Integrand size = 17, antiderivative size = 90 \[ \int \frac {x}{a+b \sqrt {c+d x}} \, dx=-\frac {a x}{b^2 d}+\frac {2 \left (a^2-b^2 c\right ) \sqrt {c+d x}}{b^3 d^2}+\frac {2 (c+d x)^{3/2}}{3 b d^2}-\frac {2 a \left (a^2-b^2 c\right ) \log \left (a+b \sqrt {c+d x}\right )}{b^4 d^2} \]
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Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {378, 1412, 786} \[ \int \frac {x}{a+b \sqrt {c+d x}} \, dx=-\frac {2 a \left (a^2-b^2 c\right ) \log \left (a+b \sqrt {c+d x}\right )}{b^4 d^2}+\frac {2 \left (a^2-b^2 c\right ) \sqrt {c+d x}}{b^3 d^2}-\frac {a x}{b^2 d}+\frac {2 (c+d x)^{3/2}}{3 b d^2} \]
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Rule 378
Rule 786
Rule 1412
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {-c+x}{a+b \sqrt {x}} \, dx,x,c+d x\right )}{d^2} \\ & = \frac {2 \text {Subst}\left (\int \frac {x \left (-c+x^2\right )}{a+b x} \, dx,x,\sqrt {c+d x}\right )}{d^2} \\ & = \frac {2 \text {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*}
Time = 0.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.94 \[ \int \frac {x}{a+b \sqrt {c+d x}} \, dx=\frac {b \left (6 a^2 \sqrt {c+d x}+2 b^2 (-2 c+d x) \sqrt {c+d x}-3 a b (c+d x)\right )-6 \left (a^3-a b^2 c\right ) \log \left (a+b \sqrt {c+d x}\right )}{3 b^4 d^2} \]
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Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (\frac {\left (d x +c \right )^{\frac {3}{2}} b^{2}}{3}-\frac {a \left (d x +c \right ) b}{2}-b^{2} c \sqrt {d x +c}+a^{2} \sqrt {d x +c}\right )}{b^{3}}-\frac {2 a \left (-b^{2} c +a^{2}\right ) \ln \left (a +b \sqrt {d x +c}\right )}{b^{4}}}{d^{2}}\) | \(85\) |
default | \(\frac {\frac {2 \left (\frac {\left (d x +c \right )^{\frac {3}{2}} b^{2}}{3}-\frac {a \left (d x +c \right ) b}{2}-b^{2} c \sqrt {d x +c}+a^{2} \sqrt {d x +c}\right )}{b^{3}}-\frac {2 a \left (-b^{2} c +a^{2}\right ) \ln \left (a +b \sqrt {d x +c}\right )}{b^{4}}}{d^{2}}\) | \(85\) |
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Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.79 \[ \int \frac {x}{a+b \sqrt {c+d x}} \, dx=-\frac {3 \, a b^{2} d x - 6 \, {\left (a b^{2} c - a^{3}\right )} \log \left (\sqrt {d x + c} b + a\right ) - 2 \, {\left (b^{3} d x - 2 \, b^{3} c + 3 \, a^{2} b\right )} \sqrt {d x + c}}{3 \, b^{4} d^{2}} \]
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Time = 1.07 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.16 \[ \int \frac {x}{a+b \sqrt {c+d x}} \, dx=\begin {cases} \frac {2 \left (- \frac {a \left (c + d x\right )}{2 b^{2}} - \frac {a \left (a^{2} - b^{2} c\right ) \left (\begin {cases} \frac {\sqrt {c + d x}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b \sqrt {c + d x} \right )}}{b} & \text {otherwise} \end {cases}\right )}{b^{3}} + \frac {\left (c + d x\right )^{\frac {3}{2}}}{3 b} + \frac {\left (a^{2} - b^{2} c\right ) \sqrt {c + d x}}{b^{3}}\right )}{d^{2}} & \text {for}\: d \neq 0 \\\frac {x^{2}}{2 \left (a + b \sqrt {c}\right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.90 \[ \int \frac {x}{a+b \sqrt {c+d x}} \, dx=\frac {\frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} - 3 \, {\left (d x + c\right )} a b - 6 \, {\left (b^{2} c - a^{2}\right )} \sqrt {d x + c}}{b^{3}} + \frac {6 \, {\left (a b^{2} c - a^{3}\right )} \log \left (\sqrt {d x + c} b + a\right )}{b^{4}}}{3 \, d^{2}} \]
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Time = 0.34 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.17 \[ \int \frac {x}{a+b \sqrt {c+d x}} \, dx=\frac {\frac {6 \, {\left (a b^{2} c - a^{3}\right )} \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{4} d} + \frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} d^{2} - 6 \, \sqrt {d x + c} b^{2} c d^{2} - 3 \, {\left (d x + c\right )} a b d^{2} + 6 \, \sqrt {d x + c} a^{2} d^{2}}{b^{3} d^{3}}}{3 \, d} \]
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Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.99 \[ \int \frac {x}{a+b \sqrt {c+d x}} \, dx=\frac {2\,{\left (c+d\,x\right )}^{3/2}}{3\,b\,d^2}-\left (\frac {2\,c}{b\,d^2}-\frac {2\,a^2}{b^3\,d^2}\right )\,\sqrt {c+d\,x}-\frac {\ln \left (a+b\,\sqrt {c+d\,x}\right )\,\left (2\,a^3-2\,a\,b^2\,c\right )}{b^4\,d^2}-\frac {a\,x}{b^2\,d} \]
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