Optimal. Leaf size=210 \[ -\frac {10}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}+\frac {4}{231} \sqrt {x} (84 x+65) \left (3 x^2+5 x+2\right )^{3/2}-\frac {4}{385} \sqrt {x} (39 x+55) \sqrt {3 x^2+5 x+2}-\frac {424 \sqrt {x} (3 x+2)}{1155 \sqrt {3 x^2+5 x+2}}-\frac {36 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{77 \sqrt {3 x^2+5 x+2}}+\frac {424 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{1155 \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.15, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ -\frac {10}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}+\frac {4}{231} \sqrt {x} (84 x+65) \left (3 x^2+5 x+2\right )^{3/2}-\frac {4}{385} \sqrt {x} (39 x+55) \sqrt {3 x^2+5 x+2}-\frac {424 \sqrt {x} (3 x+2)}{1155 \sqrt {3 x^2+5 x+2}}-\frac {36 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{77 \sqrt {3 x^2+5 x+2}}+\frac {424 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{1155 \sqrt {3 x^2+5 x+2}} \]
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) \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac {10}{33} \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}+\frac {2}{33} \int \frac {(5+108 x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {x}} \, dx\\ &=\frac {4}{231} \sqrt {x} (65+84 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}-\frac {4 \int \frac {\left (270+\frac {1053 x}{2}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {x}} \, dx}{2079}\\ &=-\frac {4}{385} \sqrt {x} (55+39 x) \sqrt {2+5 x+3 x^2}+\frac {4}{231} \sqrt {x} (65+84 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}+\frac {8 \int \frac {-\frac {10935}{2}-\frac {12879 x}{2}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx}{93555}\\ &=-\frac {4}{385} \sqrt {x} (55+39 x) \sqrt {2+5 x+3 x^2}+\frac {4}{231} \sqrt {x} (65+84 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}+\frac {16 \operatorname {Subst}\left (\int \frac {-\frac {10935}{2}-\frac {12879 x^2}{2}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{93555}\\ &=-\frac {4}{385} \sqrt {x} (55+39 x) \sqrt {2+5 x+3 x^2}+\frac {4}{231} \sqrt {x} (65+84 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}-\frac {72}{77} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )-\frac {424}{385} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {424 \sqrt {x} (2+3 x)}{1155 \sqrt {2+5 x+3 x^2}}-\frac {4}{385} \sqrt {x} (55+39 x) \sqrt {2+5 x+3 x^2}+\frac {4}{231} \sqrt {x} (65+84 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}+\frac {424 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{1155 \sqrt {2+5 x+3 x^2}}-\frac {36 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{77 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.19, size = 173, normalized size = 0.82 \[ \frac {-424 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-2 \left (58 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+4725 x^7+16065 x^6+17775 x^5+3497 x^4-6140 x^3-3106 x^2+520 x+424\right )}{1155 \sqrt {x} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (15 \, x^{3} + 19 \, x^{2} - 4\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}, x\right ) \]
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 \[ \int -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )} \sqrt {x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 132, normalized size = 0.63 \[ \frac {-\frac {90 x^{7}}{11}-\frac {306 x^{6}}{11}-\frac {2370 x^{5}}{77}-\frac {6994 x^{4}}{1155}+\frac {2456 x^{3}}{231}+\frac {7484 x^{2}}{1155}+\frac {72 x}{77}-\frac {212 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3465}+\frac {32 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{1155}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}} \]
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 \[ -\int {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )} \sqrt {x}\,{d x} \]
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 \[ -\int \sqrt {x}\,\left (5\,x-2\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2} \,d x \]
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 \[ - \int \left (- 4 \sqrt {x} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 19 x^{\frac {5}{2}} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 15 x^{\frac {7}{2}} \sqrt {3 x^{2} + 5 x + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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