Optimal. Leaf size=118 \[ \frac{2 a}{b \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}+\frac{4 \sqrt{a+b x} (3 a d+b c)}{3 \sqrt{c+d x} (b c-a d)^3}+\frac{2 \sqrt{a+b x} (3 a d+b c)}{3 b (c+d x)^{3/2} (b c-a d)^2} \]
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Rubi [A] time = 0.0375537, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ \frac{2 a}{b \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}+\frac{4 \sqrt{a+b x} (3 a d+b c)}{3 \sqrt{c+d x} (b c-a d)^3}+\frac{2 \sqrt{a+b x} (3 a d+b c)}{3 b (c+d x)^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{x}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx &=\frac{2 a}{b (b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}+\frac{(b c+3 a d) \int \frac{1}{\sqrt{a+b x} (c+d x)^{5/2}} \, dx}{b (b c-a d)}\\ &=\frac{2 a}{b (b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}+\frac{2 (b c+3 a d) \sqrt{a+b x}}{3 b (b c-a d)^2 (c+d x)^{3/2}}+\frac{(2 (b c+3 a d)) \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/2}} \, dx}{3 (b c-a d)^2}\\ &=\frac{2 a}{b (b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}+\frac{2 (b c+3 a d) \sqrt{a+b x}}{3 b (b c-a d)^2 (c+d x)^{3/2}}+\frac{4 (b c+3 a d) \sqrt{a+b x}}{3 (b c-a d)^3 \sqrt{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.0283627, size = 83, normalized size = 0.7 \[ \frac{2 \left (a^2 d (2 c+3 d x)+2 a b \left (3 c^2+5 c d x+3 d^2 x^2\right )+b^2 c x (3 c+2 d x)\right )}{3 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 115, normalized size = 1. \begin{align*} -{\frac{12\,ab{d}^{2}{x}^{2}+4\,{b}^{2}cd{x}^{2}+6\,{a}^{2}{d}^{2}x+20\,abcdx+6\,{b}^{2}{c}^{2}x+4\,{a}^{2}cd+12\,ab{c}^{2}}{3\,{a}^{3}{d}^{3}-9\,{a}^{2}cb{d}^{2}+9\,a{b}^{2}{c}^{2}d-3\,{b}^{3}{c}^{3}}{\frac{1}{\sqrt{bx+a}}} \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 = 5.62413, size = 567, normalized size = 4.81 \begin{align*} \frac{2 \,{\left (6 \, a b c^{2} + 2 \, a^{2} c d + 2 \,{\left (b^{2} c d + 3 \, a b d^{2}\right )} x^{2} +{\left (3 \, b^{2} c^{2} + 10 \, a b c d + 3 \, a^{2} d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} +{\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} +{\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} +{\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\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{3}{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 = 1.87349, size = 421, normalized size = 3.57 \begin{align*} \frac{\frac{48 \, \sqrt{b d} a b^{3}}{{\left (b^{2} c^{2}{\left | b \right |} - 2 \, a b c d{\left | b \right |} + a^{2} d^{2}{\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 )}} - \frac{\sqrt{b x + a}{\left (\frac{{\left (2 \, b^{7} c^{3} d^{2}{\left | b \right |} - a b^{6} c^{2} d^{3}{\left | b \right |} - 4 \, a^{2} b^{5} c d^{4}{\left | b \right |} + 3 \, a^{3} b^{4} d^{5}{\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 (b^{8} c^{4} d{\left | b \right |} - 2 \, a b^{7} c^{3} d^{2}{\left | b \right |} + 2 \, a^{3} b^{5} c d^{4}{\left | b \right |} - a^{4} b^{4} d^{5}{\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}}}}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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