Optimal. Leaf size=223 \[ -\frac {8192 a^6 \sqrt {a x+b \sqrt {x}}}{429 b^8 \sqrt {x}}+\frac {4096 a^5 \sqrt {a x+b \sqrt {x}}}{429 b^7 x}-\frac {1024 a^4 \sqrt {a x+b \sqrt {x}}}{143 b^6 x^{3/2}}+\frac {2560 a^3 \sqrt {a x+b \sqrt {x}}}{429 b^5 x^2}-\frac {2240 a^2 \sqrt {a x+b \sqrt {x}}}{429 b^4 x^{5/2}}+\frac {672 a \sqrt {a x+b \sqrt {x}}}{143 b^3 x^3}-\frac {56 \sqrt {a x+b \sqrt {x}}}{13 b^2 x^{7/2}}+\frac {4}{b x^3 \sqrt {a x+b \sqrt {x}}} \]
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Rubi [A] time = 0.35, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2015, 2016, 2014} \[ -\frac {1024 a^4 \sqrt {a x+b \sqrt {x}}}{143 b^6 x^{3/2}}+\frac {2560 a^3 \sqrt {a x+b \sqrt {x}}}{429 b^5 x^2}-\frac {2240 a^2 \sqrt {a x+b \sqrt {x}}}{429 b^4 x^{5/2}}-\frac {8192 a^6 \sqrt {a x+b \sqrt {x}}}{429 b^8 \sqrt {x}}+\frac {4096 a^5 \sqrt {a x+b \sqrt {x}}}{429 b^7 x}+\frac {672 a \sqrt {a x+b \sqrt {x}}}{143 b^3 x^3}-\frac {56 \sqrt {a x+b \sqrt {x}}}{13 b^2 x^{7/2}}+\frac {4}{b x^3 \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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Rule 2014
Rule 2015
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx &=\frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}+\frac {14 \int \frac {1}{x^4 \sqrt {b \sqrt {x}+a x}} \, dx}{b}\\ &=\frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}-\frac {(168 a) \int \frac {1}{x^{7/2} \sqrt {b \sqrt {x}+a x}} \, dx}{13 b^2}\\ &=\frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}+\frac {\left (1680 a^2\right ) \int \frac {1}{x^3 \sqrt {b \sqrt {x}+a x}} \, dx}{143 b^3}\\ &=\frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}-\frac {2240 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^4 x^{5/2}}-\frac {\left (4480 a^3\right ) \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx}{429 b^4}\\ &=\frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}-\frac {2240 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^4 x^{5/2}}+\frac {2560 a^3 \sqrt {b \sqrt {x}+a x}}{429 b^5 x^2}+\frac {\left (1280 a^4\right ) \int \frac {1}{x^2 \sqrt {b \sqrt {x}+a x}} \, dx}{143 b^5}\\ &=\frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}-\frac {2240 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^4 x^{5/2}}+\frac {2560 a^3 \sqrt {b \sqrt {x}+a x}}{429 b^5 x^2}-\frac {1024 a^4 \sqrt {b \sqrt {x}+a x}}{143 b^6 x^{3/2}}-\frac {\left (1024 a^5\right ) \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx}{143 b^6}\\ &=\frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}-\frac {2240 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^4 x^{5/2}}+\frac {2560 a^3 \sqrt {b \sqrt {x}+a x}}{429 b^5 x^2}-\frac {1024 a^4 \sqrt {b \sqrt {x}+a x}}{143 b^6 x^{3/2}}+\frac {4096 a^5 \sqrt {b \sqrt {x}+a x}}{429 b^7 x}+\frac {\left (2048 a^6\right ) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{429 b^7}\\ &=\frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}-\frac {2240 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^4 x^{5/2}}+\frac {2560 a^3 \sqrt {b \sqrt {x}+a x}}{429 b^5 x^2}-\frac {1024 a^4 \sqrt {b \sqrt {x}+a x}}{143 b^6 x^{3/2}}+\frac {4096 a^5 \sqrt {b \sqrt {x}+a x}}{429 b^7 x}-\frac {8192 a^6 \sqrt {b \sqrt {x}+a x}}{429 b^8 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 107, normalized size = 0.48 \[ -\frac {4 \left (2048 a^7 x^{7/2}+1024 a^6 b x^3-256 a^5 b^2 x^{5/2}+128 a^4 b^3 x^2-80 a^3 b^4 x^{3/2}+56 a^2 b^5 x-42 a b^6 \sqrt {x}+33 b^7\right )}{429 b^8 x^3 \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.42, size = 123, normalized size = 0.55 \[ \frac {4 \, {\left (1024 \, a^{7} b x^{4} - 384 \, a^{5} b^{3} x^{3} - 136 \, a^{3} b^{5} x^{2} - 75 \, a b^{7} x - {\left (2048 \, a^{8} x^{4} - 1280 \, a^{6} b^{2} x^{3} - 208 \, a^{4} b^{4} x^{2} - 98 \, a^{2} b^{6} x - 33 \, b^{8}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{429 \, {\left (a^{2} b^{8} x^{5} - b^{10} x^{4}\right )}} \]
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 \[ \int \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 636, normalized size = 2.85 \[ \frac {2 \sqrt {a x +b \sqrt {x}}\, \left (-1287 a^{9} b \,x^{\frac {17}{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 )+1287 a^{9} b \,x^{\frac {17}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-2574 a^{8} b^{2} x^{8} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+2574 a^{8} b^{2} x^{8} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-1287 a^{7} b^{3} 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 )+1287 a^{7} b^{3} 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 )+2574 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {19}{2}} x^{\frac {17}{2}}+2574 \sqrt {a x +b \sqrt {x}}\, a^{\frac {19}{2}} x^{\frac {17}{2}}+5148 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {17}{2}} b \,x^{8}+5148 \sqrt {a x +b \sqrt {x}}\, a^{\frac {17}{2}} b \,x^{8}+2574 \sqrt {a x +b \sqrt {x}}\, a^{\frac {15}{2}} b^{2} x^{\frac {15}{2}}+2574 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {15}{2}} b^{2} x^{\frac {15}{2}}-6006 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {17}{2}} x^{\frac {15}{2}}+858 \left (\left (a \sqrt {x}+b \right ) \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {17}{2}} x^{\frac {15}{2}}-9244 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {15}{2}} b \,x^{7}-2048 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {13}{2}} b^{2} x^{\frac {13}{2}}+512 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {11}{2}} b^{3} x^{6}-256 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {9}{2}} b^{4} x^{\frac {11}{2}}+160 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{5} x^{5}-112 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{6} x^{\frac {9}{2}}+84 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{7} x^{4}-66 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} \sqrt {a}\, b^{8} x^{\frac {7}{2}}\right )}{429 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \left (a \sqrt {x}+b \right )^{2} \sqrt {a}\, b^{9} 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 \[ \int \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^{7/2}\,{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x^{\frac {7}{2}} \left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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