Optimal. Leaf size=200 \[ -\frac {4096 a^6 \sqrt {a x+b \sqrt {x}}}{3003 b^7 \sqrt {x}}+\frac {2048 a^5 \sqrt {a x+b \sqrt {x}}}{3003 b^6 x}-\frac {512 a^4 \sqrt {a x+b \sqrt {x}}}{1001 b^5 x^{3/2}}+\frac {1280 a^3 \sqrt {a x+b \sqrt {x}}}{3003 b^4 x^2}-\frac {160 a^2 \sqrt {a x+b \sqrt {x}}}{429 b^3 x^{5/2}}+\frac {48 a \sqrt {a x+b \sqrt {x}}}{143 b^2 x^3}-\frac {4 \sqrt {a x+b \sqrt {x}}}{13 b x^{7/2}} \]
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Rubi [A] time = 0.30, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2016, 2014} \[ -\frac {512 a^4 \sqrt {a x+b \sqrt {x}}}{1001 b^5 x^{3/2}}+\frac {1280 a^3 \sqrt {a x+b \sqrt {x}}}{3003 b^4 x^2}-\frac {160 a^2 \sqrt {a x+b \sqrt {x}}}{429 b^3 x^{5/2}}-\frac {4096 a^6 \sqrt {a x+b \sqrt {x}}}{3003 b^7 \sqrt {x}}+\frac {2048 a^5 \sqrt {a x+b \sqrt {x}}}{3003 b^6 x}+\frac {48 a \sqrt {a x+b \sqrt {x}}}{143 b^2 x^3}-\frac {4 \sqrt {a x+b \sqrt {x}}}{13 b x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt {b \sqrt {x}+a x}} \, dx &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{13 b x^{7/2}}-\frac {(12 a) \int \frac {1}{x^{7/2} \sqrt {b \sqrt {x}+a x}} \, dx}{13 b}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{13 b x^{7/2}}+\frac {48 a \sqrt {b \sqrt {x}+a x}}{143 b^2 x^3}+\frac {\left (120 a^2\right ) \int \frac {1}{x^3 \sqrt {b \sqrt {x}+a x}} \, dx}{143 b^2}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{13 b x^{7/2}}+\frac {48 a \sqrt {b \sqrt {x}+a x}}{143 b^2 x^3}-\frac {160 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^3 x^{5/2}}-\frac {\left (320 a^3\right ) \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx}{429 b^3}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{13 b x^{7/2}}+\frac {48 a \sqrt {b \sqrt {x}+a x}}{143 b^2 x^3}-\frac {160 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^3 x^{5/2}}+\frac {1280 a^3 \sqrt {b \sqrt {x}+a x}}{3003 b^4 x^2}+\frac {\left (640 a^4\right ) \int \frac {1}{x^2 \sqrt {b \sqrt {x}+a x}} \, dx}{1001 b^4}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{13 b x^{7/2}}+\frac {48 a \sqrt {b \sqrt {x}+a x}}{143 b^2 x^3}-\frac {160 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^3 x^{5/2}}+\frac {1280 a^3 \sqrt {b \sqrt {x}+a x}}{3003 b^4 x^2}-\frac {512 a^4 \sqrt {b \sqrt {x}+a x}}{1001 b^5 x^{3/2}}-\frac {\left (512 a^5\right ) \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx}{1001 b^5}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{13 b x^{7/2}}+\frac {48 a \sqrt {b \sqrt {x}+a x}}{143 b^2 x^3}-\frac {160 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^3 x^{5/2}}+\frac {1280 a^3 \sqrt {b \sqrt {x}+a x}}{3003 b^4 x^2}-\frac {512 a^4 \sqrt {b \sqrt {x}+a x}}{1001 b^5 x^{3/2}}+\frac {2048 a^5 \sqrt {b \sqrt {x}+a x}}{3003 b^6 x}+\frac {\left (1024 a^6\right ) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{3003 b^6}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{13 b x^{7/2}}+\frac {48 a \sqrt {b \sqrt {x}+a x}}{143 b^2 x^3}-\frac {160 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^3 x^{5/2}}+\frac {1280 a^3 \sqrt {b \sqrt {x}+a x}}{3003 b^4 x^2}-\frac {512 a^4 \sqrt {b \sqrt {x}+a x}}{1001 b^5 x^{3/2}}+\frac {2048 a^5 \sqrt {b \sqrt {x}+a x}}{3003 b^6 x}-\frac {4096 a^6 \sqrt {b \sqrt {x}+a x}}{3003 b^7 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 96, normalized size = 0.48 \[ -\frac {4 \sqrt {a x+b \sqrt {x}} \left (1024 a^6 x^3-512 a^5 b x^{5/2}+384 a^4 b^2 x^2-320 a^3 b^3 x^{3/2}+280 a^2 b^4 x-252 a b^5 \sqrt {x}+231 b^6\right )}{3003 b^7 x^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 86, normalized size = 0.43 \[ \frac {4 \, {\left (512 \, a^{5} b x^{3} + 320 \, a^{3} b^{3} x^{2} + 252 \, a b^{5} x - {\left (1024 \, a^{6} x^{3} + 384 \, a^{4} b^{2} x^{2} + 280 \, a^{2} b^{4} x + 231 \, b^{6}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{3003 \, b^{7} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 208, normalized size = 1.04 \[ \frac {4 \, {\left (27456 \, a^{3} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{6} + 72072 \, a^{\frac {5}{2}} b {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{5} + 80080 \, a^{2} b^{2} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{4} + 48048 \, a^{\frac {3}{2}} b^{3} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{3} + 16380 \, a b^{4} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{2} + 3003 \, \sqrt {a} b^{5} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + 231 \, b^{6}\right )}}{3003 \, {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 306, normalized size = 1.53 \[ \frac {\sqrt {a x +b \sqrt {x}}\, \left (-3003 a^{7} b \,x^{\frac {15}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+3003 a^{7} b \,x^{\frac {15}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+6006 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {15}{2}} x^{\frac {15}{2}}+6006 \sqrt {a x +b \sqrt {x}}\, a^{\frac {15}{2}} x^{\frac {15}{2}}-12012 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {13}{2}} x^{\frac {13}{2}}+7916 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {11}{2}} b \,x^{6}-5868 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {9}{2}} b^{2} x^{\frac {11}{2}}+4332 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{3} x^{5}-3052 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{4} x^{\frac {9}{2}}+1932 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{5} x^{4}-924 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} \sqrt {a}\, b^{6} x^{\frac {7}{2}}\right )}{3003 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}\, b^{8} x^{\frac {15}{2}}} \]
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 \[ \int \frac {1}{\sqrt {a x + b \sqrt {x}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^4\,\sqrt {a\,x+b\,\sqrt {x}}} \,d x \]
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 \[ \int \frac {1}{x^{4} \sqrt {a x + b \sqrt {x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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