Optimal. Leaf size=300 \[ -\frac {(5-4 x) (2 x+1)^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}-\frac {(957-592 x) \sqrt {2 x+1}}{9610 \left (5 x^2+3 x+2\right )}-\frac {\sqrt {\frac {1}{310} \left (1806875 \sqrt {35}-9651062\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{9610}+\frac {\sqrt {\frac {1}{310} \left (1806875 \sqrt {35}-9651062\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{9610}-\frac {\sqrt {\frac {1}{310} \left (9651062+1806875 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{4805}+\frac {\sqrt {\frac {1}{310} \left (9651062+1806875 \sqrt {35}\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{4805} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.44, antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {738, 818, 826, 1169, 634, 618, 204, 628} \[ -\frac {(5-4 x) (2 x+1)^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}-\frac {(957-592 x) \sqrt {2 x+1}}{9610 \left (5 x^2+3 x+2\right )}-\frac {\sqrt {\frac {1}{310} \left (1806875 \sqrt {35}-9651062\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{9610}+\frac {\sqrt {\frac {1}{310} \left (1806875 \sqrt {35}-9651062\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{9610}-\frac {\sqrt {\frac {1}{310} \left (9651062+1806875 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{4805}+\frac {\sqrt {\frac {1}{310} \left (9651062+1806875 \sqrt {35}\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{4805} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 618
Rule 628
Rule 634
Rule 738
Rule 818
Rule 826
Rule 1169
Rubi steps
\begin {align*} \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^3} \, dx &=-\frac {(5-4 x) (1+2 x)^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {1}{62} \int \frac {(1+2 x)^{3/2} (37+4 x)}{\left (2+3 x+5 x^2\right )^2} \, dx\\ &=-\frac {(5-4 x) (1+2 x)^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(957-592 x) \sqrt {1+2 x}}{9610 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-1797-1088 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{9610}\\ &=-\frac {(5-4 x) (1+2 x)^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(957-592 x) \sqrt {1+2 x}}{9610 \left (2+3 x+5 x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-2506-1088 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{4805}\\ &=-\frac {(5-4 x) (1+2 x)^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(957-592 x) \sqrt {1+2 x}}{9610 \left (2+3 x+5 x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-2506 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (-2506+1088 \sqrt {\frac {7}{5}}\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 )}{9610 \sqrt {14 \left (2+\sqrt {35}\right )}}-\frac {\operatorname {Subst}\left (\int \frac {-2506 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (-2506+1088 \sqrt {\frac {7}{5}}\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 )}{9610 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=-\frac {(5-4 x) (1+2 x)^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(957-592 x) \sqrt {1+2 x}}{9610 \left (2+3 x+5 x^2\right )}+\frac {\sqrt {1417371+194752 \sqrt {35}} \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 )}{48050}+\frac {\sqrt {1417371+194752 \sqrt {35}} \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 )}{48050}-\frac {\sqrt {\frac {1}{310} \left (-9651062+1806875 \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 )}{9610}+\frac {\sqrt {\frac {1}{310} \left (-9651062+1806875 \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 )}{9610}\\ &=-\frac {(5-4 x) (1+2 x)^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(957-592 x) \sqrt {1+2 x}}{9610 \left (2+3 x+5 x^2\right )}-\frac {\sqrt {\frac {1}{310} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{9610}+\frac {\sqrt {\frac {1}{310} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{9610}-\frac {\sqrt {1417371+194752 \sqrt {35}} \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 )}{24025}-\frac {\sqrt {1417371+194752 \sqrt {35}} \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 )}{24025}\\ &=-\frac {(5-4 x) (1+2 x)^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(957-592 x) \sqrt {1+2 x}}{9610 \left (2+3 x+5 x^2\right )}-\frac {\sqrt {\frac {1}{310} \left (9651062+1806875 \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 )}{4805}+\frac {\sqrt {\frac {1}{310} \left (9651062+1806875 \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 )}{4805}-\frac {\sqrt {\frac {1}{310} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{9610}+\frac {\sqrt {\frac {1}{310} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{9610}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.67, size = 223, normalized size = 0.74 \[ \frac {-\frac {5 (400 x+89) (2 x+1)^{9/2}}{5 x^2+3 x+2}+\frac {217 (20 x+37) (2 x+1)^{9/2}}{\left (5 x^2+3 x+2\right )^2}+800 (2 x+1)^{7/2}+196 (2 x+1)^{5/2}-2352 (2 x+1)^{3/2}-\frac {35084}{5} \sqrt {2 x+1}+\frac {98 \left (\sqrt {2-i \sqrt {31}} \left (5549-902 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2-i \sqrt {31}}}\right )+\sqrt {2+i \sqrt {31}} \left (5549+902 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2+i \sqrt {31}}}\right )\right )}{155 \sqrt {5}}}{94178} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.15, size = 652, normalized size = 2.17 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.43, size = 642, normalized size = 2.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.15, size = 662, normalized size = 2.21 \[ -\frac {7353 \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 )}{297910 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {451 \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 )}{148955 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {358 \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 )}{4805 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {7353 \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 )}{297910 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {451 \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 )}{148955 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {358 \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 )}{4805 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {7353 \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 )}{2979100}+\frac {451 \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 )}{297910}+\frac {7353 \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 )}{2979100}-\frac {451 \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 )}{297910}+\frac {\frac {1088 \left (2 x +1\right )^{\frac {7}{2}}}{961}-\frac {23578 \left (2 x +1\right )^{\frac {5}{2}}}{4805}+\frac {14168 \left (2 x +1\right )^{\frac {3}{2}}}{4805}-\frac {17542 \sqrt {2 x +1}}{4805}}{\left (-8 x +5 \left (2 x +1\right )^{2}+3\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (2 \, x + 1\right )}^{\frac {7}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.06, size = 245, normalized size = 0.82 \[ \frac {\frac {17542\,\sqrt {2\,x+1}}{120125}-\frac {14168\,{\left (2\,x+1\right )}^{3/2}}{120125}+\frac {23578\,{\left (2\,x+1\right )}^{5/2}}{120125}-\frac {1088\,{\left (2\,x+1\right )}^{7/2}}{24025}}{\frac {112\,x}{25}-\frac {86\,{\left (2\,x+1\right )}^2}{25}+\frac {8\,{\left (2\,x+1\right )}^3}{5}-{\left (2\,x+1\right )}^4+\frac {7}{25}}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}\,13744{}\mathrm {i}}{360750390625\,\left (\frac {86779616}{72150078125}+\frac {\sqrt {31}\,17221232{}\mathrm {i}}{72150078125}\right )}+\frac {27488\,\sqrt {31}\,\sqrt {155}\,\sqrt {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}}{11183262109375\,\left (\frac {86779616}{72150078125}+\frac {\sqrt {31}\,17221232{}\mathrm {i}}{72150078125}\right )}\right )\,\sqrt {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,1{}\mathrm {i}}{744775}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}\,13744{}\mathrm {i}}{360750390625\,\left (-\frac {86779616}{72150078125}+\frac {\sqrt {31}\,17221232{}\mathrm {i}}{72150078125}\right )}-\frac {27488\,\sqrt {31}\,\sqrt {155}\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}}{11183262109375\,\left (-\frac {86779616}{72150078125}+\frac {\sqrt {31}\,17221232{}\mathrm {i}}{72150078125}\right )}\right )\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,1{}\mathrm {i}}{744775} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________