Optimal. Leaf size=233 \[ \frac {556 \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}}{1287}-\frac {4 \sqrt {x} (8575 x+6959) \left (3 x^2+5 x+2\right )^{3/2}}{27027}+\frac {8 \sqrt {x} (6381 x+6908) \sqrt {3 x^2+5 x+2}}{243243}+\frac {55112 \sqrt {x} (3 x+2)}{729729 \sqrt {3 x^2+5 x+2}}+\frac {25448 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{243243 \sqrt {3 x^2+5 x+2}}-\frac {55112 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{729729 \sqrt {3 x^2+5 x+2}}-\frac {10}{39} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2} \]
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Rubi [A] time = 0.17, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {832, 814, 839, 1189, 1100, 1136} \[ -\frac {10}{39} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac {556 \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}}{1287}-\frac {4 \sqrt {x} (8575 x+6959) \left (3 x^2+5 x+2\right )^{3/2}}{27027}+\frac {8 \sqrt {x} (6381 x+6908) \sqrt {3 x^2+5 x+2}}{243243}+\frac {55112 \sqrt {x} (3 x+2)}{729729 \sqrt {3 x^2+5 x+2}}+\frac {25448 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{243243 \sqrt {3 x^2+5 x+2}}-\frac {55112 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{729729 \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 814
Rule 832
Rule 839
Rule 1100
Rule 1136
Rule 1189
Rubi steps
\begin {align*} \int (2-5 x) x^{3/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac {10}{39} x^{3/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {2}{39} \int \sqrt {x} (15+139 x) \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac {556 \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac {10}{39} x^{3/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {4 \int \frac {\left (-139-\frac {3675 x}{2}\right ) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {x}} \, dx}{1287}\\ &=-\frac {4 \sqrt {x} (6959+8575 x) \left (2+5 x+3 x^2\right )^{3/2}}{27027}+\frac {556 \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac {10}{39} x^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {8 \int \frac {\left (-\frac {3363}{2}-\frac {10635 x}{2}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {x}} \, dx}{81081}\\ &=\frac {8 \sqrt {x} (6908+6381 x) \sqrt {2+5 x+3 x^2}}{243243}-\frac {4 \sqrt {x} (6959+8575 x) \left (2+5 x+3 x^2\right )^{3/2}}{27027}+\frac {556 \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac {10}{39} x^{3/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {16 \int \frac {\frac {47715}{2}+\frac {103335 x}{4}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx}{3648645}\\ &=\frac {8 \sqrt {x} (6908+6381 x) \sqrt {2+5 x+3 x^2}}{243243}-\frac {4 \sqrt {x} (6959+8575 x) \left (2+5 x+3 x^2\right )^{3/2}}{27027}+\frac {556 \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac {10}{39} x^{3/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {32 \operatorname {Subst}\left (\int \frac {\frac {47715}{2}+\frac {103335 x^2}{4}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{3648645}\\ &=\frac {8 \sqrt {x} (6908+6381 x) \sqrt {2+5 x+3 x^2}}{243243}-\frac {4 \sqrt {x} (6959+8575 x) \left (2+5 x+3 x^2\right )^{3/2}}{27027}+\frac {556 \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac {10}{39} x^{3/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {50896 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{243243}+\frac {55112 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{243243}\\ &=\frac {55112 \sqrt {x} (2+3 x)}{729729 \sqrt {2+5 x+3 x^2}}+\frac {8 \sqrt {x} (6908+6381 x) \sqrt {2+5 x+3 x^2}}{243243}-\frac {4 \sqrt {x} (6959+8575 x) \left (2+5 x+3 x^2\right )^{3/2}}{27027}+\frac {556 \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac {10}{39} x^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {55112 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{729729 \sqrt {2+5 x+3 x^2}}+\frac {25448 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{243243 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.19, size = 178, normalized size = 0.76 \[ \frac {21232 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+55112 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-2 \left (2525985 x^8+8374023 x^7+8989785 x^6+1830195 x^5-2497986 x^4-1171602 x^3+8508 x^2-61436 x-55112\right )}{729729 \sqrt {x} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (15 \, x^{4} + 19 \, x^{3} - 4 \, x\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}, x\right ) \]
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 \[ \int -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )} x^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 137, normalized size = 0.59 \[ -\frac {2 \left (7577955 x^{8}+25122069 x^{7}+26969355 x^{6}+5490585 x^{5}-7493958 x^{4}-3514806 x^{3}+273528 x^{2}+229032 x -13778 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+3162 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{2189187 \sqrt {3 x^{2}+5 x +2}\, \sqrt {x}} \]
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 \[ -\int {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )} x^{\frac {3}{2}}\,{d x} \]
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 \[ -\int x^{3/2}\,\left (5\,x-2\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2} \,d x \]
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 \[ - \int \left (- 4 x^{\frac {3}{2}} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 19 x^{\frac {7}{2}} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 15 x^{\frac {9}{2}} \sqrt {3 x^{2} + 5 x + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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