Optimal. Leaf size=206 \[ \frac{x^{5/2} (A b-a B)}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{3/2} (A b-5 a B)}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 \sqrt{x} (a+b x) (A b-5 a B)}{4 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 (a+b x) (A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 \sqrt{a} b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.111264, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {770, 78, 47, 50, 63, 205} \[ \frac{x^{5/2} (A b-a B)}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{3/2} (A b-5 a B)}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 \sqrt{x} (a+b x) (A b-5 a B)}{4 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 (a+b x) (A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 \sqrt{a} b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 78
Rule 47
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac{x^{3/2} (A+B x)}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-a B) x^{5/2}}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left ((A b-5 a B) \left (a b+b^2 x\right )\right ) \int \frac{x^{3/2}}{\left (a b+b^2 x\right )^2} \, dx}{4 a \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-5 a B) x^{3/2}}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{5/2}}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (3 (A b-5 a B) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{x}}{a b+b^2 x} \, dx}{8 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-5 a B) x^{3/2}}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{5/2}}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 (A b-5 a B) \sqrt{x} (a+b x)}{4 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (3 (A b-5 a B) \left (a b+b^2 x\right )\right ) \int \frac{1}{\sqrt{x} \left (a b+b^2 x\right )} \, dx}{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-5 a B) x^{3/2}}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{5/2}}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 (A b-5 a B) \sqrt{x} (a+b x)}{4 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (3 (A b-5 a B) \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 )}{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-5 a B) x^{3/2}}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{5/2}}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 (A b-5 a B) \sqrt{x} (a+b x)}{4 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 (A b-5 a B) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 \sqrt{a} b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0331139, size = 79, normalized size = 0.38 \[ \frac{x^{5/2} \left (5 a^2 (A b-a B)+(a+b x)^2 (5 a B-A b) \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};-\frac{b x}{a}\right )\right )}{10 a^3 b (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 208, normalized size = 1. \begin{align*} -{\frac{bx+a}{4\,{b}^{3}} \left ( 5\,A\sqrt{ab}{x}^{3/2}{b}^{2}-3\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{2}{b}^{3}-25\,B\sqrt{ab}{x}^{3/2}ab-8\,B\sqrt{ab}{x}^{5/2}{b}^{2}+15\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{2}a{b}^{2}-6\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) xa{b}^{2}+30\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) x{a}^{2}b+3\,A\sqrt{ab}\sqrt{x}ab-3\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{2}b-15\,B\sqrt{ab}\sqrt{x}{a}^{2}+15\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{3} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{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 = 1.36017, size = 687, normalized size = 3.33 \begin{align*} \left [\frac{3 \,{\left (5 \, B a^{3} - A a^{2} b +{\left (5 \, B a b^{2} - A b^{3}\right )} x^{2} + 2 \,{\left (5 \, B a^{2} b - A a b^{2}\right )} x\right )} \sqrt{-a b} \log \left (\frac{b x - a - 2 \, \sqrt{-a b} \sqrt{x}}{b x + a}\right ) + 2 \,{\left (8 \, B a b^{3} x^{2} + 15 \, B a^{3} b - 3 \, A a^{2} b^{2} + 5 \,{\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} x\right )} \sqrt{x}}{8 \,{\left (a b^{6} x^{2} + 2 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}, \frac{3 \,{\left (5 \, B a^{3} - A a^{2} b +{\left (5 \, B a b^{2} - A b^{3}\right )} x^{2} + 2 \,{\left (5 \, B a^{2} b - A a b^{2}\right )} x\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right ) +{\left (8 \, B a b^{3} x^{2} + 15 \, B a^{3} b - 3 \, A a^{2} b^{2} + 5 \,{\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} x\right )} \sqrt{x}}{4 \,{\left (a b^{6} x^{2} + 2 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}\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 = 1.16441, size = 150, normalized size = 0.73 \begin{align*} \frac{2 \, B \sqrt{x}}{b^{3} \mathrm{sgn}\left (b x + a\right )} - \frac{3 \,{\left (5 \, B a - A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{3} \mathrm{sgn}\left (b x + a\right )} + \frac{9 \, B a b x^{\frac{3}{2}} - 5 \, A b^{2} x^{\frac{3}{2}} + 7 \, B a^{2} \sqrt{x} - 3 \, A a b \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} b^{3} \mathrm{sgn}\left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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