Optimal. Leaf size=144 \[ \frac{2 \sqrt{x} (a+b x) (A b-a B)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 \sqrt{a} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{3/2} (a+b x)}{3 b \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0703647, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {770, 80, 50, 63, 205} \[ \frac{2 \sqrt{x} (a+b x) (A b-a B)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 \sqrt{a} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{3/2} (a+b x)}{3 b \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 80
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{x} (A+B x)}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{\sqrt{x} (A+B x)}{a b+b^2 x} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 B x^{3/2} (a+b x)}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 \left (\frac{3 A b^2}{2}-\frac{3 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{x}}{a b+b^2 x} \, dx}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (A b-a B) \sqrt{x} (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{3/2} (a+b x)}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (2 a \left (\frac{3 A b^2}{2}-\frac{3 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{1}{\sqrt{x} \left (a b+b^2 x\right )} \, dx}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (A b-a B) \sqrt{x} (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{3/2} (a+b x)}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (4 a \left (\frac{3 A b^2}{2}-\frac{3 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b+b^2 x^2} \, dx,x,\sqrt{x}\right )}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (A b-a B) \sqrt{x} (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{3/2} (a+b x)}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 \sqrt{a} (A b-a B) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0440493, size = 82, normalized size = 0.57 \[ \frac{2 (a+b x) \left (\sqrt{b} \sqrt{x} (-3 a B+3 A b+b B x)+3 \sqrt{a} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{3 b^{5/2} \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 94, normalized size = 0.7 \begin{align*}{\frac{2\,bx+2\,a}{3\,{b}^{2}} \left ( B\sqrt{ab}{x}^{{\frac{3}{2}}}b+3\,A\sqrt{ab}\sqrt{x}b-3\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) ab-3\,B\sqrt{ab}\sqrt{x}a+3\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{2} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{ab}}}} \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 = 1.39636, size = 305, normalized size = 2.12 \begin{align*} \left [-\frac{3 \,{\left (B a - A b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (B b x - 3 \, B a + 3 \, A b\right )} \sqrt{x}}{3 \, b^{2}}, \frac{2 \,{\left (3 \,{\left (B a - A b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) +{\left (B b x - 3 \, B a + 3 \, A b\right )} \sqrt{x}\right )}}{3 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16416, size = 127, normalized size = 0.88 \begin{align*} \frac{2 \,{\left (B a^{2} \mathrm{sgn}\left (b x + a\right ) - A a b \mathrm{sgn}\left (b x + a\right )\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{2 \,{\left (B b^{2} x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) - 3 \, B a b \sqrt{x} \mathrm{sgn}\left (b x + a\right ) + 3 \, A b^{2} \sqrt{x} \mathrm{sgn}\left (b x + a\right )\right )}}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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