Optimal. Leaf size=300 \[ -\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}-\frac {3}{961} \sqrt {\frac {1}{310} \left (15082+2705 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {3}{961} \sqrt {\frac {1}{310} \left (15082+2705 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}-\frac {3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922} \]
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Rubi [A]
time = 0.30, 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 = {752, 834, 840,
1183, 648, 632, 210, 642} \begin {gather*} -\frac {3}{961} \sqrt {\frac {1}{310} \left (15082+2705 \sqrt {35}\right )} \text {ArcTan}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {3}{961} \sqrt {\frac {1}{310} \left (15082+2705 \sqrt {35}\right )} \text {ArcTan}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {(5-4 x) (2 x+1)^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}+\frac {3 (78 x+11) \sqrt {2 x+1}}{1922 \left (5 x^2+3 x+2\right )}+\frac {3 \sqrt {\frac {1}{310} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922}-\frac {3 \sqrt {\frac {1}{310} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 752
Rule 834
Rule 840
Rule 1183
Rubi steps
\begin {align*} \int \frac {(1+2 x)^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx &=-\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {1}{62} \int \frac {\sqrt {1+2 x} (27+12 x)}{\left (2+3 x+5 x^2\right )^2} \, dx\\ &=-\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {201+234 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{1922}\\ &=-\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {1}{961} \text {Subst}\left (\int \frac {168+234 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=-\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {168 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (168-234 \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 )}{1922 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {168 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (168-234 \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 )}{1922 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=-\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\left (3 \left (39+4 \sqrt {35}\right )\right ) \text {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 )}{9610}+\frac {\left (3 \left (39+4 \sqrt {35}\right )\right ) \text {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 )}{9610}+\frac {\left (3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )}\right ) \text {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 )}{1922}-\frac {\left (3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )}\right ) \text {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 )}{1922}\\ &=-\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}-\frac {3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}-\frac {\left (3 \left (39+4 \sqrt {35}\right )\right ) \text {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 )}{4805}-\frac {\left (3 \left (39+4 \sqrt {35}\right )\right ) \text {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 )}{4805}\\ &=-\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}-\frac {3}{961} \sqrt {\frac {7541}{155}+\frac {541 \sqrt {35}}{62}} \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 {3}{961} \sqrt {\frac {7541}{155}+\frac {541 \sqrt {35}}{62}} \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 {3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}-\frac {3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.36, size = 143, normalized size = 0.48 \begin {gather*} \frac {\frac {155 \sqrt {1+2 x} \left (-89+381 x+1115 x^2+1170 x^3\right )}{2 \left (2+3 x+5 x^2\right )^2}+3 \sqrt {155 \left (15082-961 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+3 \sqrt {155 \left (15082+961 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{148955} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(434\) vs.
\(2(210)=420\).
time = 1.82, size = 435, normalized size = 1.45 Too large to display
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 617 vs.
\(2 (213) = 426\).
time = 3.34, size = 617, normalized size = 2.06 \begin {gather*} \frac {357492 \cdot 256095875^{\frac {1}{4}} \sqrt {155} \sqrt {35} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} \arctan \left (\frac {1}{694138872776299934375} \cdot 256095875^{\frac {3}{4}} \sqrt {2705} \sqrt {217} \sqrt {155} \sqrt {256095875^{\frac {1}{4}} \sqrt {155} {\left (39 \, \sqrt {35} \sqrt {31} - 140 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} + 28204629250 \, x + 2820462925 \, \sqrt {35} + 14102314625} \sqrt {81593620 \, \sqrt {35} + 512191750} {\left (4 \, \sqrt {35} - 39\right )} - \frac {1}{7629352212125} \cdot 256095875^{\frac {3}{4}} \sqrt {155} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} {\left (4 \, \sqrt {35} - 39\right )} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + 357492 \cdot 256095875^{\frac {1}{4}} \sqrt {155} \sqrt {35} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} \arctan \left (\frac {1}{2082416618328899803125} \cdot 256095875^{\frac {3}{4}} \sqrt {217} \sqrt {155} \sqrt {-24345 \cdot 256095875^{\frac {1}{4}} \sqrt {155} {\left (39 \, \sqrt {35} \sqrt {31} - 140 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} + 686641699091250 \, x + 68664169909125 \, \sqrt {35} + 343320849545625} \sqrt {81593620 \, \sqrt {35} + 512191750} {\left (4 \, \sqrt {35} - 39\right )} - \frac {1}{7629352212125} \cdot 256095875^{\frac {3}{4}} \sqrt {155} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} {\left (4 \, \sqrt {35} - 39\right )} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) + 3 \cdot 256095875^{\frac {1}{4}} \sqrt {155} {\left (15082 \, \sqrt {35} \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} - 94675 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} \log \left (\frac {24345}{217} \cdot 256095875^{\frac {1}{4}} \sqrt {155} {\left (39 \, \sqrt {35} \sqrt {31} - 140 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} + 3164247461250 \, x + 316424746125 \, \sqrt {35} + 1582123730625\right ) - 3 \cdot 256095875^{\frac {1}{4}} \sqrt {155} {\left (15082 \, \sqrt {35} \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} - 94675 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} \log \left (-\frac {24345}{217} \cdot 256095875^{\frac {1}{4}} \sqrt {155} {\left (39 \, \sqrt {35} \sqrt {31} - 140 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} + 3164247461250 \, x + 316424746125 \, \sqrt {35} + 1582123730625\right ) + 874343506750 \, {\left (1170 \, x^{3} + 1115 \, x^{2} + 381 \, x - 89\right )} \sqrt {2 \, x + 1}}{1680488219973500 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 731 vs.
