Optimal. Leaf size=152 \[ -\frac{10 b^4 (c+d x)^{7/2} (b c-a d)}{7 d^6}+\frac{4 b^3 (c+d x)^{5/2} (b c-a d)^2}{d^6}-\frac{20 b^2 (c+d x)^{3/2} (b c-a d)^3}{3 d^6}+\frac{10 b \sqrt{c+d x} (b c-a d)^4}{d^6}+\frac{2 (b c-a d)^5}{d^6 \sqrt{c+d x}}+\frac{2 b^5 (c+d x)^{9/2}}{9 d^6} \]
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Rubi [A] time = 0.0486974, antiderivative size = 152, 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} \[ -\frac{10 b^4 (c+d x)^{7/2} (b c-a d)}{7 d^6}+\frac{4 b^3 (c+d x)^{5/2} (b c-a d)^2}{d^6}-\frac{20 b^2 (c+d x)^{3/2} (b c-a d)^3}{3 d^6}+\frac{10 b \sqrt{c+d x} (b c-a d)^4}{d^6}+\frac{2 (b c-a d)^5}{d^6 \sqrt{c+d x}}+\frac{2 b^5 (c+d x)^{9/2}}{9 d^6} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^5}{(c+d x)^{3/2}} \, dx &=\int \left (\frac{(-b c+a d)^5}{d^5 (c+d x)^{3/2}}+\frac{5 b (b c-a d)^4}{d^5 \sqrt{c+d x}}-\frac{10 b^2 (b c-a d)^3 \sqrt{c+d x}}{d^5}+\frac{10 b^3 (b c-a d)^2 (c+d x)^{3/2}}{d^5}-\frac{5 b^4 (b c-a d) (c+d x)^{5/2}}{d^5}+\frac{b^5 (c+d x)^{7/2}}{d^5}\right ) \, dx\\ &=\frac{2 (b c-a d)^5}{d^6 \sqrt{c+d x}}+\frac{10 b (b c-a d)^4 \sqrt{c+d x}}{d^6}-\frac{20 b^2 (b c-a d)^3 (c+d x)^{3/2}}{3 d^6}+\frac{4 b^3 (b c-a d)^2 (c+d x)^{5/2}}{d^6}-\frac{10 b^4 (b c-a d) (c+d x)^{7/2}}{7 d^6}+\frac{2 b^5 (c+d x)^{9/2}}{9 d^6}\\ \end{align*}
Mathematica [A] time = 0.116278, size = 123, normalized size = 0.81 \[ \frac{2 \left (-210 b^2 (c+d x)^2 (b c-a d)^3+126 b^3 (c+d x)^3 (b c-a d)^2-45 b^4 (c+d x)^4 (b c-a d)+315 b (c+d x) (b c-a d)^4+63 (b c-a d)^5+7 b^5 (c+d x)^5\right )}{63 d^6 \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 273, normalized size = 1.8 \begin{align*} -{\frac{-14\,{b}^{5}{x}^{5}{d}^{5}-90\,a{b}^{4}{d}^{5}{x}^{4}+20\,{b}^{5}c{d}^{4}{x}^{4}-252\,{a}^{2}{b}^{3}{d}^{5}{x}^{3}+144\,a{b}^{4}c{d}^{4}{x}^{3}-32\,{b}^{5}{c}^{2}{d}^{3}{x}^{3}-420\,{a}^{3}{b}^{2}{d}^{5}{x}^{2}+504\,{a}^{2}{b}^{3}c{d}^{4}{x}^{2}-288\,a{b}^{4}{c}^{2}{d}^{3}{x}^{2}+64\,{b}^{5}{c}^{3}{d}^{2}{x}^{2}-630\,{a}^{4}b{d}^{5}x+1680\,{a}^{3}{b}^{2}c{d}^{4}x-2016\,{a}^{2}{b}^{3}{c}^{2}{d}^{3}x+1152\,a{b}^{4}{c}^{3}{d}^{2}x-256\,{b}^{5}{c}^{4}dx+126\,{a}^{5}{d}^{5}-1260\,{a}^{4}bc{d}^{4}+3360\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-4032\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+2304\,a{b}^{4}{c}^{4}d-512\,{b}^{5}{c}^{5}}{63\,{d}^{6}}{\frac{1}{\sqrt{dx+c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.969638, size = 360, normalized size = 2.37 \begin{align*} \frac{2 \,{\left (\frac{7 \,{\left (d x + c\right )}^{\frac{9}{2}} b^{5} - 45 \,{\left (b^{5} c - a b^{4} d\right )}{\left (d x + c\right )}^{\frac{7}{2}} + 126 \,{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )}{\left (d x + c\right )}^{\frac{5}{2}} - 210 \,{\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{3}{2}} + 315 \,{\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 )} \sqrt{d x + c}}{d^{5}} + \frac{63 \,{\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 )}}{\sqrt{d x + c} d^{5}}\right )}}{63 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.112, size = 590, normalized size = 3.88 \begin{align*} \frac{2 \,{\left (7 \, b^{5} d^{5} x^{5} + 256 \, b^{5} c^{5} - 1152 \, a b^{4} c^{4} d + 2016 \, a^{2} b^{3} c^{3} d^{2} - 1680 \, a^{3} b^{2} c^{2} d^{3} + 630 \, a^{4} b c d^{4} - 63 \, a^{5} d^{5} - 5 \,{\left (2 \, b^{5} c d^{4} - 9 \, a b^{4} d^{5}\right )} x^{4} + 2 \,{\left (8 \, b^{5} c^{2} d^{3} - 36 \, a b^{4} c d^{4} + 63 \, a^{2} b^{3} d^{5}\right )} x^{3} - 2 \,{\left (16 \, b^{5} c^{3} d^{2} - 72 \, a b^{4} c^{2} d^{3} + 126 \, a^{2} b^{3} c d^{4} - 105 \, a^{3} b^{2} d^{5}\right )} x^{2} +{\left (128 \, b^{5} c^{4} d - 576 \, a b^{4} c^{3} d^{2} + 1008 \, a^{2} b^{3} c^{2} d^{3} - 840 \, a^{3} b^{2} c d^{4} + 315 \, a^{4} b d^{5}\right )} x\right )} \sqrt{d x + c}}{63 \,{\left (d^{7} x + c d^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 30.5541, size = 243, normalized size = 1.6 \begin{align*} \frac{2 b^{5} \left (c + d x\right )^{\frac{9}{2}}}{9 d^{6}} + \frac{\left (c + d x\right )^{\frac{7}{2}} \left (10 a b^{4} d - 10 b^{5} c\right )}{7 d^{6}} + \frac{\left (c + d x\right )^{\frac{5}{2}} \left (20 a^{2} b^{3} d^{2} - 40 a b^{4} c d + 20 b^{5} c^{2}\right )}{5 d^{6}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (20 a^{3} b^{2} d^{3} - 60 a^{2} b^{3} c d^{2} + 60 a b^{4} c^{2} d - 20 b^{5} c^{3}\right )}{3 d^{6}} + \frac{\sqrt{c + d x} \left (10 a^{4} b d^{4} - 40 a^{3} b^{2} c d^{3} + 60 a^{2} b^{3} c^{2} d^{2} - 40 a b^{4} c^{3} d + 10 b^{5} c^{4}\right )}{d^{6}} - \frac{2 \left (a d - b c\right )^{5}}{d^{6} \sqrt{c + d x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.07866, size = 473, normalized size = 3.11 \begin{align*} \frac{2 \,{\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 )}}{\sqrt{d x + c} d^{6}} + \frac{2 \,{\left (7 \,{\left (d x + c\right )}^{\frac{9}{2}} b^{5} d^{48} - 45 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{5} c d^{48} + 126 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{5} c^{2} d^{48} - 210 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{5} c^{3} d^{48} + 315 \, \sqrt{d x + c} b^{5} c^{4} d^{48} + 45 \,{\left (d x + c\right )}^{\frac{7}{2}} a b^{4} d^{49} - 252 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{4} c d^{49} + 630 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{4} c^{2} d^{49} - 1260 \, \sqrt{d x + c} a b^{4} c^{3} d^{49} + 126 \,{\left (d x + c\right )}^{\frac{5}{2}} a^{2} b^{3} d^{50} - 630 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{3} c d^{50} + 1890 \, \sqrt{d x + c} a^{2} b^{3} c^{2} d^{50} + 210 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} b^{2} d^{51} - 1260 \, \sqrt{d x + c} a^{3} b^{2} c d^{51} + 315 \, \sqrt{d x + c} a^{4} b d^{52}\right )}}{63 \, d^{54}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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