Optimal. Leaf size=302 \[ \frac{x^{9/2} (A b-a B)}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{7/2} (5 A b-9 a B)}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 x^{5/2} (a+b x) (5 A b-9 a B)}{20 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 x^{3/2} (a+b x) (5 A b-9 a B)}{12 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 a \sqrt{x} (a+b x) (5 A b-9 a B)}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 a^{3/2} (a+b x) (5 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.150586, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{x^{9/2} (A b-a B)}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{7/2} (5 A b-9 a B)}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 x^{5/2} (a+b x) (5 A b-9 a B)}{20 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 x^{3/2} (a+b x) (5 A b-9 a B)}{12 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 a \sqrt{x} (a+b x) (5 A b-9 a B)}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 a^{3/2} (a+b x) (5 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]
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^{7/2} (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^{7/2} (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^{9/2}}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left ((5 A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac{x^{7/2}}{\left (a b+b^2 x\right )^2} \, dx}{4 a \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(5 A b-9 a B) x^{7/2}}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{9/2}}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (7 (5 A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac{x^{5/2}}{a b+b^2 x} \, dx}{8 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(5 A b-9 a B) x^{7/2}}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{9/2}}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 (5 A b-9 a B) x^{5/2} (a+b x)}{20 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (7 (5 A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac{x^{3/2}}{a b+b^2 x} \, dx}{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(5 A b-9 a B) x^{7/2}}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{9/2}}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (5 A b-9 a B) x^{3/2} (a+b x)}{12 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 (5 A b-9 a B) x^{5/2} (a+b x)}{20 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (7 a (5 A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{x}}{a b+b^2 x} \, dx}{8 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(5 A b-9 a B) x^{7/2}}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{9/2}}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 a (5 A b-9 a B) \sqrt{x} (a+b x)}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (5 A b-9 a B) x^{3/2} (a+b x)}{12 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 (5 A b-9 a B) x^{5/2} (a+b x)}{20 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (7 a^2 (5 A b-9 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^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(5 A b-9 a B) x^{7/2}}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{9/2}}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 a (5 A b-9 a B) \sqrt{x} (a+b x)}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (5 A b-9 a B) x^{3/2} (a+b x)}{12 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 (5 A b-9 a B) x^{5/2} (a+b x)}{20 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (7 a^2 (5 A b-9 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^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(5 A b-9 a B) x^{7/2}}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{9/2}}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 a (5 A b-9 a B) \sqrt{x} (a+b x)}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (5 A b-9 a B) x^{3/2} (a+b x)}{12 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 (5 A b-9 a B) x^{5/2} (a+b x)}{20 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 a^{3/2} (5 A b-9 a B) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0370976, size = 79, normalized size = 0.26 \[ \frac{x^{9/2} \left (9 a^2 (A b-a B)+(a+b x)^2 (9 a B-5 A b) \, _2F_1\left (2,\frac{9}{2};\frac{11}{2};-\frac{b x}{a}\right )\right )}{18 a^3 b (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 283, normalized size = 0.9 \begin{align*}{\frac{bx+a}{60\,{b}^{5}} \left ( 24\,B\sqrt{ab}{x}^{9/2}{b}^{4}-72\,B\sqrt{ab}{x}^{7/2}a{b}^{3}+40\,A\sqrt{ab}{x}^{7/2}{b}^{4}+504\,B\sqrt{ab}{x}^{5/2}{a}^{2}{b}^{2}-280\,A\sqrt{ab}{x}^{5/2}a{b}^{3}-875\,A\sqrt{ab}{x}^{3/2}{a}^{2}{b}^{2}+525\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{2}{a}^{2}{b}^{3}+1575\,B\sqrt{ab}{x}^{3/2}{a}^{3}b-945\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{2}{a}^{3}{b}^{2}+1050\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) x{a}^{3}{b}^{2}-1890\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) x{a}^{4}b-525\,A\sqrt{ab}\sqrt{x}{a}^{3}b+525\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{4}b+945\,B\sqrt{ab}\sqrt{x}{a}^{4}-945\,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{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 = 1.30185, size = 905, normalized size = 3. \begin{align*} \left [-\frac{105 \,{\left (9 \, B a^{4} - 5 \, A a^{3} b +{\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 2 \,{\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\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 (24 \, B b^{4} x^{4} + 945 \, B a^{4} - 525 \, A a^{3} b - 8 \,{\left (9 \, B a b^{3} - 5 \, A b^{4}\right )} x^{3} + 56 \,{\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 175 \,{\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{120 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}, -\frac{105 \,{\left (9 \, B a^{4} - 5 \, A a^{3} b +{\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 2 \,{\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) -{\left (24 \, B b^{4} x^{4} + 945 \, B a^{4} - 525 \, A a^{3} b - 8 \,{\left (9 \, B a b^{3} - 5 \, A b^{4}\right )} x^{3} + 56 \,{\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 175 \,{\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{60 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\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 = 1.18586, size = 230, normalized size = 0.76 \begin{align*} -\frac{7 \,{\left (9 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{5} \mathrm{sgn}\left (b x + a\right )} + \frac{17 \, B a^{3} b x^{\frac{3}{2}} - 13 \, A a^{2} b^{2} x^{\frac{3}{2}} + 15 \, B a^{4} \sqrt{x} - 11 \, A a^{3} b \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} b^{5} \mathrm{sgn}\left (b x + a\right )} + \frac{2 \,{\left (3 \, B b^{12} x^{\frac{5}{2}} - 15 \, B a b^{11} x^{\frac{3}{2}} + 5 \, A b^{12} x^{\frac{3}{2}} + 90 \, B a^{2} b^{10} \sqrt{x} - 45 \, A a b^{11} \sqrt{x}\right )}}{15 \, b^{15} \mathrm{sgn}\left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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