\(2 (252) = 504\).
time = 167.13, size = 731, normalized size = 2.44 \begin {gather*} - \frac {5441600 \left (2 x + 1\right )^{\frac {7}{2}}}{- 602739200 x + 134540000 \left (2 x + 1\right )^{4} - 215264000 \left (2 x + 1\right )^{3} + 462817600 \left (2 x + 1\right )^{2} - 37671200} - \frac {35481600 \left (2 x + 1\right )^{\frac {7}{2}}}{- 4219174400 x + 941780000 \left (2 x + 1\right )^{4} - 1506848000 \left (2 x + 1\right )^{3} + 3239723200 \left (2 x + 1\right )^{2} - 263698400} + \frac {6152960 \left (2 x + 1\right )^{\frac {5}{2}}}{- 602739200 x + 134540000 \left (2 x + 1\right )^{4} - 215264000 \left (2 x + 1\right )^{3} + 462817600 \left (2 x + 1\right )^{2} - 37671200} - \frac {18807040 \left (2 x + 1\right )^{\frac {5}{2}}}{- 4219174400 x + 941780000 \left (2 x + 1\right )^{4} - 1506848000 \left (2 x + 1\right )^{3} + 3239723200 \left (2 x + 1\right )^{2} - 263698400} - \frac {14335424 \left (2 x + 1\right )^{\frac {3}{2}}}{- 602739200 x + 134540000 \left (2 x + 1\right )^{4} - 215264000 \left (2 x + 1\right )^{3} + 462817600 \left (2 x + 1\right )^{2} - 37671200} - \frac {27474944 \left (2 x + 1\right )^{\frac {3}{2}}}{- 4219174400 x + 941780000 \left (2 x + 1\right )^{4} - 1506848000 \left (2 x + 1\right )^{3} + 3239723200 \left (2 x + 1\right )^{2} - 263698400} + \frac {320 \left (2 x + 1\right )^{\frac {3}{2}}}{- 4960 x + 3100 \left (2 x + 1\right )^{2} + 1860} + \frac {2560 \left (2 x + 1\right )^{\frac {3}{2}}}{- 173600 x + 108500 \left (2 x + 1\right )^{2} + 65100} + \frac {4630528 \sqrt {2 x + 1}}{- 602739200 x + 134540000 \left (2 x + 1\right )^{4} - 215264000 \left (2 x + 1\right )^{3} + 462817600 \left (2 x + 1\right )^{2} - 37671200} - \frac {108066560 \sqrt {2 x + 1}}{- 4219174400 x + 941780000 \left (2 x + 1\right )^{4} - 1506848000 \left (2 x + 1\right )^{3} + 3239723200 \left (2 x + 1\right )^{2} - 263698400} - \frac {128 \sqrt {2 x + 1}}{- 4960 x + 3100 \left (2 x + 1\right )^{2} + 1860} + \frac {6912 \sqrt {2 x + 1}}{- 173600 x + 108500 \left (2 x + 1\right )^{2} + 65100} + 64 \operatorname {RootSum} {\left (75465931487403231630327808 t^{4} + 9053854476152406016 t^{2} + 333142578125, \left ( t \mapsto t \log {\left (\frac {21632117045402271744 t^{3}}{158378125} + \frac {10865340674816 t}{1108646875} + \sqrt {2 x + 1} \right )} \right )\right )} - \frac {2176 \operatorname {RootSum} {\left (75465931487403231630327808 t^{4} + 9053854476152406016 t^{2} + 333142578125, \left ( t \mapsto t \log {\left (\frac {21632117045402271744 t^{3}}{158378125} + \frac {10865340674816 t}{1108646875} + \sqrt {2 x + 1} \right )} \right )\right )}}{25} - \frac {896 \operatorname {RootSum} {\left (3697830642882758349886062592 t^{4} + 2111968303753265086464 t^{2} + 705698730253125, \left ( t \mapsto t \log {\left (- \frac {3459438283411209322496 t^{3}}{1377792122625} + \frac {251494140770688 t}{357205365125} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} - \frac {384 \operatorname {RootSum} {\left (19950060344639488 t^{4} + 498437272576 t^{2} + 10878125, \left ( t \mapsto t \log {\left (- \frac {11049511452672 t^{3}}{2205125} + \frac {307918256 t}{2205125} + \sqrt {2 x + 1} \right )} \right )\right )}}{25} - \frac {128 \operatorname {RootSum} {\left (75465931487403231630327808 t^{4} + 9053854476152406016 t^{2} + 333142578125, \left ( t \mapsto t \log {\left (\frac {21632117045402271744 t^{3}}{158378125} + \frac {10865340674816 t}{1108646875} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} + \frac {64 \operatorname {RootSum} {\left (407144088666112 t^{4} + 3325152256 t^{2} + 11045, \left ( t \mapsto t \log {\left (\frac {33312534528 t^{3}}{235} + \frac {166784 t}{235} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} + \frac {128 \operatorname {RootSum} {\left (19950060344639488 t^{4} + 498437272576 t^{2} + 10878125, \left ( t \mapsto t \log {\left (- \frac {11049511452672 t^{3}}{2205125} + \frac {307918256 t}{2205125} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} + \frac {2688 \operatorname {RootSum} {\left (3697830642882758349886062592 t^{4} + 2111968303753265086464 t^{2} + 705698730253125, \left ( t \mapsto t \log {\left (- \frac {3459438283411209322496 t^{3}}{1377792122625} + \frac {251494140770688 t}{357205365125} + \sqrt {2 x + 1} \right )} \right )\right )}}{25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 642 vs.
