Optimal. Leaf size=444 \[ -\frac {\left (176-23 \left (1-\frac {6}{x}\right )^2\right ) x^2}{51759 \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}}+\frac {\left (45401-3722 \left (1-\frac {6}{x}\right )^2\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}}+\frac {3722 \left (\left (\frac {6}{x}-1\right )^4-182 \left (1-\frac {6}{x}\right )^2+613\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right ) \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}}-\frac {\left (7444-145 \sqrt {613}\right ) \sqrt {\frac {\left (\frac {6}{x}-1\right )^4-182 \left (1-\frac {6}{x}\right )^2+613}{\left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right )^2}} \left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right ) x^2 F\left (2 \tan ^{-1}\left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{207036\ 613^{3/4} \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}}+\frac {3722 \sqrt {\frac {\left (\frac {6}{x}-1\right )^4-182 \left (1-\frac {6}{x}\right )^2+613}{\left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right )^2}} \left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right ) x^2 E\left (2 \tan ^{-1}\left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{51759\ 613^{3/4} \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}} \]
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Rubi [A] time = 0.47, antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2069, 12, 6719, 1673, 1678, 1183, 1096, 1182, 1247, 636} \[ -\frac {\left (176-23 \left (1-\frac {6}{x}\right )^2\right ) x^2}{51759 \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}}+\frac {\left (45401-3722 \left (1-\frac {6}{x}\right )^2\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}}+\frac {3722 \left (\left (\frac {6}{x}-1\right )^4-182 \left (1-\frac {6}{x}\right )^2+613\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right ) \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}}-\frac {\left (7444-145 \sqrt {613}\right ) \sqrt {\frac {\left (\frac {6}{x}-1\right )^4-182 \left (1-\frac {6}{x}\right )^2+613}{\left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right )^2}} \left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right ) x^2 F\left (2 \tan ^{-1}\left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{207036\ 613^{3/4} \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}}+\frac {3722 \sqrt {\frac {\left (\frac {6}{x}-1\right )^4-182 \left (1-\frac {6}{x}\right )^2+613}{\left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right )^2}} \left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right ) x^2 E\left (2 \tan ^{-1}\left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{51759\ 613^{3/4} \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 636
Rule 1096
Rule 1182
Rule 1183
Rule 1247
Rule 1673
Rule 1678
Rule 2069
Rule 6719
Rubi steps
\begin {align*} \int \frac {1}{\left (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx &=-\left (1296 \operatorname {Subst}\left (\int \frac {1}{27 (-6-36 x)^2 \left (\frac {794448-8491392 x^2+1679616 x^4}{(-6-36 x)^4}\right )^{3/2}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )\right )\\ &=-\left (48 \operatorname {Subst}\left (\int \frac {1}{(-6-36 x)^2 \left (\frac {794448-8491392 x^2+1679616 x^4}{(-6-36 x)^4}\right )^{3/2}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )\right )\\ &=-\frac {\left (\sqrt {794448-8491392 \left (-\frac {1}{6}+\frac {1}{x}\right )^2+1679616 \left (-\frac {1}{6}+\frac {1}{x}\right )^4} x^2\right ) \operatorname {Subst}\left (\int \frac {(-6-36 x)^4}{\left (794448-8491392 x^2+1679616 x^4\right )^{3/2}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )}{9 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}\\ &=-\frac {\left (\sqrt {794448-8491392 \left (-\frac {1}{6}+\frac {1}{x}\right )^2+1679616 \left (-\frac {1}{6}+\frac {1}{x}\right )^4} x^2\right ) \operatorname {Subst}\left (\int \frac {x \left (31104+1119744 x^2\right )}{\left (794448-8491392 x^2+1679616 x^4\right )^{3/2}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )}{9 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}-\frac {\left (\sqrt {794448-8491392 \left (-\frac {1}{6}+\frac {1}{x}\right )^2+1679616 \left (-\frac {1}{6}+\frac {1}{x}\right )^4} x^2\right ) \operatorname {Subst}\left (\int \frac {1296+279936 x^2+1679616 x^4}{\left (794448-8491392 x^2+1679616 x^4\right )^{3/2}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )}{9 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}\\ &=\frac {\left (45401-3722 \left (1-\frac {6}{x}\right )^2\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {\left (\sqrt {794448-8491392 \left (-\frac {1}{6}+\frac {1}{x}\right )^2+1679616 \left (-\frac {1}{6}+\frac {1}{x}\right )^4} x^2\right ) \operatorname {Subst}\left (\int \frac {4012069665987624960-12096197079035019264 x^2}{\sqrt {794448-8491392 x^2+1679616 x^4}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )}{477380951360582713344 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}-\frac {\left (\sqrt {794448-8491392 \left (-\frac {1}{6}+\frac {1}{x}\right )^2+1679616 \left (-\frac {1}{6}+\frac {1}{x}\right )^4} x^2\right ) \operatorname {Subst}\left (\int \frac {31104+1119744 x}{\left (794448-8491392 x+1679616 x^2\right )^{3/2}} \, dx,x,\left (-\frac {1}{6}+\frac {1}{x}\right )^2\right )}{18 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}\\ &=-\frac {\left (176-23 \left (1-\frac {6}{x}\right )^2\right ) x^2}{51759 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {\left (45401-3722 \left (1-\frac {6}{x}\right )^2\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {\left (7444 \sqrt {794448-8491392 \left (-\frac {1}{6}+\frac {1}{x}\right )^2+1679616 \left (-\frac {1}{6}+\frac {1}{x}\right )^4} x^2\right ) \operatorname {Subst}\left (\int \frac {1-\frac {36 x^2}{\sqrt {613}}}{\sqrt {794448-8491392 x^2+1679616 x^4}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )}{17253 \sqrt {613} \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {\left (\left (88885-7444 \sqrt {613}\right ) \sqrt {794448-8491392 \left (-\frac {1}{6}+\frac {1}{x}\right )^2+1679616 \left (-\frac {1}{6}+\frac {1}{x}\right )^4} x^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {794448-8491392 x^2+1679616 x^4}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )}{10576089 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}\\ &=-\frac {\left (176-23 \left (1-\frac {6}{x}\right )^2\right ) x^2}{51759 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {\left (45401-3722 \left (1-\frac {6}{x}\right )^2\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {3722 \left (613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right ) \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {3722 \sqrt {\frac {613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4}{\left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right )^2}} \left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right ) x^2 E\left (2 \tan ^{-1}\left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{51759\ 613^{3/4} \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}-\frac {\left (7444-145 \sqrt {613}\right ) \sqrt {\frac {613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4}{\left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right )^2}} \left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right ) x^2 F\left (2 \tan ^{-1}\left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{207036\ 613^{3/4} \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}\\ \end {align*}
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Mathematica [C] time = 6.05, size = 5428, normalized size = 12.23 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9}}{9 \, x^{8} + 90 \, x^{7} - 39 \, x^{6} - 1356 \, x^{5} + 1810 \, x^{4} + 798 \, x^{3} - 756 \, x^{2} - 108 \, x + 81}, x\right ) \]
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 \[ \int \frac {1}{{\left (3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 5427, normalized size = 12.22 \[ \text {output too large to display} \]
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 \[ \int \frac {1}{{\left (3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (3\,x^4+15\,x^3-44\,x^2-6\,x+9\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (3 x^{4} + 15 x^{3} - 44 x^{2} - 6 x + 9\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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