Optimal. Leaf size=100 \[ -\frac {36 \sqrt {1-2 x} (2+3 x)^2}{605 (3+5 x)}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} (3+5 x)}+\frac {27 \sqrt {1-2 x} (792+265 x)}{3025}-\frac {54 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}} \]
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Rubi [A]
time = 0.02, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 154, 152,
65, 212} \begin {gather*} \frac {7 (3 x+2)^3}{11 \sqrt {1-2 x} (5 x+3)}-\frac {36 \sqrt {1-2 x} (3 x+2)^2}{605 (5 x+3)}+\frac {27 \sqrt {1-2 x} (265 x+792)}{3025}-\frac {54 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 152
Rule 154
Rule 212
Rubi steps
\begin {align*} \int \frac {(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^2} \, dx &=\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} (3+5 x)}-\frac {1}{11} \int \frac {(2+3 x)^2 (117+207 x)}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=-\frac {36 \sqrt {1-2 x} (2+3 x)^2}{605 (3+5 x)}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} (3+5 x)}-\frac {1}{605} \int \frac {(2+3 x) (4266+7155 x)}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {36 \sqrt {1-2 x} (2+3 x)^2}{605 (3+5 x)}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} (3+5 x)}+\frac {27 \sqrt {1-2 x} (792+265 x)}{3025}+\frac {27 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{3025}\\ &=-\frac {36 \sqrt {1-2 x} (2+3 x)^2}{605 (3+5 x)}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} (3+5 x)}+\frac {27 \sqrt {1-2 x} (792+265 x)}{3025}-\frac {27 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3025}\\ &=-\frac {36 \sqrt {1-2 x} (2+3 x)^2}{605 (3+5 x)}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} (3+5 x)}+\frac {27 \sqrt {1-2 x} (792+265 x)}{3025}-\frac {54 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 63, normalized size = 0.63 \begin {gather*} \frac {-\frac {55 \left (-78832-68661 x+114345 x^2+16335 x^3\right )}{\sqrt {1-2 x} (3+5 x)}-54 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{166375} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 63, normalized size = 0.63
method | result | size |
risch | \(-\frac {16335 x^{3}+114345 x^{2}-68661 x -78832}{3025 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {54 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{166375}\) | \(51\) |
derivativedivides | \(-\frac {27 \left (1-2 x \right )^{\frac {3}{2}}}{100}+\frac {999 \sqrt {1-2 x}}{250}+\frac {2 \sqrt {1-2 x}}{75625 \left (-\frac {6}{5}-2 x \right )}-\frac {54 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{166375}+\frac {2401}{484 \sqrt {1-2 x}}\) | \(63\) |
default | \(-\frac {27 \left (1-2 x \right )^{\frac {3}{2}}}{100}+\frac {999 \sqrt {1-2 x}}{250}+\frac {2 \sqrt {1-2 x}}{75625 \left (-\frac {6}{5}-2 x \right )}-\frac {54 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{166375}+\frac {2401}{484 \sqrt {1-2 x}}\) | \(63\) |
trager | \(\frac {\left (16335 x^{3}+114345 x^{2}-68661 x -78832\right ) \sqrt {1-2 x}}{30250 x^{2}+3025 x -9075}+\frac {27 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{166375}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 83, normalized size = 0.83 \begin {gather*} -\frac {27}{100} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {27}{166375} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {999}{250} \, \sqrt {-2 \, x + 1} - \frac {1500633 \, x + 900371}{30250 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.47, size = 75, normalized size = 0.75 \begin {gather*} \frac {27 \, \sqrt {55} {\left (10 \, x^{2} + x - 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (16335 \, x^{3} + 114345 \, x^{2} - 68661 \, x - 78832\right )} \sqrt {-2 \, x + 1}}{166375 \, {\left (10 \, x^{2} + x - 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.92, size = 86, normalized size = 0.86 \begin {gather*} -\frac {27}{100} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {27}{166375} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {999}{250} \, \sqrt {-2 \, x + 1} - \frac {1500633 \, x + 900371}{30250 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 66, normalized size = 0.66 \begin {gather*} \frac {\frac {1500633\,x}{151250}+\frac {900371}{151250}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}+\frac {999\,\sqrt {1-2\,x}}{250}-\frac {27\,{\left (1-2\,x\right )}^{3/2}}{100}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,54{}\mathrm {i}}{166375} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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