\(2 (213) = 426\).
time = 1.85, size = 642, normalized size = 2.14 \begin {gather*} \frac {3}{71528191000} \, \sqrt {31} {\left (8190 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 39 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 78 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 16380 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 137200 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 274400 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450}\right )} \arctan \left (\frac {5 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (\left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + \sqrt {2 \, x + 1}\right )}}{7 \, \sqrt {-\frac {1}{35} \, \sqrt {35} + \frac {1}{2}}}\right ) + \frac {3}{71528191000} \, \sqrt {31} {\left (8190 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 39 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 78 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 16380 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 137200 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 274400 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450}\right )} \arctan \left (-\frac {5 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (\left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} - \sqrt {2 \, x + 1}\right )}}{7 \, \sqrt {-\frac {1}{35} \, \sqrt {35} + \frac {1}{2}}}\right ) + \frac {3}{143056382000} \, \sqrt {31} {\left (39 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 8190 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 16380 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 78 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 137200 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} - 274400 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450}\right )} \log \left (2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {2 \, x + 1} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + 2 \, x + \sqrt {\frac {7}{5}} + 1\right ) - \frac {3}{143056382000} \, \sqrt {31} {\left (39 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 8190 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 16380 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 78 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 137200 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} - 274400 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450}\right )} \log \left (-2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {2 \, x + 1} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + 2 \, x + \sqrt {\frac {7}{5}} + 1\right ) + \frac {2 \, {\left (585 \, {\left (2 \, x + 1\right )}^{\frac {7}{2}} - 640 \, {\left (2 \, x + 1\right )}^{\frac {5}{2}} + 287 \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} - 588 \, \sqrt {2 \, x + 1}\right )}}{961 \, {\left (5 \, {\left (2 \, x + 1\right )}^{2} - 8 \, x + 3\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.05, size = 245, normalized size = 0.82 \begin {gather*} \frac {\frac {1176\,\sqrt {2\,x+1}}{24025}-\frac {574\,{\left (2\,x+1\right )}^{3/2}}{24025}+\frac {256\,{\left (2\,x+1\right )}^{5/2}}{4805}-\frac {234\,{\left (2\,x+1\right )}^{7/2}}{4805}}{\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 {-15082-\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}\,432{}\mathrm {i}}{2886003125\,\left (-\frac {142128}{577200625}+\frac {\sqrt {31}\,12096{}\mathrm {i}}{577200625}\right )}-\frac {864\,\sqrt {31}\,\sqrt {155}\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}}{89466096875\,\left (-\frac {142128}{577200625}+\frac {\sqrt {31}\,12096{}\mathrm {i}}{577200625}\right )}\right )\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,3{}\mathrm {i}}{148955}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}\,432{}\mathrm {i}}{2886003125\,\left (\frac {142128}{577200625}+\frac {\sqrt {31}\,12096{}\mathrm {i}}{577200625}\right )}+\frac {864\,\sqrt {31}\,\sqrt {155}\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}}{89466096875\,\left (\frac {142128}{577200625}+\frac {\sqrt {31}\,12096{}\mathrm {i}}{577200625}\right )}\right )\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,3{}\mathrm {i}}{148955} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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