Optimal. Leaf size=95 \[ \frac{(a+b x)^{3/2} (2 a B+3 A b)}{3 a}+\sqrt{a+b x} (2 a B+3 A b)-\sqrt{a} (2 a B+3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{A (a+b x)^{5/2}}{a x} \]
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Rubi [A] time = 0.0422989, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {78, 50, 63, 208} \[ \frac{(a+b x)^{3/2} (2 a B+3 A b)}{3 a}+\sqrt{a+b x} (2 a B+3 A b)-\sqrt{a} (2 a B+3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{A (a+b x)^{5/2}}{a x} \]
Antiderivative was successfully verified.
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Rule 78
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2} (A+B x)}{x^2} \, dx &=-\frac{A (a+b x)^{5/2}}{a x}+\frac{\left (\frac{3 A b}{2}+a B\right ) \int \frac{(a+b x)^{3/2}}{x} \, dx}{a}\\ &=\frac{(3 A b+2 a B) (a+b x)^{3/2}}{3 a}-\frac{A (a+b x)^{5/2}}{a x}+\frac{1}{2} (3 A b+2 a B) \int \frac{\sqrt{a+b x}}{x} \, dx\\ &=(3 A b+2 a B) \sqrt{a+b x}+\frac{(3 A b+2 a B) (a+b x)^{3/2}}{3 a}-\frac{A (a+b x)^{5/2}}{a x}+\frac{1}{2} (a (3 A b+2 a B)) \int \frac{1}{x \sqrt{a+b x}} \, dx\\ &=(3 A b+2 a B) \sqrt{a+b x}+\frac{(3 A b+2 a B) (a+b x)^{3/2}}{3 a}-\frac{A (a+b x)^{5/2}}{a x}+\frac{(a (3 A b+2 a B)) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{b}\\ &=(3 A b+2 a B) \sqrt{a+b x}+\frac{(3 A b+2 a B) (a+b x)^{3/2}}{3 a}-\frac{A (a+b x)^{5/2}}{a x}-\sqrt{a} (3 A b+2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0405195, size = 71, normalized size = 0.75 \[ \frac{\sqrt{a+b x} (a (8 B x-3 A)+2 b x (3 A+B x))}{3 x}-\sqrt{a} (2 a B+3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 77, normalized size = 0.8 \begin{align*}{\frac{2\,B}{3} \left ( bx+a \right ) ^{{\frac{3}{2}}}}+2\,Ab\sqrt{bx+a}+2\,Ba\sqrt{bx+a}+2\,a \left ( -1/2\,{\frac{A\sqrt{bx+a}}{x}}-1/2\,{\frac{3\,Ab+2\,Ba}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \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 = 2.37682, size = 371, normalized size = 3.91 \begin{align*} \left [\frac{3 \,{\left (2 \, B a + 3 \, A b\right )} \sqrt{a} x \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (2 \, B b x^{2} - 3 \, A a + 2 \,{\left (4 \, B a + 3 \, A b\right )} x\right )} \sqrt{b x + a}}{6 \, x}, \frac{3 \,{\left (2 \, B a + 3 \, A b\right )} \sqrt{-a} x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (2 \, B b x^{2} - 3 \, A a + 2 \,{\left (4 \, B a + 3 \, A b\right )} x\right )} \sqrt{b x + a}}{3 \, x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 18.2614, size = 202, normalized size = 2.13 \begin{align*} - \frac{A a^{2} b \sqrt{\frac{1}{a^{3}}} \log{\left (- a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} + \frac{A a^{2} b \sqrt{\frac{1}{a^{3}}} \log{\left (a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} + \frac{4 A a b \operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} - \frac{A a \sqrt{a + b x}}{x} + 2 A b \sqrt{a + b x} + \frac{2 B a^{2} \operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} + 2 B a \sqrt{a + b x} + B b \left (\begin{cases} \sqrt{a} x & \text{for}\: b = 0 \\\frac{2 \left (a + b x\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21214, size = 126, normalized size = 1.33 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}} B b + 6 \, \sqrt{b x + a} B a b + 6 \, \sqrt{b x + a} A b^{2} - \frac{3 \, \sqrt{b x + a} A a b}{x} + \frac{3 \,{\left (2 \, B a^{2} b + 3 \, A a b^{2}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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