Integrand size = 15, antiderivative size = 107 \[ \int \frac {x^3}{a+b \sqrt {x}} \, dx=\frac {2 a^6 \sqrt {x}}{b^7}-\frac {a^5 x}{b^6}+\frac {2 a^4 x^{3/2}}{3 b^5}-\frac {a^3 x^2}{2 b^4}+\frac {2 a^2 x^{5/2}}{5 b^3}-\frac {a x^3}{3 b^2}+\frac {2 x^{7/2}}{7 b}-\frac {2 a^7 \log \left (a+b \sqrt {x}\right )}{b^8} \]
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Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {x^3}{a+b \sqrt {x}} \, dx=-\frac {2 a^7 \log \left (a+b \sqrt {x}\right )}{b^8}+\frac {2 a^6 \sqrt {x}}{b^7}-\frac {a^5 x}{b^6}+\frac {2 a^4 x^{3/2}}{3 b^5}-\frac {a^3 x^2}{2 b^4}+\frac {2 a^2 x^{5/2}}{5 b^3}-\frac {a x^3}{3 b^2}+\frac {2 x^{7/2}}{7 b} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^7}{a+b x} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {a^6}{b^7}-\frac {a^5 x}{b^6}+\frac {a^4 x^2}{b^5}-\frac {a^3 x^3}{b^4}+\frac {a^2 x^4}{b^3}-\frac {a x^5}{b^2}+\frac {x^6}{b}-\frac {a^7}{b^7 (a+b x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 a^6 \sqrt {x}}{b^7}-\frac {a^5 x}{b^6}+\frac {2 a^4 x^{3/2}}{3 b^5}-\frac {a^3 x^2}{2 b^4}+\frac {2 a^2 x^{5/2}}{5 b^3}-\frac {a x^3}{3 b^2}+\frac {2 x^{7/2}}{7 b}-\frac {2 a^7 \log \left (a+b \sqrt {x}\right )}{b^8} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {x^3}{a+b \sqrt {x}} \, dx=\frac {\sqrt {x} \left (420 a^6-210 a^5 b \sqrt {x}+140 a^4 b^2 x-105 a^3 b^3 x^{3/2}+84 a^2 b^4 x^2-70 a b^5 x^{5/2}+60 b^6 x^3\right )}{210 b^7}-\frac {2 a^7 \log \left (a+b \sqrt {x}\right )}{b^8} \]
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Time = 3.63 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {2 x^{\frac {7}{2}} b^{6}}{7}-\frac {a \,x^{3} b^{5}}{3}+\frac {2 a^{2} x^{\frac {5}{2}} b^{4}}{5}-\frac {a^{3} b^{3} x^{2}}{2}+\frac {2 a^{4} x^{\frac {3}{2}} b^{2}}{3}-a^{5} x b +2 a^{6} \sqrt {x}}{b^{7}}-\frac {2 a^{7} \ln \left (a +b \sqrt {x}\right )}{b^{8}}\) | \(88\) |
default | \(\frac {\frac {2 x^{\frac {7}{2}} b^{6}}{7}-\frac {a \,x^{3} b^{5}}{3}+\frac {2 a^{2} x^{\frac {5}{2}} b^{4}}{5}-\frac {a^{3} b^{3} x^{2}}{2}+\frac {2 a^{4} x^{\frac {3}{2}} b^{2}}{3}-a^{5} x b +2 a^{6} \sqrt {x}}{b^{7}}-\frac {2 a^{7} \ln \left (a +b \sqrt {x}\right )}{b^{8}}\) | \(88\) |
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Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.82 \[ \int \frac {x^3}{a+b \sqrt {x}} \, dx=-\frac {70 \, a b^{6} x^{3} + 105 \, a^{3} b^{4} x^{2} + 210 \, a^{5} b^{2} x + 420 \, a^{7} \log \left (b \sqrt {x} + a\right ) - 4 \, {\left (15 \, b^{7} x^{3} + 21 \, a^{2} b^{5} x^{2} + 35 \, a^{4} b^{3} x + 105 \, a^{6} b\right )} \sqrt {x}}{210 \, b^{8}} \]
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Time = 0.25 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.02 \[ \int \frac {x^3}{a+b \sqrt {x}} \, dx=\begin {cases} - \frac {2 a^{7} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{b^{8}} + \frac {2 a^{6} \sqrt {x}}{b^{7}} - \frac {a^{5} x}{b^{6}} + \frac {2 a^{4} x^{\frac {3}{2}}}{3 b^{5}} - \frac {a^{3} x^{2}}{2 b^{4}} + \frac {2 a^{2} x^{\frac {5}{2}}}{5 b^{3}} - \frac {a x^{3}}{3 b^{2}} + \frac {2 x^{\frac {7}{2}}}{7 b} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.21 \[ \int \frac {x^3}{a+b \sqrt {x}} \, dx=-\frac {2 \, a^{7} \log \left (b \sqrt {x} + a\right )}{b^{8}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{7}}{7 \, b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{6} a}{3 \, b^{8}} + \frac {42 \, {\left (b \sqrt {x} + a\right )}^{5} a^{2}}{5 \, b^{8}} - \frac {35 \, {\left (b \sqrt {x} + a\right )}^{4} a^{3}}{2 \, b^{8}} + \frac {70 \, {\left (b \sqrt {x} + a\right )}^{3} a^{4}}{3 \, b^{8}} - \frac {21 \, {\left (b \sqrt {x} + a\right )}^{2} a^{5}}{b^{8}} + \frac {14 \, {\left (b \sqrt {x} + a\right )} a^{6}}{b^{8}} \]
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Time = 0.26 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.83 \[ \int \frac {x^3}{a+b \sqrt {x}} \, dx=-\frac {2 \, a^{7} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{8}} + \frac {60 \, b^{6} x^{\frac {7}{2}} - 70 \, a b^{5} x^{3} + 84 \, a^{2} b^{4} x^{\frac {5}{2}} - 105 \, a^{3} b^{3} x^{2} + 140 \, a^{4} b^{2} x^{\frac {3}{2}} - 210 \, a^{5} b x + 420 \, a^{6} \sqrt {x}}{210 \, b^{7}} \]
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Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.81 \[ \int \frac {x^3}{a+b \sqrt {x}} \, dx=\frac {2\,x^{7/2}}{7\,b}-\frac {a\,x^3}{3\,b^2}-\frac {a^5\,x}{b^6}-\frac {2\,a^7\,\ln \left (a+b\,\sqrt {x}\right )}{b^8}-\frac {a^3\,x^2}{2\,b^4}+\frac {2\,a^2\,x^{5/2}}{5\,b^3}+\frac {2\,a^4\,x^{3/2}}{3\,b^5}+\frac {2\,a^6\,\sqrt {x}}{b^7} \]
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