Optimal. Leaf size=133 \[ -\frac {26352 \sqrt {1-2 x} (2+3 x)^2}{34375}-\frac {1717 \sqrt {1-2 x} (2+3 x)^3}{9625}-\frac {8}{275} \sqrt {1-2 x} (2+3 x)^4-\frac {\sqrt {1-2 x} (2+3 x)^5}{55 (3+5 x)}-\frac {3 \sqrt {1-2 x} (1847824+615875 x)}{171875}-\frac {398 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{171875 \sqrt {55}} \]
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Rubi [A]
time = 0.03, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 158, 152,
65, 212} \begin {gather*} -\frac {\sqrt {1-2 x} (3 x+2)^5}{55 (5 x+3)}-\frac {8}{275} \sqrt {1-2 x} (3 x+2)^4-\frac {1717 \sqrt {1-2 x} (3 x+2)^3}{9625}-\frac {26352 \sqrt {1-2 x} (3 x+2)^2}{34375}-\frac {3 \sqrt {1-2 x} (615875 x+1847824)}{171875}-\frac {398 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{171875 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 152
Rule 158
Rule 212
Rubi steps
\begin {align*} \int \frac {(2+3 x)^6}{\sqrt {1-2 x} (3+5 x)^2} \, dx &=-\frac {\sqrt {1-2 x} (2+3 x)^5}{55 (3+5 x)}-\frac {1}{55} \int \frac {(-83-72 x) (2+3 x)^4}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {8}{275} \sqrt {1-2 x} (2+3 x)^4-\frac {\sqrt {1-2 x} (2+3 x)^5}{55 (3+5 x)}+\frac {\int \frac {(2+3 x)^3 (9630+15453 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{2475}\\ &=-\frac {1717 \sqrt {1-2 x} (2+3 x)^3}{9625}-\frac {8}{275} \sqrt {1-2 x} (2+3 x)^4-\frac {\sqrt {1-2 x} (2+3 x)^5}{55 (3+5 x)}-\frac {\int \frac {(-998613-1660176 x) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx}{86625}\\ &=-\frac {26352 \sqrt {1-2 x} (2+3 x)^2}{34375}-\frac {1717 \sqrt {1-2 x} (2+3 x)^3}{9625}-\frac {8}{275} \sqrt {1-2 x} (2+3 x)^4-\frac {\sqrt {1-2 x} (2+3 x)^5}{55 (3+5 x)}+\frac {\int \frac {(2+3 x) (69852762+116400375 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{2165625}\\ &=-\frac {26352 \sqrt {1-2 x} (2+3 x)^2}{34375}-\frac {1717 \sqrt {1-2 x} (2+3 x)^3}{9625}-\frac {8}{275} \sqrt {1-2 x} (2+3 x)^4-\frac {\sqrt {1-2 x} (2+3 x)^5}{55 (3+5 x)}-\frac {3 \sqrt {1-2 x} (1847824+615875 x)}{171875}+\frac {199 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{171875}\\ &=-\frac {26352 \sqrt {1-2 x} (2+3 x)^2}{34375}-\frac {1717 \sqrt {1-2 x} (2+3 x)^3}{9625}-\frac {8}{275} \sqrt {1-2 x} (2+3 x)^4-\frac {\sqrt {1-2 x} (2+3 x)^5}{55 (3+5 x)}-\frac {3 \sqrt {1-2 x} (1847824+615875 x)}{171875}-\frac {199 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{171875}\\ &=-\frac {26352 \sqrt {1-2 x} (2+3 x)^2}{34375}-\frac {1717 \sqrt {1-2 x} (2+3 x)^3}{9625}-\frac {8}{275} \sqrt {1-2 x} (2+3 x)^4-\frac {\sqrt {1-2 x} (2+3 x)^5}{55 (3+5 x)}-\frac {3 \sqrt {1-2 x} (1847824+615875 x)}{171875}-\frac {398 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{171875 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 73, normalized size = 0.55 \begin {gather*} \frac {-\frac {55 \sqrt {1-2 x} \left (135011752+334366065 x+273540465 x^2+200942775 x^3+92998125 x^4+19490625 x^5\right )}{3+5 x}-2786 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{66171875} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 81, normalized size = 0.61
method | result | size |
risch | \(\frac {38981250 x^{6}+166505625 x^{5}+308887425 x^{4}+346138155 x^{3}+395191665 x^{2}-64342561 x -135011752}{1203125 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {398 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{9453125}\) | \(66\) |
derivativedivides | \(-\frac {81 \left (1-2 x \right )^{\frac {9}{2}}}{400}+\frac {2187 \left (1-2 x \right )^{\frac {7}{2}}}{875}-\frac {315171 \left (1-2 x \right )^{\frac {5}{2}}}{25000}+\frac {105228 \left (1-2 x \right )^{\frac {3}{2}}}{3125}-\frac {607689 \sqrt {1-2 x}}{10000}+\frac {2 \sqrt {1-2 x}}{859375 \left (-\frac {6}{5}-2 x \right )}-\frac {398 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{9453125}\) | \(81\) |
default | \(-\frac {81 \left (1-2 x \right )^{\frac {9}{2}}}{400}+\frac {2187 \left (1-2 x \right )^{\frac {7}{2}}}{875}-\frac {315171 \left (1-2 x \right )^{\frac {5}{2}}}{25000}+\frac {105228 \left (1-2 x \right )^{\frac {3}{2}}}{3125}-\frac {607689 \sqrt {1-2 x}}{10000}+\frac {2 \sqrt {1-2 x}}{859375 \left (-\frac {6}{5}-2 x \right )}-\frac {398 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{9453125}\) | \(81\) |
trager | \(-\frac {\left (19490625 x^{5}+92998125 x^{4}+200942775 x^{3}+273540465 x^{2}+334366065 x +135011752\right ) \sqrt {1-2 x}}{1203125 \left (3+5 x \right )}-\frac {199 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{9453125}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 98, normalized size = 0.74 \begin {gather*} -\frac {81}{400} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {2187}{875} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {315171}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {105228}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {199}{9453125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {607689}{10000} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{171875 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.00, size = 79, normalized size = 0.59 \begin {gather*} \frac {1393 \, \sqrt {55} {\left (5 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (19490625 \, x^{5} + 92998125 \, x^{4} + 200942775 \, x^{3} + 273540465 \, x^{2} + 334366065 \, x + 135011752\right )} \sqrt {-2 \, x + 1}}{66171875 \, {\left (5 \, 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 = 0.76, size = 122, normalized size = 0.92 \begin {gather*} -\frac {81}{400} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {2187}{875} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {315171}{25000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {105228}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {199}{9453125} \, \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 {607689}{10000} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{171875 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 82, normalized size = 0.62 \begin {gather*} \frac {105228\,{\left (1-2\,x\right )}^{3/2}}{3125}-\frac {607689\,\sqrt {1-2\,x}}{10000}-\frac {2\,\sqrt {1-2\,x}}{859375\,\left (2\,x+\frac {6}{5}\right )}-\frac {315171\,{\left (1-2\,x\right )}^{5/2}}{25000}+\frac {2187\,{\left (1-2\,x\right )}^{7/2}}{875}-\frac {81\,{\left (1-2\,x\right )}^{9/2}}{400}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,398{}\mathrm {i}}{9453125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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