Optimal. Leaf size=187 \[ -\frac {2 (5 x+14) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {x}}+\frac {2}{105} \sqrt {x} (531 x+1045) \sqrt {3 x^2+5 x+2}+\frac {5848 \sqrt {x} (3 x+2)}{315 \sqrt {3 x^2+5 x+2}}+\frac {482 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{21 \sqrt {3 x^2+5 x+2}}-\frac {5848 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{315 \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.12, antiderivative size = 187, 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 = {812, 814, 839, 1189, 1100, 1136} \[ -\frac {2 (5 x+14) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {x}}+\frac {2}{105} \sqrt {x} (531 x+1045) \sqrt {3 x^2+5 x+2}+\frac {5848 \sqrt {x} (3 x+2)}{315 \sqrt {3 x^2+5 x+2}}+\frac {482 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{21 \sqrt {3 x^2+5 x+2}}-\frac {5848 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{315 \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 812
Rule 814
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^{3/2}} \, dx &=-\frac {2 (14+5 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt {x}}-\frac {6}{7} \int \frac {\left (-25-\frac {59 x}{2}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {x}} \, dx\\ &=\frac {2}{105} \sqrt {x} (1045+531 x) \sqrt {2+5 x+3 x^2}-\frac {2 (14+5 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt {x}}+\frac {4}{105} \int \frac {\frac {1205}{2}+731 x}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2}{105} \sqrt {x} (1045+531 x) \sqrt {2+5 x+3 x^2}-\frac {2 (14+5 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt {x}}+\frac {8}{105} \operatorname {Subst}\left (\int \frac {\frac {1205}{2}+731 x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{105} \sqrt {x} (1045+531 x) \sqrt {2+5 x+3 x^2}-\frac {2 (14+5 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt {x}}+\frac {964}{21} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )+\frac {5848}{105} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {5848 \sqrt {x} (2+3 x)}{315 \sqrt {2+5 x+3 x^2}}+\frac {2}{105} \sqrt {x} (1045+531 x) \sqrt {2+5 x+3 x^2}-\frac {2 (14+5 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt {x}}-\frac {5848 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{315 \sqrt {2+5 x+3 x^2}}+\frac {482 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{21 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 163, normalized size = 0.87 \[ \frac {1382 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+5848 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-2 \left (2025 x^5+7641 x^4+9855 x^3+177 x^2-7390 x-3328\right )}{315 \sqrt {x} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (15 \, x^{3} + 19 \, x^{2} - 4\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{x^{\frac {3}{2}}}, x\right ) \]
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 \[ \int -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 123, normalized size = 0.66 \[ -\frac {2 \left (6075 x^{5}+22923 x^{4}+29565 x^{3}+26847 x^{2}+21690 x -1462 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+771 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+7560\right )}{945 \sqrt {3 x^{2}+5 x +2}\, \sqrt {x}} \]
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 \[ -\int \frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int -\frac {\left (5\,x-2\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{x^{3/2}} \,d x \]
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 \[ - \int \left (- \frac {4 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {3}{2}}}\right )\, dx - \int 19 \sqrt {x} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 15 x^{\frac {3}{2}} \sqrt {3 x^{2} + 5 x + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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