Optimal. Leaf size=262 \[ \frac{5 a^4 A b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a^3 A b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{10 a^2 A b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{5 a A b^4 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{A b^5 x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{a^5 A \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{B (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b} \]
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Rubi [A] time = 0.0771679, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {770, 80, 43} \[ \frac{5 a^4 A b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a^3 A b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{10 a^2 A b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{5 a A b^4 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{A b^5 x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{a^5 A \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{B (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b} \]
Antiderivative was successfully verified.
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Rule 770
Rule 80
Rule 43
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5 (A+B x)}{x} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{B (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b}+\frac{\left (A \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{\left (a b+b^2 x\right )^5}{x} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{B (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b}+\frac{\left (A \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (5 a^4 b^6+\frac{a^5 b^5}{x}+10 a^3 b^7 x+10 a^2 b^8 x^2+5 a b^9 x^3+b^{10} x^4\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{5 a^4 A b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a^3 A b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{10 a^2 A b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{5 a A b^4 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{A b^5 x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{B (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b}+\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end{align*}
Mathematica [A] time = 0.0595506, size = 122, normalized size = 0.47 \[ \frac{\sqrt{(a+b x)^2} \left (x \left (50 a^2 b^3 x^2 (4 A+3 B x)+100 a^3 b^2 x (3 A+2 B x)+150 a^4 b (2 A+B x)+60 a^5 B+15 a b^4 x^3 (5 A+4 B x)+2 b^5 x^4 (6 A+5 B x)\right )+60 a^5 A \log (x)\right )}{60 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 139, normalized size = 0.5 \begin{align*}{\frac{10\,B{b}^{5}{x}^{6}+12\,A{x}^{5}{b}^{5}+60\,B{x}^{5}a{b}^{4}+75\,A{x}^{4}a{b}^{4}+150\,B{x}^{4}{a}^{2}{b}^{3}+200\,A{x}^{3}{a}^{2}{b}^{3}+200\,B{x}^{3}{a}^{3}{b}^{2}+300\,A{x}^{2}{a}^{3}{b}^{2}+150\,B{x}^{2}{a}^{4}b+60\,A{a}^{5}\ln \left ( x \right ) +300\,A{a}^{4}bx+60\,B{a}^{5}x}{60\, \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 = 1.47739, size = 252, normalized size = 0.96 \begin{align*} \frac{1}{6} \, B b^{5} x^{6} + A a^{5} \log \left (x\right ) + \frac{1}{5} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + \frac{5}{4} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + \frac{10}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + \frac{5}{2} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} +{\left (B a^{5} + 5 \, A a^{4} b\right )} x \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 \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25082, size = 257, normalized size = 0.98 \begin{align*} \frac{1}{6} \, B b^{5} x^{6} \mathrm{sgn}\left (b x + a\right ) + B a b^{4} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{5} \, A b^{5} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, B a^{2} b^{3} x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{4} \, A a b^{4} x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{3} \, B a^{3} b^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{3} \, A a^{2} b^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, B a^{4} b x^{2} \mathrm{sgn}\left (b x + a\right ) + 5 \, A a^{3} b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + B a^{5} x \mathrm{sgn}\left (b x + a\right ) + 5 \, A a^{4} b x \mathrm{sgn}\left (b x + a\right ) + A a^{5} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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