Optimal. Leaf size=101 \[ \frac{16 d^2 \sqrt{a+b x}}{3 \sqrt{c+d x} (b c-a d)^3}+\frac{8 d}{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}-\frac{2}{3 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)} \]
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Rubi [A] time = 0.0178873, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{16 d^2 \sqrt{a+b x}}{3 \sqrt{c+d x} (b c-a d)^3}+\frac{8 d}{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}-\frac{2}{3 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx &=-\frac{2}{3 (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}-\frac{(4 d) \int \frac{1}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx}{3 (b c-a d)}\\ &=-\frac{2}{3 (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}+\frac{8 d}{3 (b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}+\frac{\left (8 d^2\right ) \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/2}} \, dx}{3 (b c-a d)^2}\\ &=-\frac{2}{3 (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}+\frac{8 d}{3 (b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}+\frac{16 d^2 \sqrt{a+b x}}{3 (b c-a d)^3 \sqrt{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.029055, size = 75, normalized size = 0.74 \[ \frac{2 \left (3 a^2 d^2+6 a b d (c+2 d x)+b^2 \left (-c^2+4 c d x+8 d^2 x^2\right )\right )}{3 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 105, normalized size = 1. \begin{align*} -{\frac{16\,{b}^{2}{d}^{2}{x}^{2}+24\,ab{d}^{2}x+8\,{b}^{2}cdx+6\,{a}^{2}{d}^{2}+12\,abcd-2\,{b}^{2}{c}^{2}}{3\,{a}^{3}{d}^{3}-9\,{a}^{2}bc{d}^{2}+9\,a{b}^{2}{c}^{2}d-3\,{b}^{3}{c}^{3}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{dx+c}}}} \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 = 4.70748, size = 544, normalized size = 5.39 \begin{align*} \frac{2 \,{\left (8 \, b^{2} d^{2} x^{2} - b^{2} c^{2} + 6 \, a b c d + 3 \, a^{2} d^{2} + 4 \,{\left (b^{2} c d + 3 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} +{\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{3} +{\left (b^{5} c^{4} - a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 5 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} x^{2} +{\left (2 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3} - a^{5} 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{1}{\left (a + b x\right )^{\frac{5}{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 = 1.32288, size = 497, normalized size = 4.92 \begin{align*} \frac{2 \, \sqrt{b x + a} b^{2} d^{2}}{{\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 )} \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} + \frac{4 \,{\left (5 \, \sqrt{b d} b^{6} c^{2} d - 10 \, \sqrt{b d} a b^{5} c d^{2} + 5 \, \sqrt{b d} a^{2} b^{4} d^{3} - 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} b^{4} 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 b^{3} 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^{2} d\right )}}{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 )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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