Optimal. Leaf size=223 \[ \frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}-\frac {2240 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^4 x^{5/2}}+\frac {2560 a^3 \sqrt {b \sqrt {x}+a x}}{429 b^5 x^2}-\frac {1024 a^4 \sqrt {b \sqrt {x}+a x}}{143 b^6 x^{3/2}}+\frac {4096 a^5 \sqrt {b \sqrt {x}+a x}}{429 b^7 x}-\frac {8192 a^6 \sqrt {b \sqrt {x}+a x}}{429 b^8 \sqrt {x}} \]
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Rubi [A]
time = 0.23, 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 = {2040, 2041,
2039} \begin {gather*} -\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}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rule 2040
Rule 2041
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 = 0.21, size = 120, normalized size = 0.54 \begin {gather*} -\frac {4 \sqrt {b \sqrt {x}+a x} \left (33 b^7-42 a b^6 \sqrt {x}+56 a^2 b^5 x-80 a^3 b^4 x^{3/2}+128 a^4 b^3 x^2-256 a^5 b^2 x^{5/2}+1024 a^6 b x^3+2048 a^7 x^{7/2}\right )}{429 b^8 \left (b+a \sqrt {x}\right ) x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order
2.
time = 0.38, size = 636, normalized size = 2.85
method | result | size |
derivativedivides | \(-\frac {4}{13 b \,x^{3} \sqrt {b \sqrt {x}+a x}}-\frac {28 a \left (-\frac {2}{11 b \,x^{\frac {5}{2}} \sqrt {b \sqrt {x}+a x}}-\frac {12 a \left (-\frac {2}{9 b \,x^{2} \sqrt {b \sqrt {x}+a x}}-\frac {10 a \left (-\frac {2}{7 b \,x^{\frac {3}{2}} \sqrt {b \sqrt {x}+a x}}-\frac {8 a \left (-\frac {2}{5 b x \sqrt {b \sqrt {x}+a x}}-\frac {6 a \left (-\frac {2}{3 b \sqrt {x}\, \sqrt {b \sqrt {x}+a x}}+\frac {8 a \left (b +2 a \sqrt {x}\right )}{3 b^{3} \sqrt {b \sqrt {x}+a x}}\right )}{5 b}\right )}{7 b}\right )}{9 b}\right )}{11 b}\right )}{13 b}\) | \(176\) |
default | \(\frac {2 \sqrt {b \sqrt {x}+a x}\, \left (2574 x^{\frac {17}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {19}{2}}+2574 x^{\frac {17}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {19}{2}}-6006 x^{\frac {15}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {17}{2}}+858 x^{\frac {15}{2}} \left (\sqrt {x}\, \left (a \sqrt {x}+b \right )\right )^{\frac {3}{2}} a^{\frac {17}{2}}+2574 x^{\frac {15}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {15}{2}} b^{2}+2574 x^{\frac {15}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {15}{2}} b^{2}-2048 x^{\frac {13}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {13}{2}} b^{2}+1287 x^{\frac {17}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{9} b -1287 x^{\frac {17}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{9} b -256 x^{\frac {11}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {9}{2}} b^{4}+5148 x^{8} \sqrt {b \sqrt {x}+a x}\, a^{\frac {17}{2}} b +5148 x^{8} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {17}{2}} b +1287 x^{\frac {15}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{7} b^{3}-1287 x^{\frac {15}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{7} b^{3}-9244 x^{7} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {15}{2}} b -112 x^{\frac {9}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{6}+512 x^{6} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {11}{2}} b^{3}+160 x^{5} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{5}-66 x^{\frac {7}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} \sqrt {a}\, b^{8}+2574 x^{8} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{8} b^{2}-2574 x^{8} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{8} b^{2}+84 x^{4} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{7}\right )}{429 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{9} x^{\frac {15}{2}} \left (a \sqrt {x}+b \right )^{2} \sqrt {a}}\) | \(636\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.53, size = 123, normalized size = 0.55 \begin {gather*} \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 )}} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{\frac {7}{2}} \left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{7/2}\,{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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