Optimal. Leaf size=256 \[ -\frac{3693 (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{2002 \sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}+\frac{(3445 x+1834) \sqrt{3 x^2+5 x+2}}{1001 x^{9/2}}+\frac{6907 \sqrt{3 x^2+5 x+2}}{10010 \sqrt{x}}-\frac{1231 \sqrt{3 x^2+5 x+2}}{2002 x^{3/2}}+\frac{204 \sqrt{3 x^2+5 x+2}}{385 x^{5/2}}-\frac{6907 \sqrt{x} (3 x+2)}{10010 \sqrt{3 x^2+5 x+2}}+\frac{6907 (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5005 \sqrt{2} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.179204, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {810, 834, 839, 1189, 1100, 1136} \[ -\frac{4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}+\frac{(3445 x+1834) \sqrt{3 x^2+5 x+2}}{1001 x^{9/2}}+\frac{6907 \sqrt{3 x^2+5 x+2}}{10010 \sqrt{x}}-\frac{1231 \sqrt{3 x^2+5 x+2}}{2002 x^{3/2}}+\frac{204 \sqrt{3 x^2+5 x+2}}{385 x^{5/2}}-\frac{6907 \sqrt{x} (3 x+2)}{10010 \sqrt{3 x^2+5 x+2}}-\frac{3693 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2002 \sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{6907 (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5005 \sqrt{2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 810
Rule 834
Rule 839
Rule 1189
Rule 1100
Rule 1136
Rubi steps
\begin{align*} \int \frac{(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx &=-\frac{4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}-\frac{3}{143} \int \frac{(393+465 x) \sqrt{2+5 x+3 x^2}}{x^{11/2}} \, dx\\ &=\frac{(1834+3445 x) \sqrt{2+5 x+3 x^2}}{1001 x^{9/2}}-\frac{4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}+\frac{\int \frac{-7956-\frac{20745 x}{2}}{x^{7/2} \sqrt{2+5 x+3 x^2}} \, dx}{3003}\\ &=\frac{204 \sqrt{2+5 x+3 x^2}}{385 x^{5/2}}+\frac{(1834+3445 x) \sqrt{2+5 x+3 x^2}}{1001 x^{9/2}}-\frac{4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}-\frac{\int \frac{-\frac{55395}{2}-35802 x}{x^{5/2} \sqrt{2+5 x+3 x^2}} \, dx}{15015}\\ &=\frac{204 \sqrt{2+5 x+3 x^2}}{385 x^{5/2}}-\frac{1231 \sqrt{2+5 x+3 x^2}}{2002 x^{3/2}}+\frac{(1834+3445 x) \sqrt{2+5 x+3 x^2}}{1001 x^{9/2}}-\frac{4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}+\frac{\int \frac{-\frac{62163}{2}-\frac{166185 x}{4}}{x^{3/2} \sqrt{2+5 x+3 x^2}} \, dx}{45045}\\ &=\frac{204 \sqrt{2+5 x+3 x^2}}{385 x^{5/2}}-\frac{1231 \sqrt{2+5 x+3 x^2}}{2002 x^{3/2}}+\frac{6907 \sqrt{2+5 x+3 x^2}}{10010 \sqrt{x}}+\frac{(1834+3445 x) \sqrt{2+5 x+3 x^2}}{1001 x^{9/2}}-\frac{4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}-\frac{\int \frac{\frac{166185}{4}+\frac{186489 x}{4}}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx}{45045}\\ &=\frac{204 \sqrt{2+5 x+3 x^2}}{385 x^{5/2}}-\frac{1231 \sqrt{2+5 x+3 x^2}}{2002 x^{3/2}}+\frac{6907 \sqrt{2+5 x+3 x^2}}{10010 \sqrt{x}}+\frac{(1834+3445 x) \sqrt{2+5 x+3 x^2}}{1001 x^{9/2}}-\frac{4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}-\frac{2 \operatorname{Subst}\left (\int \frac{\frac{166185}{4}+\frac{186489 x^2}{4}}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{45045}\\ &=\frac{204 \sqrt{2+5 x+3 x^2}}{385 x^{5/2}}-\frac{1231 \sqrt{2+5 x+3 x^2}}{2002 x^{3/2}}+\frac{6907 \sqrt{2+5 x+3 x^2}}{10010 \sqrt{x}}+\frac{(1834+3445 x) \sqrt{2+5 x+3 x^2}}{1001 x^{9/2}}-\frac{4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}-\frac{3693 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{2002}-\frac{20721 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{10010}\\ &=-\frac{6907 \sqrt{x} (2+3 x)}{10010 \sqrt{2+5 x+3 x^2}}+\frac{204 \sqrt{2+5 x+3 x^2}}{385 x^{5/2}}-\frac{1231 \sqrt{2+5 x+3 x^2}}{2002 x^{3/2}}+\frac{6907 \sqrt{2+5 x+3 x^2}}{10010 \sqrt{x}}+\frac{(1834+3445 x) \sqrt{2+5 x+3 x^2}}{1001 x^{9/2}}-\frac{4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}+\frac{6907 (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5005 \sqrt{2} \sqrt{2+5 x+3 x^2}}-\frac{3693 (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2002 \sqrt{2} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.188138, size = 170, normalized size = 0.66 \[ \frac{-4651 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{15/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )-36930 x^7-29726 x^6+361120 x^5+840316 x^4+654400 x^3+125440 x^2-13814 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{15/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-67200 x-24640}{20020 x^{13/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 144, normalized size = 0.6 \begin{align*}{\frac{1}{60060} \left ( 2256\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{6}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{6}-6907\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{6}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{6}+124326\,{x}^{8}+96420\,{x}^{7}-6294\,{x}^{6}+1083360\,{x}^{5}+2520948\,{x}^{4}+1963200\,{x}^{3}+376320\,{x}^{2}-201600\,x-73920 \right ){\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}{x}^{-{\frac{13}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (5 \, x - 2\right )}}{x^{\frac{15}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (15 \, x^{3} + 19 \, x^{2} - 4\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{x^{\frac{15}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (5 \, x - 2\right )}}{x^{\frac{15}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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