Optimal. Leaf size=210 \[ \frac {5438 \sqrt {3 x^2+5 x+2}}{315 \sqrt {x}}-\frac {5438 \sqrt {x} (3 x+2)}{315 \sqrt {3 x^2+5 x+2}}-\frac {899 (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{21 \sqrt {2} \sqrt {3 x^2+5 x+2}}+\frac {5438 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{315 \sqrt {3 x^2+5 x+2}}-\frac {4 (7-15 x) \left (3 x^2+5 x+2\right )^{3/2}}{63 x^{9/2}}+\frac {(4055 x+1446) \sqrt {3 x^2+5 x+2}}{315 x^{5/2}} \]
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Rubi [A] time = 0.13, 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 = {810, 834, 839, 1189, 1100, 1136} \[ -\frac {4 (7-15 x) \left (3 x^2+5 x+2\right )^{3/2}}{63 x^{9/2}}+\frac {(4055 x+1446) \sqrt {3 x^2+5 x+2}}{315 x^{5/2}}+\frac {5438 \sqrt {3 x^2+5 x+2}}{315 \sqrt {x}}-\frac {5438 \sqrt {x} (3 x+2)}{315 \sqrt {3 x^2+5 x+2}}-\frac {899 (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{21 \sqrt {2} \sqrt {3 x^2+5 x+2}}+\frac {5438 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{315 \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 810
Rule 834
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^{11/2}} \, dx &=-\frac {4 (7-15 x) \left (2+5 x+3 x^2\right )^{3/2}}{63 x^{9/2}}-\frac {1}{21} \int \frac {(241+285 x) \sqrt {2+5 x+3 x^2}}{x^{7/2}} \, dx\\ &=\frac {(1446+4055 x) \sqrt {2+5 x+3 x^2}}{315 x^{5/2}}-\frac {4 (7-15 x) \left (2+5 x+3 x^2\right )^{3/2}}{63 x^{9/2}}+\frac {1}{315} \int \frac {-5438-\frac {13485 x}{2}}{x^{3/2} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {5438 \sqrt {2+5 x+3 x^2}}{315 \sqrt {x}}+\frac {(1446+4055 x) \sqrt {2+5 x+3 x^2}}{315 x^{5/2}}-\frac {4 (7-15 x) \left (2+5 x+3 x^2\right )^{3/2}}{63 x^{9/2}}-\frac {1}{315} \int \frac {\frac {13485}{2}+8157 x}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {5438 \sqrt {2+5 x+3 x^2}}{315 \sqrt {x}}+\frac {(1446+4055 x) \sqrt {2+5 x+3 x^2}}{315 x^{5/2}}-\frac {4 (7-15 x) \left (2+5 x+3 x^2\right )^{3/2}}{63 x^{9/2}}-\frac {2}{315} \operatorname {Subst}\left (\int \frac {\frac {13485}{2}+8157 x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {5438 \sqrt {2+5 x+3 x^2}}{315 \sqrt {x}}+\frac {(1446+4055 x) \sqrt {2+5 x+3 x^2}}{315 x^{5/2}}-\frac {4 (7-15 x) \left (2+5 x+3 x^2\right )^{3/2}}{63 x^{9/2}}-\frac {899}{21} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )-\frac {5438}{105} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {5438 \sqrt {x} (2+3 x)}{315 \sqrt {2+5 x+3 x^2}}+\frac {5438 \sqrt {2+5 x+3 x^2}}{315 \sqrt {x}}+\frac {(1446+4055 x) \sqrt {2+5 x+3 x^2}}{315 x^{5/2}}-\frac {4 (7-15 x) \left (2+5 x+3 x^2\right )^{3/2}}{63 x^{9/2}}+\frac {5438 \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 {899 (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{21 \sqrt {2} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 160, normalized size = 0.76 \[ \frac {-2609 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{11/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-10876 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{11/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+29730 x^5+64706 x^4+44480 x^3+7424 x^2-3200 x-1120}{630 x^{9/2} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (15 \, x^{3} + 19 \, x^{2} - 4\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{x^{\frac {11}{2}}}, x\right ) \]
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 \[ \int -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 134, normalized size = 0.64 \[ \frac {97884 x^{6}+252330 x^{5}-5438 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x^{4} \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+2829 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x^{4} \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+259374 x^{4}+133440 x^{3}+22272 x^{2}-9600 x -3360}{1890 \sqrt {3 x^{2}+5 x +2}\, x^{\frac {9}{2}}} \]
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 \[ -\int \frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {11}{2}}}\,{d x} \]
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 \[ \int -\frac {\left (5\,x-2\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{x^{11/2}} \,d x \]
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 \[ - \int \left (- \frac {4 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {11}{2}}}\right )\, dx - \int \frac {19 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {7}{2}}}\, dx - \int \frac {15 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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