Optimal. Leaf size=191 \[ -\frac {2077}{50} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {(5 x+3)^{3/2} (3 x+2)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {45 (5 x+3)^{3/2} (3 x+2)^{3/2}}{11 \sqrt {1-2 x}}-\frac {807}{110} \sqrt {1-2 x} (5 x+3)^{3/2} \sqrt {3 x+2}-\frac {6231}{110} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}-\frac {37663}{100} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {97, 150, 154, 158, 113, 119} \[ \frac {(5 x+3)^{3/2} (3 x+2)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {45 (5 x+3)^{3/2} (3 x+2)^{3/2}}{11 \sqrt {1-2 x}}-\frac {807}{110} \sqrt {1-2 x} (5 x+3)^{3/2} \sqrt {3 x+2}-\frac {6231}{110} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}-\frac {2077}{50} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {37663}{100} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 97
Rule 113
Rule 119
Rule 150
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(2+3 x)^{3/2} \sqrt {3+5 x} \left (\frac {75}{2}+60 x\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {45 (2+3 x)^{3/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-\frac {7665}{2}-\frac {12105 x}{2}\right ) \sqrt {2+3 x} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {807}{110} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {45 (2+3 x)^{3/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {1}{825} \int \frac {\sqrt {3+5 x} \left (\frac {1093335}{4}+\frac {841185 x}{2}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {6231}{110} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {807}{110} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {45 (2+3 x)^{3/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {\int \frac {-8852085-\frac {55929555 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{7425}\\ &=-\frac {6231}{110} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {807}{110} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {45 (2+3 x)^{3/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {6231}{100} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {37663}{100} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {6231}{110} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {807}{110} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {45 (2+3 x)^{3/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {37663}{100} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2077}{50} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
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Mathematica [A] time = 0.32, size = 125, normalized size = 0.65 \[ -\frac {-18970 \sqrt {2-4 x} (2 x-1) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+10 \sqrt {3 x+2} \sqrt {5 x+3} \left (270 x^3+1344 x^2-8039 x+2976\right )+37663 \sqrt {2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{300 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 238, normalized size = 1.25 \[ \frac {\left (-40500 x^{5}-252900 x^{4}+934290 x^{3}+1000370 x^{2}-75326 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+37940 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-83100 x +37663 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-18970 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-178560\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \sqrt {3 x +2}}{300 \left (2 x -1\right )^{2} \left (15 x^{2}+19 x +6\right )} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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