Optimal. Leaf size=280 \[ -\frac{2257166048 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{53093313 \sqrt{33}}-\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}+\frac{230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{1287 (3 x+2)^{11/2}}+\frac{1300 (5 x+3)^{3/2} \sqrt{1-2 x}}{891 (3 x+2)^{9/2}}+\frac{75041008472 \sqrt{5 x+3} \sqrt{1-2 x}}{584026443 \sqrt{3 x+2}}+\frac{1079936248 \sqrt{5 x+3} \sqrt{1-2 x}}{83432349 (3 x+2)^{3/2}}+\frac{23210828 \sqrt{5 x+3} \sqrt{1-2 x}}{11918907 (3 x+2)^{5/2}}-\frac{3347620 \sqrt{5 x+3} \sqrt{1-2 x}}{1702701 (3 x+2)^{7/2}}-\frac{75041008472 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{53093313 \sqrt{33}} \]
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Rubi [A] time = 0.115585, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {97, 150, 152, 158, 113, 119} \[ -\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}+\frac{230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{1287 (3 x+2)^{11/2}}+\frac{1300 (5 x+3)^{3/2} \sqrt{1-2 x}}{891 (3 x+2)^{9/2}}+\frac{75041008472 \sqrt{5 x+3} \sqrt{1-2 x}}{584026443 \sqrt{3 x+2}}+\frac{1079936248 \sqrt{5 x+3} \sqrt{1-2 x}}{83432349 (3 x+2)^{3/2}}+\frac{23210828 \sqrt{5 x+3} \sqrt{1-2 x}}{11918907 (3 x+2)^{5/2}}-\frac{3347620 \sqrt{5 x+3} \sqrt{1-2 x}}{1702701 (3 x+2)^{7/2}}-\frac{2257166048 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{53093313 \sqrt{33}}-\frac{75041008472 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{53093313 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{15/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac{2}{39} \int \frac{\left (-\frac{15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^{13/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}-\frac{4 \int \frac{\sqrt{1-2 x} \sqrt{3+5 x} \left (-\frac{2895}{2}+\frac{1995 x}{2}\right )}{(2+3 x)^{11/2}} \, dx}{1287}\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac{1300 \sqrt{1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{8 \int \frac{\left (\frac{438345}{4}-149460 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{9/2}} \, dx}{34749}\\ &=-\frac{3347620 \sqrt{1-2 x} \sqrt{3+5 x}}{1702701 (2+3 x)^{7/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac{1300 \sqrt{1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{16 \int \frac{\frac{15056835}{8}-\frac{10467525 x}{4}}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx}{5108103}\\ &=-\frac{3347620 \sqrt{1-2 x} \sqrt{3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac{23210828 \sqrt{1-2 x} \sqrt{3+5 x}}{11918907 (2+3 x)^{5/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac{1300 \sqrt{1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{32 \int \frac{\frac{1154474415}{8}-\frac{1305609075 x}{8}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{178783605}\\ &=-\frac{3347620 \sqrt{1-2 x} \sqrt{3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac{23210828 \sqrt{1-2 x} \sqrt{3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac{1079936248 \sqrt{1-2 x} \sqrt{3+5 x}}{83432349 (2+3 x)^{3/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac{1300 \sqrt{1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{64 \int \frac{\frac{100204281585}{16}-\frac{30373206975 x}{8}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{3754455705}\\ &=-\frac{3347620 \sqrt{1-2 x} \sqrt{3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac{23210828 \sqrt{1-2 x} \sqrt{3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac{1079936248 \sqrt{1-2 x} \sqrt{3+5 x}}{83432349 (2+3 x)^{3/2}}+\frac{75041008472 \sqrt{1-2 x} \sqrt{3+5 x}}{584026443 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac{1300 \sqrt{1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{128 \int \frac{\frac{1336148092575}{16}+\frac{2110528363275 x}{16}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{26281189935}\\ &=-\frac{3347620 \sqrt{1-2 x} \sqrt{3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac{23210828 \sqrt{1-2 x} \sqrt{3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac{1079936248 \sqrt{1-2 x} \sqrt{3+5 x}}{83432349 (2+3 x)^{3/2}}+\frac{75041008472 \sqrt{1-2 x} \sqrt{3+5 x}}{584026443 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac{1300 \sqrt{1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{1128583024 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{53093313}+\frac{75041008472 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{584026443}\\ &=-\frac{3347620 \sqrt{1-2 x} \sqrt{3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac{23210828 \sqrt{1-2 x} \sqrt{3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac{1079936248 \sqrt{1-2 x} \sqrt{3+5 x}}{83432349 (2+3 x)^{3/2}}+\frac{75041008472 \sqrt{1-2 x} \sqrt{3+5 x}}{584026443 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac{1300 \sqrt{1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}-\frac{75041008472 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{53093313 \sqrt{33}}-\frac{2257166048 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{53093313 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.298975, size = 117, normalized size = 0.42 \[ \frac{-604764298880 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{48 \sqrt{2-4 x} \sqrt{5 x+3} \left (27352447588044 x^6+110328276131100 x^5+185457331738206 x^4+166295375376786 x^3+83893544414217 x^2+22577209892436 x+2532151719515\right )}{(3 x+2)^{13/2}}+1200656135552 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{14016634632 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.022, size = 694, normalized size = 2.5 \begin{align*} \text{result too large to display} \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 (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\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 (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256}, 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 (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{15}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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