Optimal. Leaf size=62 \[ -\frac{4 d \sqrt{a+b x}}{\sqrt{c+d x} (b c-a d)^2}-\frac{2}{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)} \]
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Rubi [A] time = 0.0104859, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ -\frac{4 d \sqrt{a+b x}}{\sqrt{c+d x} (b c-a d)^2}-\frac{2}{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx &=-\frac{2}{(b c-a d) \sqrt{a+b x} \sqrt{c+d x}}-\frac{(2 d) \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/2}} \, dx}{b c-a d}\\ &=-\frac{2}{(b c-a d) \sqrt{a+b x} \sqrt{c+d x}}-\frac{4 d \sqrt{a+b x}}{(b c-a d)^2 \sqrt{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.0150307, size = 42, normalized size = 0.68 \[ -\frac{2 (a d+b (c+2 d x))}{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 52, normalized size = 0.8 \begin{align*} -2\,{\frac{2\,bdx+ad+bc}{\sqrt{bx+a}\sqrt{dx+c} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \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 [B] time = 2.87535, size = 259, normalized size = 4.18 \begin{align*} -\frac{2 \,{\left (2 \, b d x + b c + a d\right )} \sqrt{b x + a} \sqrt{d x + c}}{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x} \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{1}{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28159, size = 192, normalized size = 3.1 \begin{align*} -\frac{2 \, \sqrt{b x + a} b^{2} d}{{\left (b^{2} c^{2}{\left | b \right |} - 2 \, a b c d{\left | b \right |} + a^{2} d^{2}{\left | b \right |}\right )} \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} - \frac{4 \, \sqrt{b d} b^{2}}{{\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 )}{\left (b c{\left | b \right |} - a d{\left | b \right |}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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