Optimal. Leaf size=151 \[ \frac{2 \sqrt{a+b x} \left (-3 a^2 d^2-6 a b c d+b^2 c^2\right )}{3 b d \sqrt{c+d x} (b c-a d)^3}-\frac{2 \sqrt{a+b x} \left (3 a^2 d^2+b^2 c^2\right )}{3 b^2 d (c+d x)^{3/2} (b c-a d)^2}-\frac{2 a^2}{b^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.127016, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {89, 78, 37} \[ \frac{2 \sqrt{a+b x} \left (-3 a^2 d^2-6 a b c d+b^2 c^2\right )}{3 b d \sqrt{c+d x} (b c-a d)^3}-\frac{2 \sqrt{a+b x} \left (3 a^2 d^2+b^2 c^2\right )}{3 b^2 d (c+d x)^{3/2} (b c-a d)^2}-\frac{2 a^2}{b^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 37
Rubi steps
\begin{align*} \int \frac{x^2}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx &=-\frac{2 a^2}{b^2 (b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}+\frac{2 \int \frac{-\frac{1}{2} a (b c+3 a d)+\frac{1}{2} b (b c-a d) x}{\sqrt{a+b x} (c+d x)^{5/2}} \, dx}{b^2 (b c-a d)}\\ &=-\frac{2 a^2}{b^2 (b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}-\frac{2 \left (b^2 c^2+3 a^2 d^2\right ) \sqrt{a+b x}}{3 b^2 d (b c-a d)^2 (c+d x)^{3/2}}+\frac{\left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/2}} \, dx}{3 b d (b c-a d)^2}\\ &=-\frac{2 a^2}{b^2 (b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}-\frac{2 \left (b^2 c^2+3 a^2 d^2\right ) \sqrt{a+b x}}{3 b^2 d (b c-a d)^2 (c+d x)^{3/2}}+\frac{2 \left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \sqrt{a+b x}}{3 b d (b c-a d)^3 \sqrt{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.0413314, size = 82, normalized size = 0.54 \[ \frac{-2 a^2 \left (8 c^2+12 c d x+3 d^2 x^2\right )-4 a b c x (2 c+3 d x)+2 b^2 c^2 x^2}{3 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 111, normalized size = 0.7 \begin{align*}{\frac{6\,{a}^{2}{d}^{2}{x}^{2}+12\,abcd{x}^{2}-2\,{b}^{2}{c}^{2}{x}^{2}+24\,{a}^{2}cdx+8\,ab{c}^{2}x+16\,{a}^{2}{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 = 5.94242, size = 548, normalized size = 3.63 \begin{align*} -\frac{2 \,{\left (8 \, a^{2} c^{2} -{\left (b^{2} c^{2} - 6 \, a b c d - 3 \, a^{2} d^{2}\right )} x^{2} + 4 \,{\left (a b c^{2} + 3 \, a^{2} c d\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^{2}}{\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 = 2.13092, size = 423, normalized size = 2.8 \begin{align*} -\frac{4 \, \sqrt{b d} a^{2} b}{{\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 (b^{6} c^{4} d{\left | b \right |} - 8 \, a b^{5} c^{3} d^{2}{\left | b \right |} + 13 \, a^{2} b^{4} c^{2} d^{3}{\left | b \right |} - 6 \, a^{3} b^{3} c d^{4}{\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{6 \,{\left (a b^{6} c^{4} d{\left | b \right |} - 3 \, a^{2} b^{5} c^{3} d^{2}{\left | b \right |} + 3 \, a^{3} b^{4} c^{2} d^{3}{\left | b \right |} - a^{4} b^{3} c d^{4}{\left | b \right |}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{24 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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