Optimal. Leaf size=158 \[ -\frac{10 b^4 (c+d x)^{13/2} (b c-a d)}{13 d^6}+\frac{20 b^3 (c+d x)^{11/2} (b c-a d)^2}{11 d^6}-\frac{20 b^2 (c+d x)^{9/2} (b c-a d)^3}{9 d^6}+\frac{10 b (c+d x)^{7/2} (b c-a d)^4}{7 d^6}-\frac{2 (c+d x)^{5/2} (b c-a d)^5}{5 d^6}+\frac{2 b^5 (c+d x)^{15/2}}{15 d^6} \]
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Rubi [A] time = 0.0526392, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ -\frac{10 b^4 (c+d x)^{13/2} (b c-a d)}{13 d^6}+\frac{20 b^3 (c+d x)^{11/2} (b c-a d)^2}{11 d^6}-\frac{20 b^2 (c+d x)^{9/2} (b c-a d)^3}{9 d^6}+\frac{10 b (c+d x)^{7/2} (b c-a d)^4}{7 d^6}-\frac{2 (c+d x)^{5/2} (b c-a d)^5}{5 d^6}+\frac{2 b^5 (c+d x)^{15/2}}{15 d^6} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x)^5 (c+d x)^{3/2} \, dx &=\int \left (\frac{(-b c+a d)^5 (c+d x)^{3/2}}{d^5}+\frac{5 b (b c-a d)^4 (c+d x)^{5/2}}{d^5}-\frac{10 b^2 (b c-a d)^3 (c+d x)^{7/2}}{d^5}+\frac{10 b^3 (b c-a d)^2 (c+d x)^{9/2}}{d^5}-\frac{5 b^4 (b c-a d) (c+d x)^{11/2}}{d^5}+\frac{b^5 (c+d x)^{13/2}}{d^5}\right ) \, dx\\ &=-\frac{2 (b c-a d)^5 (c+d x)^{5/2}}{5 d^6}+\frac{10 b (b c-a d)^4 (c+d x)^{7/2}}{7 d^6}-\frac{20 b^2 (b c-a d)^3 (c+d x)^{9/2}}{9 d^6}+\frac{20 b^3 (b c-a d)^2 (c+d x)^{11/2}}{11 d^6}-\frac{10 b^4 (b c-a d) (c+d x)^{13/2}}{13 d^6}+\frac{2 b^5 (c+d x)^{15/2}}{15 d^6}\\ \end{align*}
Mathematica [A] time = 0.142616, size = 123, normalized size = 0.78 \[ \frac{2 (c+d x)^{5/2} \left (-50050 b^2 (c+d x)^2 (b c-a d)^3+40950 b^3 (c+d x)^3 (b c-a d)^2-17325 b^4 (c+d x)^4 (b c-a d)+32175 b (c+d x) (b c-a d)^4-9009 (b c-a d)^5+3003 b^5 (c+d x)^5\right )}{45045 d^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 273, normalized size = 1.7 \begin{align*}{\frac{6006\,{b}^{5}{x}^{5}{d}^{5}+34650\,a{b}^{4}{d}^{5}{x}^{4}-4620\,{b}^{5}c{d}^{4}{x}^{4}+81900\,{a}^{2}{b}^{3}{d}^{5}{x}^{3}-25200\,a{b}^{4}c{d}^{4}{x}^{3}+3360\,{b}^{5}{c}^{2}{d}^{3}{x}^{3}+100100\,{a}^{3}{b}^{2}{d}^{5}{x}^{2}-54600\,{a}^{2}{b}^{3}c{d}^{4}{x}^{2}+16800\,a{b}^{4}{c}^{2}{d}^{3}{x}^{2}-2240\,{b}^{5}{c}^{3}{d}^{2}{x}^{2}+64350\,{a}^{4}b{d}^{5}x-57200\,{a}^{3}{b}^{2}c{d}^{4}x+31200\,{a}^{2}{b}^{3}{c}^{2}{d}^{3}x-9600\,a{b}^{4}{c}^{3}{d}^{2}x+1280\,{b}^{5}{c}^{4}dx+18018\,{a}^{5}{d}^{5}-25740\,{a}^{4}bc{d}^{4}+22880\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-12480\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+3840\,a{b}^{4}{c}^{4}d-512\,{b}^{5}{c}^{5}}{45045\,{d}^{6}} \left ( dx+c \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.962181, size = 350, normalized size = 2.22 \begin{align*} \frac{2 \,{\left (3003 \,{\left (d x + c\right )}^{\frac{15}{2}} b^{5} - 17325 \,{\left (b^{5} c - a b^{4} d\right )}{\left (d x + c\right )}^{\frac{13}{2}} + 40950 \,{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )}{\left (d x + c\right )}^{\frac{11}{2}} - 50050 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )}{\left (d x + c\right )}^{\frac{9}{2}} + 32175 \,{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}{\left (d x + c\right )}^{\frac{7}{2}} - 9009 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )}{\left (d x + c\right )}^{\frac{5}{2}}\right )}}{45045 \, d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.10938, size = 953, normalized size = 6.03 \begin{align*} \frac{2 \,{\left (3003 \, b^{5} d^{7} x^{7} - 256 \, b^{5} c^{7} + 1920 \, a b^{4} c^{6} d - 6240 \, a^{2} b^{3} c^{5} d^{2} + 11440 \, a^{3} b^{2} c^{4} d^{3} - 12870 \, a^{4} b c^{3} d^{4} + 9009 \, a^{5} c^{2} d^{5} + 231 \,{\left (16 \, b^{5} c d^{6} + 75 \, a b^{4} d^{7}\right )} x^{6} + 63 \,{\left (b^{5} c^{2} d^{5} + 350 \, a b^{4} c d^{6} + 650 \, a^{2} b^{3} d^{7}\right )} x^{5} - 35 \,{\left (2 \, b^{5} c^{3} d^{4} - 15 \, a b^{4} c^{2} d^{5} - 1560 \, a^{2} b^{3} c d^{6} - 1430 \, a^{3} b^{2} d^{7}\right )} x^{4} + 5 \,{\left (16 \, b^{5} c^{4} d^{3} - 120 \, a b^{4} c^{3} d^{4} + 390 \, a^{2} b^{3} c^{2} d^{5} + 14300 \, a^{3} b^{2} c d^{6} + 6435 \, a^{4} b d^{7}\right )} x^{3} - 3 \,{\left (32 \, b^{5} c^{5} d^{2} - 240 \, a b^{4} c^{4} d^{3} + 780 \, a^{2} b^{3} c^{3} d^{4} - 1430 \, a^{3} b^{2} c^{2} d^{5} - 17160 \, a^{4} b c d^{6} - 3003 \, a^{5} d^{7}\right )} x^{2} +{\left (128 \, b^{5} c^{6} d - 960 \, a b^{4} c^{5} d^{2} + 3120 \, a^{2} b^{3} c^{4} d^{3} - 5720 \, a^{3} b^{2} c^{3} d^{4} + 6435 \, a^{4} b c^{2} d^{5} + 18018 \, a^{5} c d^{6}\right )} x\right )} \sqrt{d x + c}}{45045 \, d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.2502, size = 763, normalized size = 4.83 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10514, size = 869, normalized size = 5.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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