Optimal. Leaf size=90 \[ \frac{2 \left (a^2-b^2 c\right ) \sqrt{c+d x}}{b^3 d^2}-\frac{2 a \left (a^2-b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}-\frac{a x}{b^2 d}+\frac{2 (c+d x)^{3/2}}{3 b d^2} \]
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Rubi [A] time = 0.0803427, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {371, 1398, 772} \[ \frac{2 \left (a^2-b^2 c\right ) \sqrt{c+d x}}{b^3 d^2}-\frac{2 a \left (a^2-b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}-\frac{a x}{b^2 d}+\frac{2 (c+d x)^{3/2}}{3 b d^2} \]
Antiderivative was successfully verified.
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Rule 371
Rule 1398
Rule 772
Rubi steps
\begin{align*} \int \frac{x}{a+b \sqrt{c+d x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-c+x}{a+b \sqrt{x}} \, dx,x,c+d x\right )}{d^2}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{x \left (-c+x^2\right )}{a+b x} \, dx,x,\sqrt{c+d x}\right )}{d^2}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{a^2-b^2 c}{b^3}-\frac{a x}{b^2}+\frac{x^2}{b}+\frac{-a^3+a b^2 c}{b^3 (a+b x)}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^2}\\ &=-\frac{a x}{b^2 d}+\frac{2 \left (a^2-b^2 c\right ) \sqrt{c+d x}}{b^3 d^2}+\frac{2 (c+d x)^{3/2}}{3 b d^2}-\frac{2 a \left (a^2-b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}\\ \end{align*}
Mathematica [A] time = 0.0651472, size = 82, normalized size = 0.91 \[ \frac{b \left (6 a^2 \sqrt{c+d x}-3 a b d x+2 b^2 (d x-2 c) \sqrt{c+d x}\right )-6 \left (a^3-a b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{3 b^4 d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 116, normalized size = 1.3 \begin{align*}{\frac{2}{3\,b{d}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{ax}{{b}^{2}d}}-{\frac{ac}{{b}^{2}{d}^{2}}}-2\,{\frac{c\sqrt{dx+c}}{b{d}^{2}}}+2\,{\frac{\sqrt{dx+c}{a}^{2}}{{b}^{3}{d}^{2}}}+2\,{\frac{a\ln \left ( a+b\sqrt{dx+c} \right ) c}{{b}^{2}{d}^{2}}}-2\,{\frac{{a}^{3}\ln \left ( a+b\sqrt{dx+c} \right ) }{{b}^{4}{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11058, size = 109, normalized size = 1.21 \begin{align*} \frac{\frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} - 3 \,{\left (d x + c\right )} a b - 6 \,{\left (b^{2} c - a^{2}\right )} \sqrt{d x + c}}{b^{3}} + \frac{6 \,{\left (a b^{2} c - a^{3}\right )} \log \left (\sqrt{d x + c} b + a\right )}{b^{4}}}{3 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64355, size = 166, normalized size = 1.84 \begin{align*} -\frac{3 \, a b^{2} d x - 6 \,{\left (a b^{2} c - a^{3}\right )} \log \left (\sqrt{d x + c} b + a\right ) - 2 \,{\left (b^{3} d x - 2 \, b^{3} c + 3 \, a^{2} b\right )} \sqrt{d x + c}}{3 \, b^{4} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.07034, size = 109, normalized size = 1.21 \begin{align*} \begin{cases} \frac{2 \left (- \frac{a \left (c + d x\right )}{2 b^{2} d} - \frac{a \left (a^{2} - b^{2} c\right ) \left (\begin{cases} \frac{\sqrt{c + d x}}{a} & \text{for}\: b = 0 \\\frac{\log{\left (a + b \sqrt{c + d x} \right )}}{b} & \text{otherwise} \end{cases}\right )}{b^{3} d} + \frac{\left (c + d x\right )^{\frac{3}{2}}}{3 b d} + \frac{\left (a^{2} - b^{2} c\right ) \sqrt{c + d x}}{b^{3} d}\right )}{d} & \text{for}\: d \neq 0 \\\frac{x^{2}}{2 \left (a + b \sqrt{c}\right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19539, size = 177, normalized size = 1.97 \begin{align*} \frac{\frac{6 \,{\left (a b^{2} c - a^{3}\right )} \log \left ({\left | \sqrt{d x + c} b + a \right |}\right )}{b^{4} d} - \frac{6 \,{\left (a b^{2} c \log \left ({\left | a \right |}\right ) - a^{3} \log \left ({\left | a \right |}\right )\right )}}{b^{4} d} + \frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} d^{2} - 6 \, \sqrt{d x + c} b^{2} c d^{2} - 3 \,{\left (d x + c\right )} a b d^{2} + 6 \, \sqrt{d x + c} a^{2} d^{2}}{b^{3} d^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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