Optimal. Leaf size=214 \[ \frac{2 b^2 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{5 (a+b x)}+\frac{2 a b x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac{2 a^2 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{a+b x}-\frac{2 a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{x} (a+b x)}+\frac{2 b^3 B x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)} \]
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Rubi [A] time = 0.0862784, antiderivative size = 214, 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} \[ \frac{2 b^2 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{5 (a+b x)}+\frac{2 a b x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac{2 a^2 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{a+b x}-\frac{2 a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{x} (a+b x)}+\frac{2 b^3 B x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)} \]
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^{3/2}} \, 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^{3/2}} \, 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^{3/2}}+\frac{a^2 b^3 (3 A b+a B)}{\sqrt{x}}+3 a b^4 (A b+a B) \sqrt{x}+b^5 (A b+3 a B) x^{3/2}+b^6 B x^{5/2}\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}}{\sqrt{x} (a+b x)}+\frac{2 a^2 (3 A b+a B) \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{2 a b (A b+a B) x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{2 b^2 (A b+3 a B) x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{2 b^3 B x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.034925, size = 85, normalized size = 0.4 \[ \frac{2 \sqrt{(a+b x)^2} \left (35 a^2 b x (3 A+B x)-35 a^3 (A-B x)+7 a b^2 x^2 (5 A+3 B x)+b^3 x^3 (7 A+5 B x)\right )}{35 \sqrt{x} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 92, normalized size = 0.4 \begin{align*} -{\frac{-10\,B{x}^{4}{b}^{3}-14\,A{b}^{3}{x}^{3}-42\,B{x}^{3}a{b}^{2}-70\,A{x}^{2}a{b}^{2}-70\,B{x}^{2}{a}^{2}b-210\,A{a}^{2}bx-70\,{a}^{3}Bx+70\,A{a}^{3}}{35\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09734, size = 180, normalized size = 0.84 \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 )} A + \frac{2}{105} \,{\left (3 \,{\left (5 \, b^{3} x^{2} + 7 \, a b^{2} x\right )} x^{\frac{3}{2}} + 14 \,{\left (3 \, a b^{2} x^{2} + 5 \, a^{2} b x\right )} \sqrt{x} + \frac{35 \,{\left (a^{2} b x^{2} + 3 \, a^{3} x\right )}}{\sqrt{x}}\right )} B \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52283, size = 166, normalized size = 0.78 \begin{align*} \frac{2 \,{\left (5 \, B b^{3} x^{4} - 35 \, A a^{3} + 7 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 35 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 35 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{35 \, \sqrt{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{3}{2}}}{x^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17764, size = 169, normalized size = 0.79 \begin{align*} \frac{2}{7} \, B b^{3} x^{\frac{7}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{6}{5} \, B a b^{2} x^{\frac{5}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{5} \, A b^{3} x^{\frac{5}{2}} \mathrm{sgn}\left (b x + a\right ) + 2 \, B a^{2} b x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) + 2 \, A a b^{2} x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) + 2 \, B a^{3} \sqrt{x} \mathrm{sgn}\left (b x + a\right ) + 6 \, A a^{2} b \sqrt{x} \mathrm{sgn}\left (b x + a\right ) - \frac{2 \, A a^{3} \mathrm{sgn}\left (b x + a\right )}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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