Optimal. Leaf size=107 \[ -\frac{(a+b x)^{3/2} (4 a B+A b)}{4 a x}+\frac{3 b \sqrt{a+b x} (4 a B+A b)}{4 a}-\frac{3 b (4 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{A (a+b x)^{5/2}}{2 a x^2} \]
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Rubi [A] time = 0.04583, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {78, 47, 50, 63, 208} \[ -\frac{(a+b x)^{3/2} (4 a B+A b)}{4 a x}+\frac{3 b \sqrt{a+b x} (4 a B+A b)}{4 a}-\frac{3 b (4 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{A (a+b x)^{5/2}}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 78
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2} (A+B x)}{x^3} \, dx &=-\frac{A (a+b x)^{5/2}}{2 a x^2}+\frac{\left (\frac{A b}{2}+2 a B\right ) \int \frac{(a+b x)^{3/2}}{x^2} \, dx}{2 a}\\ &=-\frac{(A b+4 a B) (a+b x)^{3/2}}{4 a x}-\frac{A (a+b x)^{5/2}}{2 a x^2}+\frac{(3 b (A b+4 a B)) \int \frac{\sqrt{a+b x}}{x} \, dx}{8 a}\\ &=\frac{3 b (A b+4 a B) \sqrt{a+b x}}{4 a}-\frac{(A b+4 a B) (a+b x)^{3/2}}{4 a x}-\frac{A (a+b x)^{5/2}}{2 a x^2}+\frac{1}{8} (3 b (A b+4 a B)) \int \frac{1}{x \sqrt{a+b x}} \, dx\\ &=\frac{3 b (A b+4 a B) \sqrt{a+b x}}{4 a}-\frac{(A b+4 a B) (a+b x)^{3/2}}{4 a x}-\frac{A (a+b x)^{5/2}}{2 a x^2}+\frac{1}{4} (3 (A b+4 a B)) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )\\ &=\frac{3 b (A b+4 a B) \sqrt{a+b x}}{4 a}-\frac{(A b+4 a B) (a+b x)^{3/2}}{4 a x}-\frac{A (a+b x)^{5/2}}{2 a x^2}-\frac{3 b (A b+4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 \sqrt{a}}\\ \end{align*}
Mathematica [C] time = 0.0232135, size = 55, normalized size = 0.51 \[ \frac{(a+b x)^{5/2} \left (b x^2 (4 a B+A b) \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{b x}{a}+1\right )-5 a^2 A\right )}{10 a^3 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 84, normalized size = 0.8 \begin{align*} 2\,b \left ( B\sqrt{bx+a}+{\frac{ \left ( -5/8\,Ab-1/2\,Ba \right ) \left ( bx+a \right ) ^{3/2}+ \left ( 1/2\,B{a}^{2}+3/8\,Aba \right ) \sqrt{bx+a}}{{b}^{2}{x}^{2}}}-3/8\,{\frac{Ab+4\,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 = 2.44013, size = 414, normalized size = 3.87 \begin{align*} \left [\frac{3 \,{\left (4 \, B a b + A b^{2}\right )} \sqrt{a} x^{2} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (8 \, B a b x^{2} - 2 \, A a^{2} -{\left (4 \, B a^{2} + 5 \, A a b\right )} x\right )} \sqrt{b x + a}}{8 \, a x^{2}}, \frac{3 \,{\left (4 \, B a b + A b^{2}\right )} \sqrt{-a} x^{2} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (8 \, B a b x^{2} - 2 \, A a^{2} -{\left (4 \, B a^{2} + 5 \, A a b\right )} x\right )} \sqrt{b x + a}}{4 \, a x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 44.2251, size = 428, normalized size = 4. \begin{align*} - \frac{10 A a^{3} b^{2} \sqrt{a + b x}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} + \frac{6 A a^{2} b^{2} \left (a + b x\right )^{\frac{3}{2}}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} + \frac{3 A a^{2} b^{2} \sqrt{\frac{1}{a^{5}}} \log{\left (- a^{3} \sqrt{\frac{1}{a^{5}}} + \sqrt{a + b x} \right )}}{8} - \frac{3 A a^{2} b^{2} \sqrt{\frac{1}{a^{5}}} \log{\left (a^{3} \sqrt{\frac{1}{a^{5}}} + \sqrt{a + b x} \right )}}{8} - A a b^{2} \sqrt{\frac{1}{a^{3}}} \log{\left (- a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )} + A a b^{2} \sqrt{\frac{1}{a^{3}}} \log{\left (a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )} + \frac{2 A b^{2} \operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} - \frac{2 A b \sqrt{a + b x}}{x} - \frac{B a^{2} b \sqrt{\frac{1}{a^{3}}} \log{\left (- a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} + \frac{B 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 B a b \operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} - \frac{B a \sqrt{a + b x}}{x} + 2 B b \sqrt{a + b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2423, size = 161, normalized size = 1.5 \begin{align*} \frac{8 \, \sqrt{b x + a} B b^{2} + \frac{3 \,{\left (4 \, B a b^{2} + A b^{3}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{4 \,{\left (b x + a\right )}^{\frac{3}{2}} B a b^{2} - 4 \, \sqrt{b x + a} B a^{2} b^{2} + 5 \,{\left (b x + a\right )}^{\frac{3}{2}} A b^{3} - 3 \, \sqrt{b x + a} A a b^{3}}{b^{2} x^{2}}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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