Optimal. Leaf size=80 \[ -\frac {a^3 \left (a+b \sqrt {x}\right )^6}{3 b^4}+\frac {6 a^2 \left (a+b \sqrt {x}\right )^7}{7 b^4}+\frac {2 \left (a+b \sqrt {x}\right )^9}{9 b^4}-\frac {3 a \left (a+b \sqrt {x}\right )^8}{4 b^4} \]
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Rubi [A] time = 0.04, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {6 a^2 \left (a+b \sqrt {x}\right )^7}{7 b^4}-\frac {a^3 \left (a+b \sqrt {x}\right )^6}{3 b^4}+\frac {2 \left (a+b \sqrt {x}\right )^9}{9 b^4}-\frac {3 a \left (a+b \sqrt {x}\right )^8}{4 b^4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \left (a+b \sqrt {x}\right )^5 x \, dx &=2 \operatorname {Subst}\left (\int x^3 (a+b x)^5 \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {a^3 (a+b x)^5}{b^3}+\frac {3 a^2 (a+b x)^6}{b^3}-\frac {3 a (a+b x)^7}{b^3}+\frac {(a+b x)^8}{b^3}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^3 \left (a+b \sqrt {x}\right )^6}{3 b^4}+\frac {6 a^2 \left (a+b \sqrt {x}\right )^7}{7 b^4}-\frac {3 a \left (a+b \sqrt {x}\right )^8}{4 b^4}+\frac {2 \left (a+b \sqrt {x}\right )^9}{9 b^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 73, normalized size = 0.91 \[ \frac {a^5 x^2}{2}+2 a^4 b x^{5/2}+\frac {10}{3} a^3 b^2 x^3+\frac {20}{7} a^2 b^3 x^{7/2}+\frac {5}{4} a b^4 x^4+\frac {2}{9} b^5 x^{9/2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.26, size = 63, normalized size = 0.79 \[ \frac {5}{4} \, a b^{4} x^{4} + \frac {10}{3} \, a^{3} b^{2} x^{3} + \frac {1}{2} \, a^{5} x^{2} + \frac {2}{63} \, {\left (7 \, b^{5} x^{4} + 90 \, a^{2} b^{3} x^{3} + 63 \, a^{4} b x^{2}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 57, normalized size = 0.71 \[ \frac {2}{9} \, b^{5} x^{\frac {9}{2}} + \frac {5}{4} \, a b^{4} x^{4} + \frac {20}{7} \, a^{2} b^{3} x^{\frac {7}{2}} + \frac {10}{3} \, a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{\frac {5}{2}} + \frac {1}{2} \, a^{5} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 58, normalized size = 0.72 \[ \frac {2 b^{5} x^{\frac {9}{2}}}{9}+\frac {5 a \,b^{4} x^{4}}{4}+\frac {20 a^{2} b^{3} x^{\frac {7}{2}}}{7}+\frac {10 a^{3} b^{2} x^{3}}{3}+2 a^{4} b \,x^{\frac {5}{2}}+\frac {a^{5} x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 64, normalized size = 0.80 \[ \frac {2 \, {\left (b \sqrt {x} + a\right )}^{9}}{9 \, b^{4}} - \frac {3 \, {\left (b \sqrt {x} + a\right )}^{8} a}{4 \, b^{4}} + \frac {6 \, {\left (b \sqrt {x} + a\right )}^{7} a^{2}}{7 \, b^{4}} - \frac {{\left (b \sqrt {x} + a\right )}^{6} a^{3}}{3 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 57, normalized size = 0.71 \[ \frac {a^5\,x^2}{2}+\frac {2\,b^5\,x^{9/2}}{9}+\frac {5\,a\,b^4\,x^4}{4}+2\,a^4\,b\,x^{5/2}+\frac {10\,a^3\,b^2\,x^3}{3}+\frac {20\,a^2\,b^3\,x^{7/2}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.34, size = 71, normalized size = 0.89 \[ \frac {a^{5} x^{2}}{2} + 2 a^{4} b x^{\frac {5}{2}} + \frac {10 a^{3} b^{2} x^{3}}{3} + \frac {20 a^{2} b^{3} x^{\frac {7}{2}}}{7} + \frac {5 a b^{4} x^{4}}{4} + \frac {2 b^{5} x^{\frac {9}{2}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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