Optimal. Leaf size=170 \[ \frac {1024 a^5 \sqrt {a x+b \sqrt {x}}}{693 b^6 \sqrt {x}}-\frac {512 a^4 \sqrt {a x+b \sqrt {x}}}{693 b^5 x}+\frac {128 a^3 \sqrt {a x+b \sqrt {x}}}{231 b^4 x^{3/2}}-\frac {320 a^2 \sqrt {a x+b \sqrt {x}}}{693 b^3 x^2}+\frac {40 a \sqrt {a x+b \sqrt {x}}}{99 b^2 x^{5/2}}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3} \]
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Rubi [A] time = 0.24, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2016, 2014} \[ \frac {128 a^3 \sqrt {a x+b \sqrt {x}}}{231 b^4 x^{3/2}}-\frac {320 a^2 \sqrt {a x+b \sqrt {x}}}{693 b^3 x^2}+\frac {1024 a^5 \sqrt {a x+b \sqrt {x}}}{693 b^6 \sqrt {x}}-\frac {512 a^4 \sqrt {a x+b \sqrt {x}}}{693 b^5 x}+\frac {40 a \sqrt {a x+b \sqrt {x}}}{99 b^2 x^{5/2}}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3} \]
Antiderivative was successfully verified.
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Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^{7/2} \sqrt {b \sqrt {x}+a x}} \, dx &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{11 b x^3}-\frac {(10 a) \int \frac {1}{x^3 \sqrt {b \sqrt {x}+a x}} \, dx}{11 b}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{11 b x^3}+\frac {40 a \sqrt {b \sqrt {x}+a x}}{99 b^2 x^{5/2}}+\frac {\left (80 a^2\right ) \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx}{99 b^2}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{11 b x^3}+\frac {40 a \sqrt {b \sqrt {x}+a x}}{99 b^2 x^{5/2}}-\frac {320 a^2 \sqrt {b \sqrt {x}+a x}}{693 b^3 x^2}-\frac {\left (160 a^3\right ) \int \frac {1}{x^2 \sqrt {b \sqrt {x}+a x}} \, dx}{231 b^3}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{11 b x^3}+\frac {40 a \sqrt {b \sqrt {x}+a x}}{99 b^2 x^{5/2}}-\frac {320 a^2 \sqrt {b \sqrt {x}+a x}}{693 b^3 x^2}+\frac {128 a^3 \sqrt {b \sqrt {x}+a x}}{231 b^4 x^{3/2}}+\frac {\left (128 a^4\right ) \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx}{231 b^4}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{11 b x^3}+\frac {40 a \sqrt {b \sqrt {x}+a x}}{99 b^2 x^{5/2}}-\frac {320 a^2 \sqrt {b \sqrt {x}+a x}}{693 b^3 x^2}+\frac {128 a^3 \sqrt {b \sqrt {x}+a x}}{231 b^4 x^{3/2}}-\frac {512 a^4 \sqrt {b \sqrt {x}+a x}}{693 b^5 x}-\frac {\left (256 a^5\right ) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{693 b^5}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{11 b x^3}+\frac {40 a \sqrt {b \sqrt {x}+a x}}{99 b^2 x^{5/2}}-\frac {320 a^2 \sqrt {b \sqrt {x}+a x}}{693 b^3 x^2}+\frac {128 a^3 \sqrt {b \sqrt {x}+a x}}{231 b^4 x^{3/2}}-\frac {512 a^4 \sqrt {b \sqrt {x}+a x}}{693 b^5 x}+\frac {1024 a^5 \sqrt {b \sqrt {x}+a x}}{693 b^6 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 83, normalized size = 0.49 \[ \frac {4 \sqrt {a x+b \sqrt {x}} \left (256 a^5 x^{5/2}-128 a^4 b x^2+96 a^3 b^2 x^{3/2}-80 a^2 b^3 x+70 a b^4 \sqrt {x}-63 b^5\right )}{693 b^6 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.41, size = 72, normalized size = 0.42 \[ -\frac {4 \, {\left (128 \, a^{4} b x^{2} + 80 \, a^{2} b^{3} x + 63 \, b^{5} - 2 \, {\left (128 \, a^{5} x^{2} + 48 \, a^{3} b^{2} x + 35 \, a b^{4}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{693 \, b^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 177, normalized size = 1.04 \[ \frac {4 \, {\left (3696 \, a^{\frac {5}{2}} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{5} + 7920 \, a^{2} b {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{4} + 6930 \, a^{\frac {3}{2}} b^{2} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{3} + 3080 \, a b^{3} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{2} + 693 \, \sqrt {a} b^{4} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + 63 \, b^{5}\right )}}{693 \, {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 284, normalized size = 1.67 \[ \frac {\sqrt {a x +b \sqrt {x}}\, \left (693 a^{6} b \,x^{\frac {13}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-693 a^{6} b \,x^{\frac {13}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-1386 \sqrt {a x +b \sqrt {x}}\, a^{\frac {13}{2}} x^{\frac {13}{2}}-1386 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {13}{2}} x^{\frac {13}{2}}+2772 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {11}{2}} x^{\frac {11}{2}}-1748 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {9}{2}} b \,x^{5}+1236 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{2} x^{\frac {9}{2}}-852 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3} x^{4}+532 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{4} x^{\frac {7}{2}}-252 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} \sqrt {a}\, b^{5} x^{3}\right )}{693 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}\, b^{7} x^{\frac {13}{2}}} \]
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 \[ \int \frac {1}{\sqrt {a x + b \sqrt {x}} x^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {1}{x^{7/2}\,\sqrt {a\,x+b\,\sqrt {x}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{\frac {7}{2}} \sqrt {a x + b \sqrt {x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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