Optimal. Leaf size=100 \[ -\frac{6 b^2 (c+d x)^{7/2} (b c-a d)}{7 d^4}+\frac{6 b (c+d x)^{5/2} (b c-a d)^2}{5 d^4}-\frac{2 (c+d x)^{3/2} (b c-a d)^3}{3 d^4}+\frac{2 b^3 (c+d x)^{9/2}}{9 d^4} \]
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Rubi [A] time = 0.0355149, antiderivative size = 100, 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} \[ -\frac{6 b^2 (c+d x)^{7/2} (b c-a d)}{7 d^4}+\frac{6 b (c+d x)^{5/2} (b c-a d)^2}{5 d^4}-\frac{2 (c+d x)^{3/2} (b c-a d)^3}{3 d^4}+\frac{2 b^3 (c+d x)^{9/2}}{9 d^4} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x)^3 \sqrt{c+d x} \, dx &=\int \left (\frac{(-b c+a d)^3 \sqrt{c+d x}}{d^3}+\frac{3 b (b c-a d)^2 (c+d x)^{3/2}}{d^3}-\frac{3 b^2 (b c-a d) (c+d x)^{5/2}}{d^3}+\frac{b^3 (c+d x)^{7/2}}{d^3}\right ) \, dx\\ &=-\frac{2 (b c-a d)^3 (c+d x)^{3/2}}{3 d^4}+\frac{6 b (b c-a d)^2 (c+d x)^{5/2}}{5 d^4}-\frac{6 b^2 (b c-a d) (c+d x)^{7/2}}{7 d^4}+\frac{2 b^3 (c+d x)^{9/2}}{9 d^4}\\ \end{align*}
Mathematica [A] time = 0.0604461, size = 79, normalized size = 0.79 \[ \frac{2 (c+d x)^{3/2} \left (-135 b^2 (c+d x)^2 (b c-a d)+189 b (c+d x) (b c-a d)^2-105 (b c-a d)^3+35 b^3 (c+d x)^3\right )}{315 d^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 116, normalized size = 1.2 \begin{align*}{\frac{70\,{b}^{3}{x}^{3}{d}^{3}+270\,a{b}^{2}{d}^{3}{x}^{2}-60\,{b}^{3}c{d}^{2}{x}^{2}+378\,{a}^{2}b{d}^{3}x-216\,a{b}^{2}c{d}^{2}x+48\,{b}^{3}{c}^{2}dx+210\,{a}^{3}{d}^{3}-252\,{a}^{2}bc{d}^{2}+144\,a{b}^{2}{c}^{2}d-32\,{b}^{3}{c}^{3}}{315\,{d}^{4}} \left ( dx+c \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967357, size = 159, normalized size = 1.59 \begin{align*} \frac{2 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} b^{3} - 135 \,{\left (b^{3} c - a b^{2} d\right )}{\left (d x + c\right )}^{\frac{7}{2}} + 189 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}{\left (d x + c\right )}^{\frac{5}{2}} - 105 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left (d x + c\right )}^{\frac{3}{2}}\right )}}{315 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05845, size = 359, normalized size = 3.59 \begin{align*} \frac{2 \,{\left (35 \, b^{3} d^{4} x^{4} - 16 \, b^{3} c^{4} + 72 \, a b^{2} c^{3} d - 126 \, a^{2} b c^{2} d^{2} + 105 \, a^{3} c d^{3} + 5 \,{\left (b^{3} c d^{3} + 27 \, a b^{2} d^{4}\right )} x^{3} - 3 \,{\left (2 \, b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3} - 63 \, a^{2} b d^{4}\right )} x^{2} +{\left (8 \, b^{3} c^{3} d - 36 \, a b^{2} c^{2} d^{2} + 63 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} x\right )} \sqrt{d x + c}}{315 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.36295, size = 146, normalized size = 1.46 \begin{align*} \frac{2 \left (\frac{b^{3} \left (c + d x\right )^{\frac{9}{2}}}{9 d^{3}} + \frac{\left (c + d x\right )^{\frac{7}{2}} \left (3 a b^{2} d - 3 b^{3} c\right )}{7 d^{3}} + \frac{\left (c + d x\right )^{\frac{5}{2}} \left (3 a^{2} b d^{2} - 6 a b^{2} c d + 3 b^{3} c^{2}\right )}{5 d^{3}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{3 d^{3}}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06367, size = 188, normalized size = 1.88 \begin{align*} \frac{2 \,{\left (105 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} + \frac{63 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} a^{2} b}{d} + \frac{9 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} c + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2}\right )} a b^{2}}{d^{2}} + \frac{{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} - 135 \,{\left (d x + c\right )}^{\frac{7}{2}} c + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{2} - 105 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{3}\right )} b^{3}}{d^{3}}\right )}}{315 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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