Optimal. Leaf size=253 \[ -\frac {4}{7 \sqrt {2 x+1}}-\frac {1}{7} \sqrt {\frac {1}{434} \left (178+35 \sqrt {35}\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{7} \sqrt {\frac {1}{434} \left (178+35 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{7} \sqrt {\frac {2}{217} \left (35 \sqrt {35}-178\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{7} \sqrt {\frac {2}{217} \left (35 \sqrt {35}-178\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \]
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Rubi [A] time = 0.34, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {709, 826, 1169, 634, 618, 204, 628} \begin {gather*} -\frac {4}{7 \sqrt {2 x+1}}-\frac {1}{7} \sqrt {\frac {1}{434} \left (178+35 \sqrt {35}\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{7} \sqrt {\frac {1}{434} \left (178+35 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{7} \sqrt {\frac {2}{217} \left (35 \sqrt {35}-178\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{7} \sqrt {\frac {2}{217} \left (35 \sqrt {35}-178\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 709
Rule 826
Rule 1169
Rubi steps
\begin {align*} \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx &=-\frac {4}{7 \sqrt {1+2 x}}+\frac {1}{7} \int \frac {-1-10 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=-\frac {4}{7 \sqrt {1+2 x}}+\frac {2}{7} \operatorname {Subst}\left (\int \frac {8-10 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=-\frac {4}{7 \sqrt {1+2 x}}+\frac {\operatorname {Subst}\left (\int \frac {8 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (8+2 \sqrt {35}\right ) x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{7 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\operatorname {Subst}\left (\int \frac {8 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (8+2 \sqrt {35}\right ) x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{7 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=-\frac {4}{7 \sqrt {1+2 x}}-\frac {\left (4+\sqrt {35}\right ) \operatorname {Subst}\left (\int \frac {-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{7 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\left (4+\sqrt {35}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{7 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {1}{245} \left (-35+4 \sqrt {35}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )+\frac {1}{245} \left (-35+4 \sqrt {35}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )\\ &=-\frac {4}{7 \sqrt {1+2 x}}-\frac {1}{7} \sqrt {\frac {89}{217}+\frac {5 \sqrt {35}}{62}} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{7} \sqrt {\frac {89}{217}+\frac {5 \sqrt {35}}{62}} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{245} \left (2 \left (35-4 \sqrt {35}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {2}{5} \left (2-\sqrt {35}\right )-x^2} \, dx,x,-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )+\frac {1}{245} \left (2 \left (35-4 \sqrt {35}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {2}{5} \left (2-\sqrt {35}\right )-x^2} \, dx,x,\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )\\ &=-\frac {4}{7 \sqrt {1+2 x}}+\frac {1}{7} \sqrt {\frac {2}{217} \left (-178+35 \sqrt {35}\right )} \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-2 \sqrt {1+2 x}\right )\right )-\frac {1}{7} \sqrt {\frac {2}{217} \left (-178+35 \sqrt {35}\right )} \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )\right )-\frac {1}{7} \sqrt {\frac {89}{217}+\frac {5 \sqrt {35}}{62}} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{7} \sqrt {\frac {89}{217}+\frac {5 \sqrt {35}}{62}} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )\\ \end {align*}
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Mathematica [C] time = 0.43, size = 122, normalized size = 0.48 \begin {gather*} \frac {2 \left (-\frac {2170}{\sqrt {2 x+1}}+\sqrt {10-5 i \sqrt {31}} \left (124+27 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2-i \sqrt {31}}}\right )+\sqrt {10+5 i \sqrt {31}} \left (124-27 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2+i \sqrt {31}}}\right )\right )}{7595} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.65, size = 120, normalized size = 0.47 \begin {gather*} -\frac {4}{7 \sqrt {2 x+1}}-\frac {2}{7} \sqrt {\frac {1}{217} \left (-178-19 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {-\frac {2}{7}-\frac {i \sqrt {31}}{7}} \sqrt {2 x+1}\right )-\frac {2}{7} \sqrt {\frac {1}{217} \left (-178+19 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {-\frac {2}{7}+\frac {i \sqrt {31}}{7}} \sqrt {2 x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 500, normalized size = 1.98 \begin {gather*} \frac {2356 \cdot 42875^{\frac {1}{4}} \sqrt {217} \sqrt {35} {\left (2 \, x + 1\right )} \sqrt {-12460 \, \sqrt {35} + 85750} \arctan \left (\frac {1}{3644188279375} \cdot 42875^{\frac {3}{4}} \sqrt {217} \sqrt {31} \sqrt {42875^{\frac {1}{4}} \sqrt {217} {\left (\sqrt {35} \sqrt {31} + 4 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-12460 \, \sqrt {35} + 85750} + 1443050 \, x + 144305 \, \sqrt {35} + 721525} {\left (4 \, \sqrt {35} \sqrt {19} + 35 \, \sqrt {19}\right )} \sqrt {-12460 \, \sqrt {35} + 85750} - \frac {1}{176773625} \cdot 42875^{\frac {3}{4}} \sqrt {217} \sqrt {2 \, x + 1} {\left (4 \, \sqrt {35} + 35\right )} \sqrt {-12460 \, \sqrt {35} + 85750} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) + 2356 \cdot 42875^{\frac {1}{4}} \sqrt {217} \sqrt {35} {\left (2 \, x + 1\right )} \sqrt {-12460 \, \sqrt {35} + 85750} \arctan \left (\frac {1}{255093179556250} \cdot 42875^{\frac {3}{4}} \sqrt {217} \sqrt {-151900 \cdot 42875^{\frac {1}{4}} \sqrt {217} {\left (\sqrt {35} \sqrt {31} + 4 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-12460 \, \sqrt {35} + 85750} + 219199295000 \, x + 21919929500 \, \sqrt {35} + 109599647500} {\left (4 \, \sqrt {35} \sqrt {19} + 35 \, \sqrt {19}\right )} \sqrt {-12460 \, \sqrt {35} + 85750} - \frac {1}{176773625} \cdot 42875^{\frac {3}{4}} \sqrt {217} \sqrt {2 \, x + 1} {\left (4 \, \sqrt {35} + 35\right )} \sqrt {-12460 \, \sqrt {35} + 85750} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + 42875^{\frac {1}{4}} \sqrt {217} {\left (178 \, \sqrt {35} \sqrt {31} {\left (2 \, x + 1\right )} + 1225 \, \sqrt {31} {\left (2 \, x + 1\right )}\right )} \sqrt {-12460 \, \sqrt {35} + 85750} \log \left (\frac {151900}{19} \cdot 42875^{\frac {1}{4}} \sqrt {217} {\left (\sqrt {35} \sqrt {31} + 4 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-12460 \, \sqrt {35} + 85750} + 11536805000 \, x + 1153680500 \, \sqrt {35} + 5768402500\right ) - 42875^{\frac {1}{4}} \sqrt {217} {\left (178 \, \sqrt {35} \sqrt {31} {\left (2 \, x + 1\right )} + 1225 \, \sqrt {31} {\left (2 \, x + 1\right )}\right )} \sqrt {-12460 \, \sqrt {35} + 85750} \log \left (-\frac {151900}{19} \cdot 42875^{\frac {1}{4}} \sqrt {217} {\left (\sqrt {35} \sqrt {31} + 4 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-12460 \, \sqrt {35} + 85750} + 11536805000 \, x + 1153680500 \, \sqrt {35} + 5768402500\right ) - 1252567400 \, \sqrt {2 \, x + 1}}{2191992950 \, {\left (2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.82, size = 594, normalized size = 2.35
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.33, size = 616, normalized size = 2.43 \begin {gather*} -\frac {10 \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{217 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {27 \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}\, \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{1519 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {16 \sqrt {5}\, \sqrt {7}\, \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{49 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {10 \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{217 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {27 \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}\, \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{1519 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {16 \sqrt {5}\, \sqrt {7}\, \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{49 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{217}-\frac {27 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{3038}+\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{217}+\frac {27 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{3038}-\frac {4}{7 \sqrt {2 x +1}} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )} {\left (2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 182, normalized size = 0.72 \begin {gather*} -\frac {4}{7\,\sqrt {2\,x+1}}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {178-\sqrt {31}\,19{}\mathrm {i}}\,\sqrt {2\,x+1}\,2432{}\mathrm {i}}{2100875\,\left (\frac {65664}{300125}+\frac {\sqrt {31}\,9728{}\mathrm {i}}{300125}\right )}-\frac {4864\,\sqrt {31}\,\sqrt {217}\,\sqrt {178-\sqrt {31}\,19{}\mathrm {i}}\,\sqrt {2\,x+1}}{65127125\,\left (\frac {65664}{300125}+\frac {\sqrt {31}\,9728{}\mathrm {i}}{300125}\right )}\right )\,\sqrt {178-\sqrt {31}\,19{}\mathrm {i}}\,2{}\mathrm {i}}{1519}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {178+\sqrt {31}\,19{}\mathrm {i}}\,\sqrt {2\,x+1}\,2432{}\mathrm {i}}{2100875\,\left (-\frac {65664}{300125}+\frac {\sqrt {31}\,9728{}\mathrm {i}}{300125}\right )}+\frac {4864\,\sqrt {31}\,\sqrt {217}\,\sqrt {178+\sqrt {31}\,19{}\mathrm {i}}\,\sqrt {2\,x+1}}{65127125\,\left (-\frac {65664}{300125}+\frac {\sqrt {31}\,9728{}\mathrm {i}}{300125}\right )}\right )\,\sqrt {178+\sqrt {31}\,19{}\mathrm {i}}\,2{}\mathrm {i}}{1519} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (2 x + 1\right )^{\frac {3}{2}} \left (5 x^{2} + 3 x + 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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