Optimal. Leaf size=68 \[ \frac {2 a^2 \left (a+b x^n\right )^{3/2}}{3 b^3 n}+\frac {2 \left (a+b x^n\right )^{7/2}}{7 b^3 n}-\frac {4 a \left (a+b x^n\right )^{5/2}}{5 b^3 n} \]
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Rubi [A] time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {266, 43} \[ \frac {2 a^2 \left (a+b x^n\right )^{3/2}}{3 b^3 n}+\frac {2 \left (a+b x^n\right )^{7/2}}{7 b^3 n}-\frac {4 a \left (a+b x^n\right )^{5/2}}{5 b^3 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1+3 n} \sqrt {a+b x^n} \, dx &=\frac {\operatorname {Subst}\left (\int x^2 \sqrt {a+b x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^2 \sqrt {a+b x}}{b^2}-\frac {2 a (a+b x)^{3/2}}{b^2}+\frac {(a+b x)^{5/2}}{b^2}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {2 a^2 \left (a+b x^n\right )^{3/2}}{3 b^3 n}-\frac {4 a \left (a+b x^n\right )^{5/2}}{5 b^3 n}+\frac {2 \left (a+b x^n\right )^{7/2}}{7 b^3 n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 44, normalized size = 0.65 \[ \frac {2 \left (a+b x^n\right )^{3/2} \left (8 a^2-12 a b x^n+15 b^2 x^{2 n}\right )}{105 b^3 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 53, normalized size = 0.78 \[ \frac {2 \, {\left (15 \, b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} - 4 \, a^{2} b x^{n} + 8 \, a^{3}\right )} \sqrt {b x^{n} + a}}{105 \, b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x^{n} + a} x^{3 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 54, normalized size = 0.79 \[ \frac {2 \left (-4 a^{2} b \,x^{n}+3 a \,b^{2} x^{2 n}+15 b^{3} x^{3 n}+8 a^{3}\right ) \sqrt {b \,x^{n}+a}}{105 b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 53, normalized size = 0.78 \[ \frac {2 \, {\left (15 \, b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} - 4 \, a^{2} b x^{n} + 8 \, a^{3}\right )} \sqrt {b x^{n} + a}}{105 \, b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^{3\,n-1}\,\sqrt {a+b\,x^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 21.96, size = 1015, normalized size = 14.93 \[ \frac {16 a^{\frac {19}{2}} b^{\frac {9}{2}} x^{\frac {9 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} + \frac {40 a^{\frac {17}{2}} b^{\frac {11}{2}} x^{\frac {11 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} + \frac {30 a^{\frac {15}{2}} b^{\frac {13}{2}} x^{\frac {13 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} + \frac {40 a^{\frac {13}{2}} b^{\frac {15}{2}} x^{\frac {15 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} + \frac {100 a^{\frac {11}{2}} b^{\frac {17}{2}} x^{\frac {17 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} + \frac {96 a^{\frac {9}{2}} b^{\frac {19}{2}} x^{\frac {19 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} + \frac {30 a^{\frac {7}{2}} b^{\frac {21}{2}} x^{\frac {21 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} - \frac {16 a^{10} b^{4} x^{4 n}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} - \frac {48 a^{9} b^{5} x^{5 n}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} - \frac {48 a^{8} b^{6} x^{6 n}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} - \frac {16 a^{7} b^{7} x^{7 n}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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