Optimal. Leaf size=112 \[ \frac{b^2 (A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{(a+b x)^{3/2} (A b-6 a B)}{12 a x^2}+\frac{b \sqrt{a+b x} (A b-6 a B)}{8 a x}-\frac{A (a+b x)^{5/2}}{3 a x^3} \]
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Rubi [A] time = 0.0471317, antiderivative size = 112, 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, 47, 63, 208} \[ \frac{b^2 (A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{(a+b x)^{3/2} (A b-6 a B)}{12 a x^2}+\frac{b \sqrt{a+b x} (A b-6 a B)}{8 a x}-\frac{A (a+b x)^{5/2}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 78
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2} (A+B x)}{x^4} \, dx &=-\frac{A (a+b x)^{5/2}}{3 a x^3}+\frac{\left (-\frac{A b}{2}+3 a B\right ) \int \frac{(a+b x)^{3/2}}{x^3} \, dx}{3 a}\\ &=\frac{(A b-6 a B) (a+b x)^{3/2}}{12 a x^2}-\frac{A (a+b x)^{5/2}}{3 a x^3}-\frac{(b (A b-6 a B)) \int \frac{\sqrt{a+b x}}{x^2} \, dx}{8 a}\\ &=\frac{b (A b-6 a B) \sqrt{a+b x}}{8 a x}+\frac{(A b-6 a B) (a+b x)^{3/2}}{12 a x^2}-\frac{A (a+b x)^{5/2}}{3 a x^3}-\frac{\left (b^2 (A b-6 a B)\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{16 a}\\ &=\frac{b (A b-6 a B) \sqrt{a+b x}}{8 a x}+\frac{(A b-6 a B) (a+b x)^{3/2}}{12 a x^2}-\frac{A (a+b x)^{5/2}}{3 a x^3}-\frac{(b (A b-6 a B)) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{8 a}\\ &=\frac{b (A b-6 a B) \sqrt{a+b x}}{8 a x}+\frac{(A b-6 a B) (a+b x)^{3/2}}{12 a x^2}-\frac{A (a+b x)^{5/2}}{3 a x^3}+\frac{b^2 (A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0756756, size = 105, normalized size = 0.94 \[ \frac{-(a+b x) \left (4 a^2 (2 A+3 B x)+2 a b x (7 A+15 B x)+3 A b^2 x^2\right )-3 b^2 x^3 \sqrt{\frac{b x}{a}+1} (6 a B-A b) \tanh ^{-1}\left (\sqrt{\frac{b x}{a}+1}\right )}{24 a x^3 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 96, normalized size = 0.9 \begin{align*} 2\,{b}^{2} \left ({\frac{1}{{b}^{3}{x}^{3}} \left ( -1/16\,{\frac{ \left ( Ab+10\,Ba \right ) \left ( bx+a \right ) ^{5/2}}{a}}+ \left ( -1/6\,Ab+Ba \right ) \left ( bx+a \right ) ^{3/2}+ \left ( -3/8\,B{a}^{2}+1/16\,Aba \right ) \sqrt{bx+a} \right ) }+1/16\,{\frac{Ab-6\,Ba}{{a}^{3/2}}{\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 = 2.43122, size = 486, normalized size = 4.34 \begin{align*} \left [-\frac{3 \,{\left (6 \, B a b^{2} - A b^{3}\right )} \sqrt{a} x^{3} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (8 \, A a^{3} + 3 \,{\left (10 \, B a^{2} b + A a b^{2}\right )} x^{2} + 2 \,{\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x\right )} \sqrt{b x + a}}{48 \, a^{2} x^{3}}, \frac{3 \,{\left (6 \, B a b^{2} - A b^{3}\right )} \sqrt{-a} x^{3} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) -{\left (8 \, A a^{3} + 3 \,{\left (10 \, B a^{2} b + A a b^{2}\right )} x^{2} + 2 \,{\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x\right )} \sqrt{b x + a}}{24 \, a^{2} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 65.4678, size = 806, normalized size = 7.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2696, size = 196, normalized size = 1.75 \begin{align*} \frac{\frac{3 \,{\left (6 \, B a b^{3} - A b^{4}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{30 \,{\left (b x + a\right )}^{\frac{5}{2}} B a b^{3} - 48 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{2} b^{3} + 18 \, \sqrt{b x + a} B a^{3} b^{3} + 3 \,{\left (b x + a\right )}^{\frac{5}{2}} A b^{4} + 8 \,{\left (b x + a\right )}^{\frac{3}{2}} A a b^{4} - 3 \, \sqrt{b x + a} A a^{2} b^{4}}{a b^{3} x^{3}}}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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