Optimal. Leaf size=203 \[ \frac {392989907 \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{2016 \sqrt {66} \sqrt {2 x-5}}-\frac {25}{84} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2-\frac {305}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)-\frac {120355}{288} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}-\frac {5109835 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{756 \sqrt {5-2 x}} \]
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Rubi [A] time = 0.22, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {167, 1600, 1615, 158, 114, 113, 121, 119} \[ -\frac {25}{84} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2-\frac {305}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)-\frac {120355}{288} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}+\frac {392989907 \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{2016 \sqrt {66} \sqrt {2 x-5}}-\frac {5109835 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{756 \sqrt {5-2 x}} \]
Antiderivative was successfully verified.
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Rule 113
Rule 114
Rule 119
Rule 121
Rule 158
Rule 167
Rule 1600
Rule 1615
Rubi steps
\begin {align*} \int \frac {(7+5 x)^4}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx &=-\frac {25}{84} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {1}{168} \int \frac {(7+5 x) \left (48949+134855 x+128100 x^2\right )}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx\\ &=-\frac {305}{24} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)-\frac {25}{84} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2-\frac {\int \frac {-9476460-227834100 x-303294600 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{20160}\\ &=-\frac {120355}{288} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {305}{24} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)-\frac {25}{84} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2-\frac {\int \frac {8530322220-88297948800 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{2177280}\\ &=-\frac {120355}{288} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {305}{24} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)-\frac {25}{84} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {5109835}{252} \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx+\frac {392989907 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{4032}\\ &=-\frac {120355}{288} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {305}{24} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)-\frac {25}{84} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {\left (392989907 \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}{2016 \sqrt {22} \sqrt {-5+2 x}}+\frac {\left (5109835 \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}{252 \sqrt {5-2 x}}\\ &=-\frac {120355}{288} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {305}{24} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)-\frac {25}{84} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2-\frac {5109835 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{756 \sqrt {5-2 x}}+\frac {392989907 \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{2016 \sqrt {66} \sqrt {-5+2 x}}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 125, normalized size = 0.62 \[ \frac {392989907 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )-1650 \sqrt {2-3 x} \sqrt {4 x+1} \left (1200 x^3+10608 x^2+50078 x-210245\right )-449665480 \sqrt {66} \sqrt {5-2 x} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{133056 \sqrt {2 x-5}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (625 \, x^{4} + 3500 \, x^{3} + 7350 \, x^{2} + 6860 \, x + 2401\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{24 \, x^{3} - 70 \, x^{2} + 21 \, x + 10}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 7\right )}^{4}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 150, normalized size = 0.74 \[ \frac {\sqrt {-3 x +2}\, \sqrt {2 x -5}\, \sqrt {4 x +1}\, \left (-23760000 x^{5}-200138400 x^{4}-900068400 x^{3}+4611000900 x^{2}-1569263850 x -899330960 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticE \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )+1178969721 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )-693808500\right )}{3193344 x^{3}-9313920 x^{2}+2794176 x +1330560} \]
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 \[ \int \frac {{\left (5 \, x + 7\right )}^{4}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}\,{d x} \]
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 \[ \int \frac {{\left (5\,x+7\right )}^4}{\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}} \,d x \]
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 \[ \int \frac {\left (5 x + 7\right )^{4}}{\sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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