Optimal. Leaf size=185 \[ \frac {2 (89-35 x) \sqrt {3 x^2+5 x+2}}{5 \sqrt {x}}-\frac {1418 \sqrt {x} (3 x+2)}{15 \sqrt {3 x^2+5 x+2}}-\frac {117 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}+\frac {1418 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{15 \sqrt {3 x^2+5 x+2}}-\frac {4 (3-5 x) \left (3 x^2+5 x+2\right )^{3/2}}{15 x^{5/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {810, 812, 839, 1189, 1100, 1136} \[ -\frac {4 (3-5 x) \left (3 x^2+5 x+2\right )^{3/2}}{15 x^{5/2}}+\frac {2 (89-35 x) \sqrt {3 x^2+5 x+2}}{5 \sqrt {x}}-\frac {1418 \sqrt {x} (3 x+2)}{15 \sqrt {3 x^2+5 x+2}}-\frac {117 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}+\frac {1418 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{15 \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 810
Rule 812
Rule 839
Rule 1100
Rule 1136
Rule 1189
Rubi steps
\begin {align*} \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{7/2}} \, dx &=-\frac {4 (3-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{15 x^{5/2}}-\frac {1}{5} \int \frac {(89+105 x) \sqrt {2+5 x+3 x^2}}{x^{3/2}} \, dx\\ &=\frac {2 (89-35 x) \sqrt {2+5 x+3 x^2}}{5 \sqrt {x}}-\frac {4 (3-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{15 x^{5/2}}+\frac {2}{15} \int \frac {-\frac {1755}{2}-\frac {2127 x}{2}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2 (89-35 x) \sqrt {2+5 x+3 x^2}}{5 \sqrt {x}}-\frac {4 (3-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{15 x^{5/2}}+\frac {4}{15} \operatorname {Subst}\left (\int \frac {-\frac {1755}{2}-\frac {2127 x^2}{2}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 (89-35 x) \sqrt {2+5 x+3 x^2}}{5 \sqrt {x}}-\frac {4 (3-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{15 x^{5/2}}-234 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )-\frac {1418}{5} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1418 \sqrt {x} (2+3 x)}{15 \sqrt {2+5 x+3 x^2}}+\frac {2 (89-35 x) \sqrt {2+5 x+3 x^2}}{5 \sqrt {x}}-\frac {4 (3-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{15 x^{5/2}}+\frac {1418 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{15 \sqrt {2+5 x+3 x^2}}-\frac {117 \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 = 0.19, size = 163, normalized size = 0.88 \[ \frac {-337 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{7/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-1418 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{7/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-2 \left (225 x^5+1605 x^4+2230 x^3+906 x^2+80 x+24\right )}{15 x^{5/2} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (15 \, x^{3} + 19 \, x^{2} - 4\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{x^{\frac {7}{2}}}, x\right ) \]
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 \[ \int -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 129, normalized size = 0.70 \[ \frac {-1350 x^{5}+3132 x^{4}+7890 x^{3}-709 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x^{2} \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+372 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x^{2} \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+3072 x^{2}-480 x -144}{45 \sqrt {3 x^{2}+5 x +2}\, x^{\frac {5}{2}}} \]
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 \[ -\int \frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {7}{2}}}\,{d x} \]
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 \[ \int -\frac {\left (5\,x-2\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{x^{7/2}} \,d x \]
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 \[ - \int \left (- \frac {4 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {7}{2}}}\right )\, dx - \int \frac {19 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {3}{2}}}\, dx - \int \frac {15 \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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