Optimal. Leaf size=176 \[ -\frac {12 a^2}{b^4 \sqrt {a^2+2 a b \sqrt [4]{x}+b^2 \sqrt {x}}}-\frac {12 a \left (a+b \sqrt [4]{x}\right ) \log \left (a+b \sqrt [4]{x}\right )}{b^4 \sqrt {a^2+2 a b \sqrt [4]{x}+b^2 \sqrt {x}}}+\frac {4 \sqrt [4]{x} \left (a+b \sqrt [4]{x}\right )}{b^3 \sqrt {a^2+2 a b \sqrt [4]{x}+b^2 \sqrt {x}}}+\frac {2 a^3}{b^4 \left (a+b \sqrt [4]{x}\right ) \sqrt {a^2+2 a b \sqrt [4]{x}+b^2 \sqrt {x}}} \]
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Rubi [A] time = 0.10, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1341, 646, 43} \[ \frac {2 a^3}{b^4 \left (a+b \sqrt [4]{x}\right ) \sqrt {a^2+2 a b \sqrt [4]{x}+b^2 \sqrt {x}}}-\frac {12 a^2}{b^4 \sqrt {a^2+2 a b \sqrt [4]{x}+b^2 \sqrt {x}}}+\frac {4 \sqrt [4]{x} \left (a+b \sqrt [4]{x}\right )}{b^3 \sqrt {a^2+2 a b \sqrt [4]{x}+b^2 \sqrt {x}}}-\frac {12 a \left (a+b \sqrt [4]{x}\right ) \log \left (a+b \sqrt [4]{x}\right )}{b^4 \sqrt {a^2+2 a b \sqrt [4]{x}+b^2 \sqrt {x}}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rule 1341
Rubi steps
\begin {align*} \int \frac {1}{\left (a^2+2 a b \sqrt [4]{x}+b^2 \sqrt {x}\right )^{3/2}} \, dx &=4 \operatorname {Subst}\left (\int \frac {x^3}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx,x,\sqrt [4]{x}\right )\\ &=\frac {\left (4 b^3 \left (a+b \sqrt [4]{x}\right )\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (a b+b^2 x\right )^3} \, dx,x,\sqrt [4]{x}\right )}{\sqrt {a^2+2 a b \sqrt [4]{x}+b^2 \sqrt {x}}}\\ &=\frac {\left (4 b^3 \left (a+b \sqrt [4]{x}\right )\right ) \operatorname {Subst}\left (\int \left (\frac {1}{b^6}-\frac {a^3}{b^6 (a+b x)^3}+\frac {3 a^2}{b^6 (a+b x)^2}-\frac {3 a}{b^6 (a+b x)}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt {a^2+2 a b \sqrt [4]{x}+b^2 \sqrt {x}}}\\ &=-\frac {12 a^2}{b^4 \sqrt {a^2+2 a b \sqrt [4]{x}+b^2 \sqrt {x}}}+\frac {2 a^3}{b^4 \left (a+b \sqrt [4]{x}\right ) \sqrt {a^2+2 a b \sqrt [4]{x}+b^2 \sqrt {x}}}+\frac {4 \left (a+b \sqrt [4]{x}\right ) \sqrt [4]{x}}{b^3 \sqrt {a^2+2 a b \sqrt [4]{x}+b^2 \sqrt {x}}}-\frac {12 a \left (a+b \sqrt [4]{x}\right ) \log \left (a+b \sqrt [4]{x}\right )}{b^4 \sqrt {a^2+2 a b \sqrt [4]{x}+b^2 \sqrt {x}}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 93, normalized size = 0.53 \[ \frac {2 \left (-5 a^3-4 a^2 b \sqrt [4]{x}+4 a b^2 \sqrt {x}-6 a \left (a+b \sqrt [4]{x}\right )^2 \log \left (a+b \sqrt [4]{x}\right )+2 b^3 x^{3/4}\right )}{b^4 \left (a+b \sqrt [4]{x}\right ) \sqrt {\left (a+b \sqrt [4]{x}\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 9.06, size = 147, normalized size = 0.84 \[ \frac {2 \, {\left (9 \, a^{5} b^{4} x - 5 \, a^{9} - 6 \, {\left (a b^{8} x^{2} - 2 \, a^{5} b^{4} x + a^{9}\right )} \log \left (b x^{\frac {1}{4}} + a\right ) - 2 \, {\left (3 \, a^{2} b^{7} x - a^{6} b^{3}\right )} x^{\frac {3}{4}} + {\left (7 \, a^{3} b^{6} x - 3 \, a^{7} b^{2}\right )} \sqrt {x} + 2 \, {\left (b^{9} x^{2} - 6 \, a^{4} b^{5} x + 3 \, a^{8} b\right )} x^{\frac {1}{4}}\right )}}{b^{12} x^{2} - 2 \, a^{4} b^{8} x + a^{8} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 114, normalized size = 0.65 \[ \frac {2 \sqrt {b^{2} \sqrt {x}+2 a b \,x^{\frac {1}{4}}+a^{2}}\, \left (-6 a \,b^{2} \sqrt {x}\, \ln \left (b \,x^{\frac {1}{4}}+a \right )-12 a^{2} b \,x^{\frac {1}{4}} \ln \left (b \,x^{\frac {1}{4}}+a \right )-6 a^{3} \ln \left (b \,x^{\frac {1}{4}}+a \right )+2 b^{3} x^{\frac {3}{4}}+4 a \,b^{2} \sqrt {x}-4 a^{2} b \,x^{\frac {1}{4}}-5 a^{3}\right )}{\left (b \,x^{\frac {1}{4}}+a \right )^{3} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 114, normalized size = 0.65 \[ \frac {4 \, \sqrt {x}}{\sqrt {b^{2} \sqrt {x} + 2 \, a b x^{\frac {1}{4}} + a^{2}} b^{2}} - \frac {12 \, a \log \left (x^{\frac {1}{4}} + \frac {a}{b}\right )}{b^{4}} + \frac {8 \, a^{2}}{\sqrt {b^{2} \sqrt {x} + 2 \, a b x^{\frac {1}{4}} + a^{2}} b^{4}} - \frac {24 \, a^{2} x^{\frac {1}{4}}}{b^{5} {\left (x^{\frac {1}{4}} + \frac {a}{b}\right )}^{2}} - \frac {22 \, a^{3}}{b^{6} {\left (x^{\frac {1}{4}} + \frac {a}{b}\right )}^{2}} \]
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 \[ \int \frac {1}{{\left (a^2+b^2\,\sqrt {x}+2\,a\,b\,x^{1/4}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (a^{2} + 2 a b \sqrt [4]{x} + b^{2} \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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