Optimal. Leaf size=238 \[ \frac{2 x^{5/2} (a+b x) (A b-a B)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a x^{3/2} (a+b x) (A b-a B)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 \sqrt{x} (a+b x) (A b-a B)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a^{5/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{7/2} (a+b x)}{7 b \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.108314, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{2 x^{5/2} (a+b x) (A b-a B)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a x^{3/2} (a+b x) (A b-a B)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 \sqrt{x} (a+b x) (A b-a B)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a^{5/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{7/2} (a+b x)}{7 b \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 80
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{5/2} (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{x^{5/2} (A+B x)}{a b+b^2 x} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 B x^{7/2} (a+b x)}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 \left (\frac{7 A b^2}{2}-\frac{7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{x^{5/2}}{a b+b^2 x} \, dx}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (A b-a B) x^{5/2} (a+b x)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{7/2} (a+b x)}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (2 a \left (\frac{7 A b^2}{2}-\frac{7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{x^{3/2}}{a b+b^2 x} \, dx}{7 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 a (A b-a B) x^{3/2} (a+b x)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) x^{5/2} (a+b x)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{7/2} (a+b x)}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 a^2 \left (\frac{7 A b^2}{2}-\frac{7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{x}}{a b+b^2 x} \, dx}{7 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 a^2 (A b-a B) \sqrt{x} (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a (A b-a B) x^{3/2} (a+b x)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) x^{5/2} (a+b x)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{7/2} (a+b x)}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (2 a^3 \left (\frac{7 A b^2}{2}-\frac{7 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}{7 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 a^2 (A b-a B) \sqrt{x} (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a (A b-a B) x^{3/2} (a+b x)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) x^{5/2} (a+b x)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{7/2} (a+b x)}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (4 a^3 \left (\frac{7 A b^2}{2}-\frac{7 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 )}{7 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 a^2 (A b-a B) \sqrt{x} (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a (A b-a B) x^{3/2} (a+b x)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) x^{5/2} (a+b x)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{7/2} (a+b x)}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a^{5/2} (A b-a B) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0652006, size = 120, normalized size = 0.5 \[ \frac{2 (a+b x) \left (\sqrt{b} \sqrt{x} \left (35 a^2 b (3 A+B x)-105 a^3 B-7 a b^2 x (5 A+3 B x)+3 b^3 x^2 (7 A+5 B x)\right )+105 a^{5/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{105 b^{9/2} \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 163, normalized size = 0.7 \begin{align*}{\frac{2\,bx+2\,a}{105\,{b}^{4}} \left ( 15\,B\sqrt{ab}{x}^{7/2}{b}^{3}+21\,A\sqrt{ab}{x}^{5/2}{b}^{3}-21\,B\sqrt{ab}{x}^{5/2}a{b}^{2}-35\,A\sqrt{ab}{x}^{3/2}a{b}^{2}+35\,B\sqrt{ab}{x}^{3/2}{a}^{2}b+105\,A\sqrt{ab}\sqrt{x}{a}^{2}b-105\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{3}b-105\,B\sqrt{ab}\sqrt{x}{a}^{3}+105\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{4} \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 = 1.32508, size = 524, normalized size = 2.2 \begin{align*} \left [-\frac{105 \,{\left (B a^{3} - A a^{2} 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 (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \,{\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt{x}}{105 \, b^{4}}, \frac{2 \,{\left (105 \,{\left (B a^{3} - A a^{2} b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) +{\left (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \,{\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt{x}\right )}}{105 \, b^{4}}\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 = 1.11492, size = 228, normalized size = 0.96 \begin{align*} \frac{2 \,{\left (B a^{4} \mathrm{sgn}\left (b x + a\right ) - A a^{3} b \mathrm{sgn}\left (b x + a\right )\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{2 \,{\left (15 \, B b^{6} x^{\frac{7}{2}} \mathrm{sgn}\left (b x + a\right ) - 21 \, B a b^{5} x^{\frac{5}{2}} \mathrm{sgn}\left (b x + a\right ) + 21 \, A b^{6} x^{\frac{5}{2}} \mathrm{sgn}\left (b x + a\right ) + 35 \, B a^{2} b^{4} x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) - 35 \, A a b^{5} x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) - 105 \, B a^{3} b^{3} \sqrt{x} \mathrm{sgn}\left (b x + a\right ) + 105 \, A a^{2} b^{4} \sqrt{x} \mathrm{sgn}\left (b x + a\right )\right )}}{105 \, b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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