Optimal. Leaf size=153 \[ -\frac {35495 \sqrt {1-2 x}}{1078 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^2}+\frac {429 \sqrt {1-2 x}}{98 (2+3 x) (3+5 x)^2}+\frac {1177080 \sqrt {1-2 x}}{5929 (3+5 x)}+\frac {134217}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {321825}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {105, 156, 162,
65, 212} \begin {gather*} \frac {1177080 \sqrt {1-2 x}}{5929 (5 x+3)}-\frac {35495 \sqrt {1-2 x}}{1078 (5 x+3)^2}+\frac {429 \sqrt {1-2 x}}{98 (3 x+2) (5 x+3)^2}+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^2}+\frac {134217}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {321825}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 105
Rule 156
Rule 162
Rule 212
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx &=\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^2}+\frac {1}{14} \int \frac {73-105 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^2}+\frac {429 \sqrt {1-2 x}}{98 (2+3 x) (3+5 x)^2}+\frac {1}{98} \int \frac {7763-10725 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac {35495 \sqrt {1-2 x}}{1078 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^2}+\frac {429 \sqrt {1-2 x}}{98 (2+3 x) (3+5 x)^2}-\frac {\int \frac {558318-638910 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx}{2156}\\ &=-\frac {35495 \sqrt {1-2 x}}{1078 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^2}+\frac {429 \sqrt {1-2 x}}{98 (2+3 x) (3+5 x)^2}+\frac {1177080 \sqrt {1-2 x}}{5929 (3+5 x)}+\frac {\int \frac {23063874-14124960 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{23716}\\ &=-\frac {35495 \sqrt {1-2 x}}{1078 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^2}+\frac {429 \sqrt {1-2 x}}{98 (2+3 x) (3+5 x)^2}+\frac {1177080 \sqrt {1-2 x}}{5929 (3+5 x)}-\frac {402651}{98} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {1609125}{242} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {35495 \sqrt {1-2 x}}{1078 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^2}+\frac {429 \sqrt {1-2 x}}{98 (2+3 x) (3+5 x)^2}+\frac {1177080 \sqrt {1-2 x}}{5929 (3+5 x)}+\frac {402651}{98} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {1609125}{242} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {35495 \sqrt {1-2 x}}{1078 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^2}+\frac {429 \sqrt {1-2 x}}{98 (2+3 x) (3+5 x)^2}+\frac {1177080 \sqrt {1-2 x}}{5929 (3+5 x)}+\frac {134217}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {321825}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 99, normalized size = 0.65 \begin {gather*} \frac {\sqrt {1-2 x} \left (26794499+127303347 x+201297915 x^2+105937200 x^3\right )}{11858 \left (6+19 x+15 x^2\right )^2}+\frac {134217}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {321825}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 94, normalized size = 0.61
method | result | size |
risch | \(-\frac {\left (-1+2 x \right ) \left (105937200 x^{3}+201297915 x^{2}+127303347 x +26794499\right )}{11858 \left (15 x^{2}+19 x +6\right )^{2} \sqrt {1-2 x}}-\frac {321825 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {134217 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\) | \(79\) |
derivativedivides | \(\frac {-\frac {121875 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {24125 \sqrt {1-2 x}}{11}}{\left (-6-10 x \right )^{2}}-\frac {321825 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {972 \left (\frac {71 \left (1-2 x \right )^{\frac {3}{2}}}{196}-\frac {215 \sqrt {1-2 x}}{252}\right )}{\left (-4-6 x \right )^{2}}+\frac {134217 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\) | \(94\) |
default | \(\frac {-\frac {121875 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {24125 \sqrt {1-2 x}}{11}}{\left (-6-10 x \right )^{2}}-\frac {321825 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {972 \left (\frac {71 \left (1-2 x \right )^{\frac {3}{2}}}{196}-\frac {215 \sqrt {1-2 x}}{252}\right )}{\left (-4-6 x \right )^{2}}+\frac {134217 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\) | \(94\) |
trager | \(\frac {\left (105937200 x^{3}+201297915 x^{2}+127303347 x +26794499\right ) \sqrt {1-2 x}}{11858 \left (15 x^{2}+19 x +6\right )^{2}}-\frac {525 \RootOf \left (\textit {\_Z}^{2}-20667295\right ) \ln \left (\frac {-5 \RootOf \left (\textit {\_Z}^{2}-20667295\right ) x +33715 \sqrt {1-2 x}+8 \RootOf \left (\textit {\_Z}^{2}-20667295\right )}{3+5 x}\right )}{2662}-\frac {81 \RootOf \left (\textit {\_Z}^{2}-57658629\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-57658629\right ) x +34797 \sqrt {1-2 x}-5 \RootOf \left (\textit {\_Z}^{2}-57658629\right )}{2+3 x}\right )}{686}\) | \(126\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 146, normalized size = 0.95 \begin {gather*} \frac {321825}{2662} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {134217}{686} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (52968600 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 360203715 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 816108324 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 616051205 \, \sqrt {-2 \, x + 1}\right )}}{5929 \, {\left (225 \, {\left (2 \, x - 1\right )}^{4} + 2040 \, {\left (2 \, x - 1\right )}^{3} + 6934 \, {\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.69, size = 162, normalized size = 1.06 \begin {gather*} \frac {110385975 \, \sqrt {11} \sqrt {5} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 178642827 \, \sqrt {7} \sqrt {3} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (105937200 \, x^{3} + 201297915 \, x^{2} + 127303347 \, x + 26794499\right )} \sqrt {-2 \, x + 1}}{913066 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 16.21, size = 6346, normalized size = 41.48 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.17, size = 148, normalized size = 0.97 \begin {gather*} \frac {321825}{2662} \, \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 {134217}{686} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2 \, {\left (52968600 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 360203715 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 816108324 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 616051205 \, \sqrt {-2 \, x + 1}\right )}}{5929 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 107, normalized size = 0.70 \begin {gather*} \frac {134217\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {321825\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331}+\frac {\frac {3200266\,\sqrt {1-2\,x}}{3465}-\frac {544072216\,{\left (1-2\,x\right )}^{3/2}}{444675}+\frac {16009054\,{\left (1-2\,x\right )}^{5/2}}{29645}-\frac {470832\,{\left (1-2\,x\right )}^{7/2}}{5929}}{\frac {20944\,x}{225}+\frac {6934\,{\left (2\,x-1\right )}^2}{225}+\frac {136\,{\left (2\,x-1\right )}^3}{15}+{\left (2\,x-1\right )}^4-\frac {4543}{225}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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