Optimal. Leaf size=158 \[ -\frac{2 c}{3 d (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac{16 d \sqrt{a+b x} (a d+b c)}{3 \sqrt{c+d x} (b c-a d)^4}-\frac{8 (a d+b c)}{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}+\frac{2 (a d+b c)}{3 d (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^2} \]
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Rubi [A] time = 0.0550696, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ -\frac{2 c}{3 d (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac{16 d \sqrt{a+b x} (a d+b c)}{3 \sqrt{c+d x} (b c-a d)^4}-\frac{8 (a d+b c)}{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}+\frac{2 (a d+b c)}{3 d (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx &=-\frac{2 c}{3 d (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac{(b c+a d) \int \frac{1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx}{d (b c-a d)}\\ &=-\frac{2 c}{3 d (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac{2 (b c+a d)}{3 d (b c-a d)^2 (a+b x)^{3/2} \sqrt{c+d x}}+\frac{(4 (b c+a d)) \int \frac{1}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx}{3 (b c-a d)^2}\\ &=-\frac{2 c}{3 d (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac{2 (b c+a d)}{3 d (b c-a d)^2 (a+b x)^{3/2} \sqrt{c+d x}}-\frac{8 (b c+a d)}{3 (b c-a d)^3 \sqrt{a+b x} \sqrt{c+d x}}-\frac{(8 d (b c+a d)) \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/2}} \, dx}{3 (b c-a d)^3}\\ &=-\frac{2 c}{3 d (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac{2 (b c+a d)}{3 d (b c-a d)^2 (a+b x)^{3/2} \sqrt{c+d x}}-\frac{8 (b c+a d)}{3 (b c-a d)^3 \sqrt{a+b x} \sqrt{c+d x}}-\frac{16 d (b c+a d) \sqrt{a+b x}}{3 (b c-a d)^4 \sqrt{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.0407968, size = 134, normalized size = 0.85 \[ -\frac{2 \left (3 a^2 b d \left (4 c^2+7 c d x+4 d^2 x^2\right )+a^3 d^2 (2 c+3 d x)+a b^2 \left (21 c^2 d x+2 c^3+24 c d^2 x^2+8 d^3 x^3\right )+b^3 c x \left (3 c^2+12 c d x+8 d^2 x^2\right )\right )}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 198, normalized size = 1.3 \begin{align*} -{\frac{16\,a{b}^{2}{d}^{3}{x}^{3}+16\,{b}^{3}c{d}^{2}{x}^{3}+24\,{a}^{2}b{d}^{3}{x}^{2}+48\,a{b}^{2}c{d}^{2}{x}^{2}+24\,{b}^{3}{c}^{2}d{x}^{2}+6\,{a}^{3}{d}^{3}x+42\,{a}^{2}bc{d}^{2}x+42\,a{b}^{2}{c}^{2}dx+6\,{b}^{3}{c}^{3}x+4\,{a}^{3}c{d}^{2}+24\,{a}^{2}b{c}^{2}d+4\,a{b}^{2}{c}^{3}}{3\,{a}^{4}{d}^{4}-12\,{a}^{3}bc{d}^{3}+18\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-12\,a{b}^{3}{c}^{3}d+3\,{b}^{4}{c}^{4}} \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \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 [B] time = 9.48107, size = 936, normalized size = 5.92 \begin{align*} -\frac{2 \,{\left (2 \, a b^{2} c^{3} + 12 \, a^{2} b c^{2} d + 2 \, a^{3} c d^{2} + 8 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{3} + 12 \,{\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 3 \,{\left (b^{3} c^{3} + 7 \, a b^{2} c^{2} d + 7 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} +{\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \,{\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} +{\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \,{\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.23239, size = 933, normalized size = 5.91 \begin{align*} \frac{\frac{\sqrt{b x + a}{\left (\frac{{\left (5 \, b^{8} c^{4} d^{3}{\left | b \right |} - 12 \, a b^{7} c^{3} d^{4}{\left | b \right |} + 6 \, a^{2} b^{6} c^{2} d^{5}{\left | b \right |} + 4 \, a^{3} b^{5} c d^{6}{\left | b \right |} - 3 \, a^{4} b^{4} d^{7}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{3 \,{\left (2 \, b^{9} c^{5} d^{2}{\left | b \right |} - 7 \, a b^{8} c^{4} d^{3}{\left | b \right |} + 8 \, a^{2} b^{7} c^{3} d^{4}{\left | b \right |} - 2 \, a^{3} b^{6} c^{2} d^{5}{\left | b \right |} - 2 \, a^{4} b^{5} c d^{6}{\left | b \right |} + a^{5} b^{4} d^{7}{\left | b \right |}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} - \frac{16 \,{\left (3 \, \sqrt{b d} b^{8} c^{3} - \sqrt{b d} a b^{7} c^{2} d - 7 \, \sqrt{b d} a^{2} b^{6} c d^{2} + 5 \, \sqrt{b d} a^{3} b^{5} d^{3} - 6 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b^{6} c^{2} - 6 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{5} c d + 12 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{4} d^{2} + 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} b^{4} c + 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{3} d\right )}}{{\left (b^{3} c^{3}{\left | b \right |} - 3 \, a b^{2} c^{2} d{\left | b \right |} + 3 \, a^{2} b c d^{2}{\left | b \right |} - a^{3} d^{3}{\left | b \right |}\right )}{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3}}}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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