Optimal. Leaf size=113 \[ -\frac{2 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^4 \sqrt{c+d x}}-\frac{2 \sqrt{c+d x} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^4}+\frac{2 (c+d x)^{3/2} (C d-3 c D)}{3 d^4}+\frac{2 D (c+d x)^{5/2}}{5 d^4} \]
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Rubi [A] time = 0.0810817, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {1850} \[ -\frac{2 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^4 \sqrt{c+d x}}-\frac{2 \sqrt{c+d x} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^4}+\frac{2 (c+d x)^{3/2} (C d-3 c D)}{3 d^4}+\frac{2 D (c+d x)^{5/2}}{5 d^4} \]
Antiderivative was successfully verified.
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Rule 1850
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2+D x^3}{(c+d x)^{3/2}} \, dx &=\int \left (\frac{c^2 C d-B c d^2+A d^3-c^3 D}{d^3 (c+d x)^{3/2}}+\frac{-2 c C d+B d^2+3 c^2 D}{d^3 \sqrt{c+d x}}+\frac{(C d-3 c D) \sqrt{c+d x}}{d^3}+\frac{D (c+d x)^{3/2}}{d^3}\right ) \, dx\\ &=-\frac{2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^4 \sqrt{c+d x}}-\frac{2 \left (2 c C d-B d^2-3 c^2 D\right ) \sqrt{c+d x}}{d^4}+\frac{2 (C d-3 c D) (c+d x)^{3/2}}{3 d^4}+\frac{2 D (c+d x)^{5/2}}{5 d^4}\\ \end{align*}
Mathematica [A] time = 0.0800003, size = 82, normalized size = 0.73 \[ \frac{2 \left (d^3 \left (x \left (15 B+5 C x+3 D x^2\right )-15 A\right )+2 c d^2 (15 B-x (10 C+3 D x))-8 c^2 d (5 C-3 D x)+48 c^3 D\right )}{15 d^4 \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 91, normalized size = 0.8 \begin{align*} -{\frac{-6\,D{x}^{3}{d}^{3}-10\,C{d}^{3}{x}^{2}+12\,Dc{d}^{2}{x}^{2}-30\,B{d}^{3}x+40\,Cc{d}^{2}x-48\,D{c}^{2}dx+30\,A{d}^{3}-60\,Bc{d}^{2}+80\,C{c}^{2}d-96\,D{c}^{3}}{15\,{d}^{4}}{\frac{1}{\sqrt{dx+c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.07117, size = 138, normalized size = 1.22 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (d x + c\right )}^{\frac{5}{2}} D - 5 \,{\left (3 \, D c - C d\right )}{\left (d x + c\right )}^{\frac{3}{2}} + 15 \,{\left (3 \, D c^{2} - 2 \, C c d + B d^{2}\right )} \sqrt{d x + c}}{d^{3}} + \frac{15 \,{\left (D c^{3} - C c^{2} d + B c d^{2} - A d^{3}\right )}}{\sqrt{d x + c} d^{3}}\right )}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.2075, size = 114, normalized size = 1.01 \begin{align*} \frac{2 D \left (c + d x\right )^{\frac{5}{2}}}{5 d^{4}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (2 C d - 6 D c\right )}{3 d^{4}} + \frac{\sqrt{c + d x} \left (2 B d^{2} - 4 C c d + 6 D c^{2}\right )}{d^{4}} + \frac{2 \left (- A d^{3} + B c d^{2} - C c^{2} d + D c^{3}\right )}{d^{4} \sqrt{c + d x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.30635, size = 171, normalized size = 1.51 \begin{align*} \frac{2 \,{\left (D c^{3} - C c^{2} d + B c d^{2} - A d^{3}\right )}}{\sqrt{d x + c} d^{4}} + \frac{2 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} D d^{16} - 15 \,{\left (d x + c\right )}^{\frac{3}{2}} D c d^{16} + 45 \, \sqrt{d x + c} D c^{2} d^{16} + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} C d^{17} - 30 \, \sqrt{d x + c} C c d^{17} + 15 \, \sqrt{d x + c} B d^{18}\right )}}{15 \, d^{20}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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