Optimal. Leaf size=262 \[ \frac{x^{3/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\sqrt{x} (3 a B+5 A b)}{64 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\sqrt{x} (3 a B+5 A b)}{96 a^2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\sqrt{x} (3 a B+5 A b)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (3 a B+5 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{7/2} b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.139067, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {770, 78, 47, 51, 63, 205} \[ \frac{x^{3/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\sqrt{x} (3 a B+5 A b)}{64 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\sqrt{x} (3 a B+5 A b)}{96 a^2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\sqrt{x} (3 a B+5 A b)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (3 a B+5 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{7/2} b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 78
Rule 47
Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{x} (A+B x)}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-a B) x^{3/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (b^2 (5 A b+3 a B) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{x}}{\left (a b+b^2 x\right )^4} \, dx}{8 a \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-a B) x^{3/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(5 A b+3 a B) \sqrt{x}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left ((5 A b+3 a B) \left (a b+b^2 x\right )\right ) \int \frac{1}{\sqrt{x} \left (a b+b^2 x\right )^3} \, dx}{48 a \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-a B) x^{3/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(5 A b+3 a B) \sqrt{x}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(5 A b+3 a B) \sqrt{x}}{96 a^2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left ((5 A b+3 a B) \left (a b+b^2 x\right )\right ) \int \frac{1}{\sqrt{x} \left (a b+b^2 x\right )^2} \, dx}{64 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(5 A b+3 a B) \sqrt{x}}{64 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{3/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(5 A b+3 a B) \sqrt{x}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(5 A b+3 a B) \sqrt{x}}{96 a^2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left ((5 A b+3 a B) \left (a b+b^2 x\right )\right ) \int \frac{1}{\sqrt{x} \left (a b+b^2 x\right )} \, dx}{128 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(5 A b+3 a B) \sqrt{x}}{64 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{3/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(5 A b+3 a B) \sqrt{x}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(5 A b+3 a B) \sqrt{x}}{96 a^2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left ((5 A b+3 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 )}{64 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(5 A b+3 a B) \sqrt{x}}{64 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{3/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(5 A b+3 a B) \sqrt{x}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(5 A b+3 a B) \sqrt{x}}{96 a^2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(5 A b+3 a B) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{7/2} b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0352324, size = 79, normalized size = 0.3 \[ \frac{x^{3/2} \left ((a+b x)^4 (3 a B+5 A b) \, _2F_1\left (\frac{3}{2},4;\frac{5}{2};-\frac{b x}{a}\right )-3 a^4 (a B-A b)\right )}{12 a^5 b (a+b x)^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 357, normalized size = 1.4 \begin{align*}{\frac{bx+a}{192\,{a}^{3}{b}^{2}} \left ( 15\,A\sqrt{ab}{x}^{7/2}{b}^{4}+9\,B\sqrt{ab}{x}^{7/2}a{b}^{3}+55\,A\sqrt{ab}{x}^{5/2}a{b}^{3}+15\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{4}{b}^{5}+33\,B\sqrt{ab}{x}^{5/2}{a}^{2}{b}^{2}+9\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{4}a{b}^{4}+60\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{3}a{b}^{4}+36\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{3}{a}^{2}{b}^{3}+73\,A\sqrt{ab}{x}^{3/2}{a}^{2}{b}^{2}+90\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{2}{a}^{2}{b}^{3}-33\,B\sqrt{ab}{x}^{3/2}{a}^{3}b+54\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{2}{a}^{3}{b}^{2}+60\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) x{a}^{3}{b}^{2}+36\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) x{a}^{4}b-15\,A\sqrt{ab}\sqrt{x}{a}^{3}b+15\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{4}b-9\,B\sqrt{ab}\sqrt{x}{a}^{4}+9\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{5} \right ){\frac{1}{\sqrt{ab}}} \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 = 1.40393, size = 1157, normalized size = 4.42 \begin{align*} \left [-\frac{3 \,{\left (3 \, B a^{5} + 5 \, A a^{4} b +{\left (3 \, B a b^{4} + 5 \, A b^{5}\right )} x^{4} + 4 \,{\left (3 \, B a^{2} b^{3} + 5 \, A a b^{4}\right )} x^{3} + 6 \,{\left (3 \, B a^{3} b^{2} + 5 \, A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (3 \, B a^{4} b + 5 \, A a^{3} 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 (9 \, B a^{5} b + 15 \, A a^{4} b^{2} - 3 \,{\left (3 \, B a^{2} b^{4} + 5 \, A a b^{5}\right )} x^{3} - 11 \,{\left (3 \, B a^{3} b^{3} + 5 \, A a^{2} b^{4}\right )} x^{2} +{\left (33 \, B a^{4} b^{2} - 73 \, A a^{3} b^{3}\right )} x\right )} \sqrt{x}}{384 \,{\left (a^{4} b^{7} x^{4} + 4 \, a^{5} b^{6} x^{3} + 6 \, a^{6} b^{5} x^{2} + 4 \, a^{7} b^{4} x + a^{8} b^{3}\right )}}, -\frac{3 \,{\left (3 \, B a^{5} + 5 \, A a^{4} b +{\left (3 \, B a b^{4} + 5 \, A b^{5}\right )} x^{4} + 4 \,{\left (3 \, B a^{2} b^{3} + 5 \, A a b^{4}\right )} x^{3} + 6 \,{\left (3 \, B a^{3} b^{2} + 5 \, A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (3 \, B a^{4} b + 5 \, A a^{3} b^{2}\right )} x\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right ) +{\left (9 \, B a^{5} b + 15 \, A a^{4} b^{2} - 3 \,{\left (3 \, B a^{2} b^{4} + 5 \, A a b^{5}\right )} x^{3} - 11 \,{\left (3 \, B a^{3} b^{3} + 5 \, A a^{2} b^{4}\right )} x^{2} +{\left (33 \, B a^{4} b^{2} - 73 \, A a^{3} b^{3}\right )} x\right )} \sqrt{x}}{192 \,{\left (a^{4} b^{7} x^{4} + 4 \, a^{5} b^{6} x^{3} + 6 \, a^{6} b^{5} x^{2} + 4 \, a^{7} b^{4} x + a^{8} b^{3}\right )}}\right ] \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{\sqrt{x} \left (A + B x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16311, size = 200, normalized size = 0.76 \begin{align*} \frac{{\left (3 \, B a + 5 \, A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{64 \, \sqrt{a b} a^{3} b^{2} \mathrm{sgn}\left (b x + a\right )} + \frac{9 \, B a b^{3} x^{\frac{7}{2}} + 15 \, A b^{4} x^{\frac{7}{2}} + 33 \, B a^{2} b^{2} x^{\frac{5}{2}} + 55 \, A a b^{3} x^{\frac{5}{2}} - 33 \, B a^{3} b x^{\frac{3}{2}} + 73 \, A a^{2} b^{2} x^{\frac{3}{2}} - 9 \, B a^{4} \sqrt{x} - 15 \, A a^{3} b \sqrt{x}}{192 \,{\left (b x + a\right )}^{4} a^{3} b^{2} \mathrm{sgn}\left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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