Optimal. Leaf size=129 \[ \frac {32 (c x)^{-7 n/2} \left (a+b x^n\right )^{7/2}}{35 a^4 c n}-\frac {16 (c x)^{-7 n/2} \left (a+b x^n\right )^{5/2}}{5 a^3 c n}+\frac {4 (c x)^{-7 n/2} \left (a+b x^n\right )^{3/2}}{a^2 c n}-\frac {2 (c x)^{-7 n/2} \sqrt {a+b x^n}}{a c n} \]
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Rubi [A] time = 0.05, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {273, 264} \[ \frac {32 (c x)^{-7 n/2} \left (a+b x^n\right )^{7/2}}{35 a^4 c n}-\frac {16 (c x)^{-7 n/2} \left (a+b x^n\right )^{5/2}}{5 a^3 c n}+\frac {4 (c x)^{-7 n/2} \left (a+b x^n\right )^{3/2}}{a^2 c n}-\frac {2 (c x)^{-7 n/2} \sqrt {a+b x^n}}{a c n} \]
Antiderivative was successfully verified.
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Rule 264
Rule 273
Rubi steps
\begin {align*} \int \frac {(c x)^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx &=-\frac {2 (c x)^{-7 n/2} \sqrt {a+b x^n}}{a c n}-\frac {6 \int (c x)^{-1-\frac {7 n}{2}} \sqrt {a+b x^n} \, dx}{a}\\ &=-\frac {2 (c x)^{-7 n/2} \sqrt {a+b x^n}}{a c n}+\frac {4 (c x)^{-7 n/2} \left (a+b x^n\right )^{3/2}}{a^2 c n}+\frac {8 \int (c x)^{-1-\frac {7 n}{2}} \left (a+b x^n\right )^{3/2} \, dx}{a^2}\\ &=-\frac {2 (c x)^{-7 n/2} \sqrt {a+b x^n}}{a c n}+\frac {4 (c x)^{-7 n/2} \left (a+b x^n\right )^{3/2}}{a^2 c n}-\frac {16 (c x)^{-7 n/2} \left (a+b x^n\right )^{5/2}}{5 a^3 c n}-\frac {16 \int (c x)^{-1-\frac {7 n}{2}} \left (a+b x^n\right )^{5/2} \, dx}{5 a^3}\\ &=-\frac {2 (c x)^{-7 n/2} \sqrt {a+b x^n}}{a c n}+\frac {4 (c x)^{-7 n/2} \left (a+b x^n\right )^{3/2}}{a^2 c n}-\frac {16 (c x)^{-7 n/2} \left (a+b x^n\right )^{5/2}}{5 a^3 c n}+\frac {32 (c x)^{-7 n/2} \left (a+b x^n\right )^{7/2}}{35 a^4 c n}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 69, normalized size = 0.53 \[ -\frac {2 (c x)^{-7 n/2} \sqrt {a+b x^n} \left (5 a^3-6 a^2 b x^n+8 a b^2 x^{2 n}-16 b^3 x^{3 n}\right )}{35 a^4 c n} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {\left (c x\right )^{-\frac {7}{2} \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{-\frac {7 n}{2}-1}}{\sqrt {b \,x^{n}+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x\right )^{-\frac {7}{2} \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]
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 \[ \int \frac {1}{{\left (c\,x\right )}^{\frac {7\,n}{2}+1}\,\sqrt {a+b\,x^n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.91, size = 665, normalized size = 5.16 \[ - \frac {10 a^{6} b^{\frac {19}{2}} c^{- \frac {7 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{c \left (35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}\right )} - \frac {18 a^{5} b^{\frac {21}{2}} c^{- \frac {7 n}{2}} x^{n} \sqrt {\frac {a x^{- n}}{b} + 1}}{c \left (35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}\right )} - \frac {10 a^{4} b^{\frac {23}{2}} c^{- \frac {7 n}{2}} x^{2 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{c \left (35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}\right )} + \frac {10 a^{3} b^{\frac {25}{2}} c^{- \frac {7 n}{2}} x^{3 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{c \left (35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}\right )} + \frac {60 a^{2} b^{\frac {27}{2}} c^{- \frac {7 n}{2}} x^{4 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{c \left (35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}\right )} + \frac {80 a b^{\frac {29}{2}} c^{- \frac {7 n}{2}} x^{5 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{c \left (35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}\right )} + \frac {32 b^{\frac {31}{2}} c^{- \frac {7 n}{2}} x^{6 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{c \left (35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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