Optimal. Leaf size=127 \[ -\frac{68}{125} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )-\frac{22 \sqrt{3 x+2} (1-2 x)^{3/2}}{15 (5 x+3)^{3/2}}+\frac{572 \sqrt{3 x+2} \sqrt{1-2 x}}{25 \sqrt{5 x+3}}-\frac{584}{125} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0366901, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {98, 150, 158, 113, 119} \[ -\frac{22 \sqrt{3 x+2} (1-2 x)^{3/2}}{15 (5 x+3)^{3/2}}+\frac{572 \sqrt{3 x+2} \sqrt{1-2 x}}{25 \sqrt{5 x+3}}-\frac{68}{125} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{584}{125} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{\sqrt{2+3 x} (3+5 x)^{5/2}} \, dx &=-\frac{22 (1-2 x)^{3/2} \sqrt{2+3 x}}{15 (3+5 x)^{3/2}}-\frac{2}{15} \int \frac{\sqrt{1-2 x} (102+27 x)}{\sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{22 (1-2 x)^{3/2} \sqrt{2+3 x}}{15 (3+5 x)^{3/2}}+\frac{572 \sqrt{1-2 x} \sqrt{2+3 x}}{25 \sqrt{3+5 x}}-\frac{4}{75} \int \frac{-\frac{1689}{2}-1314 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{22 (1-2 x)^{3/2} \sqrt{2+3 x}}{15 (3+5 x)^{3/2}}+\frac{572 \sqrt{1-2 x} \sqrt{2+3 x}}{25 \sqrt{3+5 x}}+\frac{374}{125} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{1752}{125} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{22 (1-2 x)^{3/2} \sqrt{2+3 x}}{15 (3+5 x)^{3/2}}+\frac{572 \sqrt{1-2 x} \sqrt{2+3 x}}{25 \sqrt{3+5 x}}-\frac{584}{125} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{68}{125} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.170532, size = 97, normalized size = 0.76 \[ \frac{2}{375} \left (-315 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{55 \sqrt{1-2 x} \sqrt{3 x+2} (400 x+229)}{(5 x+3)^{3/2}}+876 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 219, normalized size = 1.7 \begin{align*}{\frac{2}{2250\,{x}^{2}+375\,x-750} \left ( 1575\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-4380\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+945\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -2628\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) +132000\,{x}^{3}+97570\,{x}^{2}-31405\,x-25190 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{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 (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{3 \, x + 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 (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{375 \, x^{4} + 925 \, x^{3} + 855 \, x^{2} + 351 \, x + 54}, 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 (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{3 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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