Optimal. Leaf size=197 \[ 20 \sqrt {3 x^2+5 x+2} \sqrt {x}-\frac {24 (3 x+2) \sqrt {x}}{\sqrt {3 x^2+5 x+2}}-\frac {20 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}+\frac {24 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}+\frac {2 (95 x+74) x^{5/2}}{3 \sqrt {3 x^2+5 x+2}}-\frac {64}{3} \sqrt {3 x^2+5 x+2} x^{3/2} \]
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Rubi [A] time = 0.14, antiderivative size = 197, 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 = {818, 832, 839, 1189, 1100, 1136} \[ \frac {2 (95 x+74) x^{5/2}}{3 \sqrt {3 x^2+5 x+2}}-\frac {64}{3} \sqrt {3 x^2+5 x+2} x^{3/2}+20 \sqrt {3 x^2+5 x+2} \sqrt {x}-\frac {24 (3 x+2) \sqrt {x}}{\sqrt {3 x^2+5 x+2}}-\frac {20 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}+\frac {24 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 818
Rule 832
Rule 839
Rule 1100
Rule 1136
Rule 1189
Rubi steps
\begin {align*} \int \frac {(2-5 x) x^{7/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx &=\frac {2 x^{5/2} (74+95 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {2}{3} \int \frac {(-185-240 x) x^{3/2}}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2 x^{5/2} (74+95 x)}{3 \sqrt {2+5 x+3 x^2}}-\frac {64}{3} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {4}{45} \int \frac {\sqrt {x} \left (720+\frac {2025 x}{2}\right )}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2 x^{5/2} (74+95 x)}{3 \sqrt {2+5 x+3 x^2}}+20 \sqrt {x} \sqrt {2+5 x+3 x^2}-\frac {64}{3} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {8}{405} \int \frac {-\frac {2025}{2}-\frac {3645 x}{2}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2 x^{5/2} (74+95 x)}{3 \sqrt {2+5 x+3 x^2}}+20 \sqrt {x} \sqrt {2+5 x+3 x^2}-\frac {64}{3} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {16}{405} \operatorname {Subst}\left (\int \frac {-\frac {2025}{2}-\frac {3645 x^2}{2}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 x^{5/2} (74+95 x)}{3 \sqrt {2+5 x+3 x^2}}+20 \sqrt {x} \sqrt {2+5 x+3 x^2}-\frac {64}{3} x^{3/2} \sqrt {2+5 x+3 x^2}-40 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )-72 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {24 \sqrt {x} (2+3 x)}{\sqrt {2+5 x+3 x^2}}+\frac {2 x^{5/2} (74+95 x)}{3 \sqrt {2+5 x+3 x^2}}+20 \sqrt {x} \sqrt {2+5 x+3 x^2}-\frac {64}{3} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {24 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {2+5 x+3 x^2}}-\frac {20 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 156, normalized size = 0.79 \[ \frac {12 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-72 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-2 \left (x^4-4 x^3+22 x^2+120 x+72\right )}{3 \sqrt {x} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (5 \, x^{4} - 2 \, x^{3}\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4}, x\right ) \]
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 \[ \int -\frac {{\left (5 \, x - 2\right )} x^{\frac {7}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 117, normalized size = 0.59 \[ \frac {-\frac {2 x^{4}}{3}+\frac {8 x^{3}}{3}+\frac {172 x^{2}}{3}+40 x -4 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+\frac {16 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}} \]
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 \[ -\int \frac {{\left (5 \, x - 2\right )} x^{\frac {7}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {x^{7/2}\,\left (5\,x-2\right )}{{\left (3\,x^2+5\,x+2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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