Optimal. Leaf size=216 \[ -\frac{2 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{\sqrt{x} (a+b x)}+\frac{6 a b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac{2 b^2 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{3 (a+b x)}-\frac{2 a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}+\frac{2 b^3 B x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]
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Rubi [A] time = 0.0834914, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {770, 76} \[ -\frac{2 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{\sqrt{x} (a+b x)}+\frac{6 a b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac{2 b^2 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{3 (a+b x)}-\frac{2 a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}+\frac{2 b^3 B x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 76
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{5/2}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^3 (A+B x)}{x^{5/2}} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{a^3 A b^3}{x^{5/2}}+\frac{a^2 b^3 (3 A b+a B)}{x^{3/2}}+\frac{3 a b^4 (A b+a B)}{\sqrt{x}}+b^5 (A b+3 a B) \sqrt{x}+b^6 B x^{3/2}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac{2 a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}-\frac{2 a^2 (3 A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{x} (a+b x)}+\frac{6 a b (A b+a B) \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{2 b^2 (A b+3 a B) x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{2 b^3 B x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0342976, size = 84, normalized size = 0.39 \[ -\frac{2 \sqrt{(a+b x)^2} \left (45 a^2 b x (A-B x)+5 a^3 (A+3 B x)-15 a b^2 x^2 (3 A+B x)-b^3 x^3 (5 A+3 B x)\right )}{15 x^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 92, normalized size = 0.4 \begin{align*} -{\frac{-6\,B{x}^{4}{b}^{3}-10\,A{b}^{3}{x}^{3}-30\,B{x}^{3}a{b}^{2}-90\,A{x}^{2}a{b}^{2}-90\,B{x}^{2}{a}^{2}b+90\,A{a}^{2}bx+30\,{a}^{3}Bx+10\,A{a}^{3}}{15\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13387, size = 176, normalized size = 0.81 \begin{align*} \frac{2}{15} \,{\left ({\left (3 \, b^{3} x^{2} + 5 \, a b^{2} x\right )} \sqrt{x} + \frac{10 \,{\left (a b^{2} x^{2} + 3 \, a^{2} b x\right )}}{\sqrt{x}} + \frac{15 \,{\left (a^{2} b x^{2} - a^{3} x\right )}}{x^{\frac{3}{2}}}\right )} B + \frac{2}{3} \, A{\left (\frac{b^{3} x^{2} + 3 \, a b^{2} x}{\sqrt{x}} + \frac{6 \,{\left (a b^{2} x^{2} - a^{2} b x\right )}}{x^{\frac{3}{2}}} - \frac{3 \, a^{2} b x^{2} + a^{3} x}{x^{\frac{5}{2}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61932, size = 165, normalized size = 0.76 \begin{align*} \frac{2 \,{\left (3 \, B b^{3} x^{4} - 5 \, A a^{3} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 45 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} - 15 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15 \, x^{\frac{3}{2}}} \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{3}{2}}}{x^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13727, size = 166, normalized size = 0.77 \begin{align*} \frac{2}{5} \, B b^{3} x^{\frac{5}{2}} \mathrm{sgn}\left (b x + a\right ) + 2 \, B a b^{2} x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{3} \, A b^{3} x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) + 6 \, B a^{2} b \sqrt{x} \mathrm{sgn}\left (b x + a\right ) + 6 \, A a b^{2} \sqrt{x} \mathrm{sgn}\left (b x + a\right ) - \frac{2 \,{\left (3 \, B a^{3} x \mathrm{sgn}\left (b x + a\right ) + 9 \, A a^{2} b x \mathrm{sgn}\left (b x + a\right ) + A a^{3} \mathrm{sgn}\left (b x + a\right )\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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