1}{11} \, B b^{5} x^{11} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, B a b^{4} x^{10} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{10} \, A b^{5} x^{10} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{9} \, B a^{2} b^{3} x^{9} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{9} \, A a b^{4} x^{9} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{4} \, B a^{3} b^{2} x^{8} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{4} \, A a^{2} b^{3} x^{8} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{7} \, B a^{4} b x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{7} \, A a^{3} b^{2} x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, B a^{5} x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{6} \, A a^{4} b x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{5} \, A a^{5} x^{5} \mathrm{sgn}\left (b x + a\right ) - \frac{{\left (5 \, B a^{11} - 11 \, A a^{10} b\right )} \mathrm{sgn}\left (b x + a\right )}{13860 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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