Optimal. Leaf size=227 \[ \frac{397 \sqrt{\frac{3}{22}} \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )}{89125 \sqrt{2 x-5}}+\frac{8953 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{556140 (5 x+7)}-\frac{\sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{10 (5 x+7)^2}-\frac{8953 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{1390350 \sqrt{5-2 x}}-\frac{14832503 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{287339000 \sqrt{11} \sqrt{2 x-5}} \]
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Rubi [A] time = 0.308533, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.314, Rules used = {160, 1604, 1607, 168, 538, 537, 158, 114, 113, 121, 119} \[ \frac{8953 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{556140 (5 x+7)}-\frac{\sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{10 (5 x+7)^2}+\frac{397 \sqrt{\frac{3}{22}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{89125 \sqrt{2 x-5}}-\frac{8953 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{1390350 \sqrt{5-2 x}}-\frac{14832503 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{287339000 \sqrt{11} \sqrt{2 x-5}} \]
Antiderivative was successfully verified.
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Rule 160
Rule 1604
Rule 1607
Rule 168
Rule 538
Rule 537
Rule 158
Rule 114
Rule 113
Rule 121
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{(7+5 x)^3} \, dx &=-\frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{10 (7+5 x)^2}+\frac{1}{20} \int \frac{-21+140 x-72 x^2}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^2} \, dx\\ &=-\frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{10 (7+5 x)^2}+\frac{8953 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{556140 (7+5 x)}+\frac{\int \frac{-106729-199200 x+214872 x^2}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)} \, dx}{1112280}\\ &=-\frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{10 (7+5 x)^2}+\frac{8953 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{556140 (7+5 x)}+\frac{\int \frac{-\frac{2500104}{25}+\frac{214872 x}{5}}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{1112280}+\frac{14832503 \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)} \, dx}{27807000}\\ &=-\frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{10 (7+5 x)^2}+\frac{8953 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{556140 (7+5 x)}+\frac{1191 \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{178250}+\frac{8953 \int \frac{\sqrt{-5+2 x}}{\sqrt{2-3 x} \sqrt{1+4 x}} \, dx}{463450}-\frac{14832503 \operatorname{Subst}\left (\int \frac{1}{\left (31-5 x^2\right ) \sqrt{\frac{11}{3}-\frac{4 x^2}{3}} \sqrt{-\frac{11}{3}-\frac{2 x^2}{3}}} \, dx,x,\sqrt{2-3 x}\right )}{13903500}\\ &=-\frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{10 (7+5 x)^2}+\frac{8953 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{556140 (7+5 x)}+\frac{\left (1191 \sqrt{5-2 x}\right ) \int \frac{1}{\sqrt{2-3 x} \sqrt{\frac{10}{11}-\frac{4 x}{11}} \sqrt{1+4 x}} \, dx}{89125 \sqrt{22} \sqrt{-5+2 x}}-\frac{\left (14832503 \sqrt{5-2 x}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (31-5 x^2\right ) \sqrt{\frac{11}{3}-\frac{4 x^2}{3}} \sqrt{1+\frac{2 x^2}{11}}} \, dx,x,\sqrt{2-3 x}\right )}{4634500 \sqrt{33} \sqrt{-5+2 x}}+\frac{\left (8953 \sqrt{-5+2 x}\right ) \int \frac{\sqrt{\frac{15}{11}-\frac{6 x}{11}}}{\sqrt{2-3 x} \sqrt{\frac{3}{11}+\frac{12 x}{11}}} \, dx}{463450 \sqrt{5-2 x}}\\ &=-\frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{10 (7+5 x)^2}+\frac{8953 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{556140 (7+5 x)}-\frac{8953 \sqrt{11} \sqrt{-5+2 x} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{1390350 \sqrt{5-2 x}}+\frac{397 \sqrt{\frac{3}{22}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{1+4 x}\right )|\frac{1}{3}\right )}{89125 \sqrt{-5+2 x}}-\frac{14832503 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{287339000 \sqrt{11} \sqrt{-5+2 x}}\\ \end{align*}
Mathematica [A] time = 0.696315, size = 136, normalized size = 0.6 \[ \frac{\sqrt{2 x-5} \left (\frac{\sqrt{11} \left (5759676 \text{EllipticF}\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right ),-\frac{1}{2}\right )-61059460 E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )-44497509 \Pi \left (\frac{55}{124};-\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )\right )}{\sqrt{5-2 x}}+\frac{17050 \sqrt{2-3 x} \sqrt{4 x+1} (44765 x+7057)}{(5 x+7)^2}\right )}{9482187000} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.025, size = 461, normalized size = 2. \begin{align*}{\frac{1}{ \left ( 227572488000\,{x}^{3}-663753090000\,{x}^{2}+199125927000\,x+94821870000 \right ) \left ( 7+5\,x \right ) ^{2}}\sqrt{2-3\,x}\sqrt{2\,x-5}\sqrt{4\,x+1} \left ( 143991900\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticF} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ){x}^{2}-1526486500\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticE} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ){x}^{2}+1112437725\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticPi} \left ( 2/11\,\sqrt{22-33\,x},{\frac{55}{124}},i/2\sqrt{2} \right ){x}^{2}+403177320\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticF} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ) x-4274162200\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticE} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ) x+3114825630\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticPi} \left ( 2/11\,\sqrt{22-33\,x},{\frac{55}{124}},i/2\sqrt{2} \right ) x+282224124\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticF} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ) -2991913540\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticE} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ) +2180377941\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticPi} \left ( 2/11\,\sqrt{22-33\,x},{\frac{55}{124}},i/2\sqrt{2} \right ) +18317838000\,{x}^{4}-50539303100\,{x}^{3}+7605578750\,{x}^{2}+10159191350\,x+1203218500 \right ) } \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{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3}}\,{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{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{125 \, x^{3} + 525 \, x^{2} + 735 \, x + 343}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 - 3 x} \sqrt{2 x - 5} \sqrt{4 x + 1}}{\left (5 x + 7\right )^{3}}\, dx \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{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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