Optimal. Leaf size=185 \[ \frac{3 \sqrt{x} (a B+A b)}{128 a^3 b^3 (a+b x)}+\frac{\sqrt{x} (a B+A b)}{64 a^2 b^3 (a+b x)^2}+\frac{3 (a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{7/2} b^{7/2}}-\frac{x^{3/2} (a B+A b)}{8 a b^2 (a+b x)^4}-\frac{\sqrt{x} (a B+A b)}{16 a b^3 (a+b x)^3}+\frac{x^{5/2} (A b-a B)}{5 a b (a+b x)^5} \]
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Rubi [A] time = 0.0925746, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {27, 78, 47, 51, 63, 205} \[ \frac{3 \sqrt{x} (a B+A b)}{128 a^3 b^3 (a+b x)}+\frac{\sqrt{x} (a B+A b)}{64 a^2 b^3 (a+b x)^2}+\frac{3 (a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{7/2} b^{7/2}}-\frac{x^{3/2} (a B+A b)}{8 a b^2 (a+b x)^4}-\frac{\sqrt{x} (a B+A b)}{16 a b^3 (a+b x)^3}+\frac{x^{5/2} (A b-a B)}{5 a b (a+b x)^5} \]
Antiderivative was successfully verified.
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Rule 27
Rule 78
Rule 47
Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{x^{3/2} (A+B x)}{(a+b x)^6} \, dx\\ &=\frac{(A b-a B) x^{5/2}}{5 a b (a+b x)^5}+\frac{(A b+a B) \int \frac{x^{3/2}}{(a+b x)^5} \, dx}{2 a b}\\ &=\frac{(A b-a B) x^{5/2}}{5 a b (a+b x)^5}-\frac{(A b+a B) x^{3/2}}{8 a b^2 (a+b x)^4}+\frac{(3 (A b+a B)) \int \frac{\sqrt{x}}{(a+b x)^4} \, dx}{16 a b^2}\\ &=\frac{(A b-a B) x^{5/2}}{5 a b (a+b x)^5}-\frac{(A b+a B) x^{3/2}}{8 a b^2 (a+b x)^4}-\frac{(A b+a B) \sqrt{x}}{16 a b^3 (a+b x)^3}+\frac{(A b+a B) \int \frac{1}{\sqrt{x} (a+b x)^3} \, dx}{32 a b^3}\\ &=\frac{(A b-a B) x^{5/2}}{5 a b (a+b x)^5}-\frac{(A b+a B) x^{3/2}}{8 a b^2 (a+b x)^4}-\frac{(A b+a B) \sqrt{x}}{16 a b^3 (a+b x)^3}+\frac{(A b+a B) \sqrt{x}}{64 a^2 b^3 (a+b x)^2}+\frac{(3 (A b+a B)) \int \frac{1}{\sqrt{x} (a+b x)^2} \, dx}{128 a^2 b^3}\\ &=\frac{(A b-a B) x^{5/2}}{5 a b (a+b x)^5}-\frac{(A b+a B) x^{3/2}}{8 a b^2 (a+b x)^4}-\frac{(A b+a B) \sqrt{x}}{16 a b^3 (a+b x)^3}+\frac{(A b+a B) \sqrt{x}}{64 a^2 b^3 (a+b x)^2}+\frac{3 (A b+a B) \sqrt{x}}{128 a^3 b^3 (a+b x)}+\frac{(3 (A b+a B)) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{256 a^3 b^3}\\ &=\frac{(A b-a B) x^{5/2}}{5 a b (a+b x)^5}-\frac{(A b+a B) x^{3/2}}{8 a b^2 (a+b x)^4}-\frac{(A b+a B) \sqrt{x}}{16 a b^3 (a+b x)^3}+\frac{(A b+a B) \sqrt{x}}{64 a^2 b^3 (a+b x)^2}+\frac{3 (A b+a B) \sqrt{x}}{128 a^3 b^3 (a+b x)}+\frac{(3 (A b+a B)) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{128 a^3 b^3}\\ &=\frac{(A b-a B) x^{5/2}}{5 a b (a+b x)^5}-\frac{(A b+a B) x^{3/2}}{8 a b^2 (a+b x)^4}-\frac{(A b+a B) \sqrt{x}}{16 a b^3 (a+b x)^3}+\frac{(A b+a B) \sqrt{x}}{64 a^2 b^3 (a+b x)^2}+\frac{3 (A b+a B) \sqrt{x}}{128 a^3 b^3 (a+b x)}+\frac{3 (A b+a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{7/2} b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0330053, size = 60, normalized size = 0.32 \[ \frac{x^{5/2} \left (\frac{5 a^5 (A b-a B)}{(a+b x)^5}+5 (a B+A b) \, _2F_1\left (\frac{5}{2},5;\frac{7}{2};-\frac{b x}{a}\right )\right )}{25 a^6 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 143, normalized size = 0.8 \begin{align*} 2\,{\frac{1}{ \left ( bx+a \right ) ^{5}} \left ({\frac{ \left ( 3\,Ab+3\,aB \right ) b{x}^{9/2}}{256\,{a}^{3}}}+{\frac{ \left ( 7\,Ab+7\,aB \right ){x}^{7/2}}{128\,{a}^{2}}}+1/10\,{\frac{ \left ( Ab-aB \right ){x}^{5/2}}{ab}}-{\frac{ \left ( 7\,Ab+7\,aB \right ){x}^{3/2}}{128\,{b}^{2}}}-{\frac{ \left ( 3\,Ab+3\,aB \right ) a\sqrt{x}}{256\,{b}^{3}}} \right ) }+{\frac{3\,A}{128\,{a}^{3}{b}^{2}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,B}{128\,{a}^{2}{b}^{3}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\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 = 1.7149, size = 1315, normalized size = 7.11 \begin{align*} \left [-\frac{15 \,{\left (B a^{6} + A a^{5} b +{\left (B a b^{5} + A b^{6}\right )} x^{5} + 5 \,{\left (B a^{2} b^{4} + A a b^{5}\right )} x^{4} + 10 \,{\left (B a^{3} b^{3} + A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (B a^{4} b^{2} + A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (B a^{5} b + A a^{4} b^{2}\right )} x\right )} \sqrt{-a b} \log \left (\frac{b x - a - 2 \, \sqrt{-a b} \sqrt{x}}{b x + a}\right ) + 2 \,{\left (15 \, B a^{6} b + 15 \, A a^{5} b^{2} - 15 \,{\left (B a^{2} b^{5} + A a b^{6}\right )} x^{4} - 70 \,{\left (B a^{3} b^{4} + A a^{2} b^{5}\right )} x^{3} + 128 \,{\left (B a^{4} b^{3} - A a^{3} b^{4}\right )} x^{2} + 70 \,{\left (B a^{5} b^{2} + A a^{4} b^{3}\right )} x\right )} \sqrt{x}}{1280 \,{\left (a^{4} b^{9} x^{5} + 5 \, a^{5} b^{8} x^{4} + 10 \, a^{6} b^{7} x^{3} + 10 \, a^{7} b^{6} x^{2} + 5 \, a^{8} b^{5} x + a^{9} b^{4}\right )}}, -\frac{15 \,{\left (B a^{6} + A a^{5} b +{\left (B a b^{5} + A b^{6}\right )} x^{5} + 5 \,{\left (B a^{2} b^{4} + A a b^{5}\right )} x^{4} + 10 \,{\left (B a^{3} b^{3} + A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (B a^{4} b^{2} + A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (B a^{5} b + A a^{4} b^{2}\right )} x\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right ) +{\left (15 \, B a^{6} b + 15 \, A a^{5} b^{2} - 15 \,{\left (B a^{2} b^{5} + A a b^{6}\right )} x^{4} - 70 \,{\left (B a^{3} b^{4} + A a^{2} b^{5}\right )} x^{3} + 128 \,{\left (B a^{4} b^{3} - A a^{3} b^{4}\right )} x^{2} + 70 \,{\left (B a^{5} b^{2} + A a^{4} b^{3}\right )} x\right )} \sqrt{x}}{640 \,{\left (a^{4} b^{9} x^{5} + 5 \, a^{5} b^{8} x^{4} + 10 \, a^{6} b^{7} x^{3} + 10 \, a^{7} b^{6} x^{2} + 5 \, a^{8} b^{5} x + a^{9} b^{4}\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 = 1.21849, size = 208, normalized size = 1.12 \begin{align*} \frac{3 \,{\left (B a + A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{128 \, \sqrt{a b} a^{3} b^{3}} + \frac{15 \, B a b^{4} x^{\frac{9}{2}} + 15 \, A b^{5} x^{\frac{9}{2}} + 70 \, B a^{2} b^{3} x^{\frac{7}{2}} + 70 \, A a b^{4} x^{\frac{7}{2}} - 128 \, B a^{3} b^{2} x^{\frac{5}{2}} + 128 \, A a^{2} b^{3} x^{\frac{5}{2}} - 70 \, B a^{4} b x^{\frac{3}{2}} - 70 \, A a^{3} b^{2} x^{\frac{3}{2}} - 15 \, B a^{5} \sqrt{x} - 15 \, A a^{4} b \sqrt{x}}{640 \,{\left (b x + a\right )}^{5} a^{3} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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