Optimal. Leaf size=111 \[ \frac {2 a^7}{b^8 \left (a+b \sqrt {x}\right )}+\frac {14 a^6 \log \left (a+b \sqrt {x}\right )}{b^8}-\frac {12 a^5 \sqrt {x}}{b^7}+\frac {5 a^4 x}{b^6}-\frac {8 a^3 x^{3/2}}{3 b^5}+\frac {3 a^2 x^2}{2 b^4}-\frac {4 a x^{5/2}}{5 b^3}+\frac {x^3}{3 b^2} \]
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Rubi [A] time = 0.08, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac {8 a^3 x^{3/2}}{3 b^5}+\frac {3 a^2 x^2}{2 b^4}+\frac {2 a^7}{b^8 \left (a+b \sqrt {x}\right )}-\frac {12 a^5 \sqrt {x}}{b^7}+\frac {5 a^4 x}{b^6}+\frac {14 a^6 \log \left (a+b \sqrt {x}\right )}{b^8}-\frac {4 a x^{5/2}}{5 b^3}+\frac {x^3}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b \sqrt {x}\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^7}{(a+b x)^2} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {6 a^5}{b^7}+\frac {5 a^4 x}{b^6}-\frac {4 a^3 x^2}{b^5}+\frac {3 a^2 x^3}{b^4}-\frac {2 a x^4}{b^3}+\frac {x^5}{b^2}-\frac {a^7}{b^7 (a+b x)^2}+\frac {7 a^6}{b^7 (a+b x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2 a^7}{b^8 \left (a+b \sqrt {x}\right )}-\frac {12 a^5 \sqrt {x}}{b^7}+\frac {5 a^4 x}{b^6}-\frac {8 a^3 x^{3/2}}{3 b^5}+\frac {3 a^2 x^2}{2 b^4}-\frac {4 a x^{5/2}}{5 b^3}+\frac {x^3}{3 b^2}+\frac {14 a^6 \log \left (a+b \sqrt {x}\right )}{b^8}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 102, normalized size = 0.92 \[ \frac {\frac {60 a^7}{a+b \sqrt {x}}+420 a^6 \log \left (a+b \sqrt {x}\right )-360 a^5 b \sqrt {x}+150 a^4 b^2 x-80 a^3 b^3 x^{3/2}+45 a^2 b^4 x^2-24 a b^5 x^{5/2}+10 b^6 x^3}{30 b^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 128, normalized size = 1.15 \[ \frac {10 \, b^{8} x^{4} + 35 \, a^{2} b^{6} x^{3} + 105 \, a^{4} b^{4} x^{2} - 150 \, a^{6} b^{2} x - 60 \, a^{8} + 420 \, {\left (a^{6} b^{2} x - a^{8}\right )} \log \left (b \sqrt {x} + a\right ) - 4 \, {\left (6 \, a b^{7} x^{3} + 14 \, a^{3} b^{5} x^{2} + 70 \, a^{5} b^{3} x - 105 \, a^{7} b\right )} \sqrt {x}}{30 \, {\left (b^{10} x - a^{2} b^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 100, normalized size = 0.90 \[ \frac {14 \, a^{6} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{8}} + \frac {2 \, a^{7}}{{\left (b \sqrt {x} + a\right )} b^{8}} + \frac {10 \, b^{10} x^{3} - 24 \, a b^{9} x^{\frac {5}{2}} + 45 \, a^{2} b^{8} x^{2} - 80 \, a^{3} b^{7} x^{\frac {3}{2}} + 150 \, a^{4} b^{6} x - 360 \, a^{5} b^{5} \sqrt {x}}{30 \, b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 94, normalized size = 0.85 \[ \frac {x^{3}}{3 b^{2}}-\frac {4 a \,x^{\frac {5}{2}}}{5 b^{3}}+\frac {3 a^{2} x^{2}}{2 b^{4}}-\frac {8 a^{3} x^{\frac {3}{2}}}{3 b^{5}}+\frac {2 a^{7}}{\left (b \sqrt {x}+a \right ) b^{8}}+\frac {14 a^{6} \ln \left (b \sqrt {x}+a \right )}{b^{8}}+\frac {5 a^{4} x}{b^{6}}-\frac {12 a^{5} \sqrt {x}}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 129, normalized size = 1.16 \[ \frac {14 \, a^{6} \log \left (b \sqrt {x} + a\right )}{b^{8}} + \frac {{\left (b \sqrt {x} + a\right )}^{6}}{3 \, b^{8}} - \frac {14 \, {\left (b \sqrt {x} + a\right )}^{5} a}{5 \, b^{8}} + \frac {21 \, {\left (b \sqrt {x} + a\right )}^{4} a^{2}}{2 \, b^{8}} - \frac {70 \, {\left (b \sqrt {x} + a\right )}^{3} a^{3}}{3 \, b^{8}} + \frac {35 \, {\left (b \sqrt {x} + a\right )}^{2} a^{4}}{b^{8}} - \frac {42 \, {\left (b \sqrt {x} + a\right )} a^{5}}{b^{8}} + \frac {2 \, a^{7}}{{\left (b \sqrt {x} + a\right )} b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 99, normalized size = 0.89 \[ \frac {x^3}{3\,b^2}+\frac {2\,a^7}{b\,\left (a\,b^7+b^8\,\sqrt {x}\right )}-\frac {4\,a\,x^{5/2}}{5\,b^3}+\frac {5\,a^4\,x}{b^6}+\frac {14\,a^6\,\ln \left (a+b\,\sqrt {x}\right )}{b^8}+\frac {3\,a^2\,x^2}{2\,b^4}-\frac {8\,a^3\,x^{3/2}}{3\,b^5}-\frac {12\,a^5\,\sqrt {x}}{b^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.61, size = 272, normalized size = 2.45 \[ \begin {cases} \frac {420 a^{7} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{30 a b^{8} + 30 b^{9} \sqrt {x}} + \frac {420 a^{7}}{30 a b^{8} + 30 b^{9} \sqrt {x}} + \frac {420 a^{6} b \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{30 a b^{8} + 30 b^{9} \sqrt {x}} - \frac {210 a^{5} b^{2} x}{30 a b^{8} + 30 b^{9} \sqrt {x}} + \frac {70 a^{4} b^{3} x^{\frac {3}{2}}}{30 a b^{8} + 30 b^{9} \sqrt {x}} - \frac {35 a^{3} b^{4} x^{2}}{30 a b^{8} + 30 b^{9} \sqrt {x}} + \frac {21 a^{2} b^{5} x^{\frac {5}{2}}}{30 a b^{8} + 30 b^{9} \sqrt {x}} - \frac {14 a b^{6} x^{3}}{30 a b^{8} + 30 b^{9} \sqrt {x}} + \frac {10 b^{7} x^{\frac {7}{2}}}{30 a b^{8} + 30 b^{9} \sqrt {x}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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