Integrand size = 15, antiderivative size = 121 \[ \int \left (3 i x+4 x^2\right )^{7/2} \, dx=\frac {25515 (3 i+8 x) \sqrt {3 i x+4 x^2}}{4194304}+\frac {945 (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}}{131072}+\frac {21 (3 i+8 x) \left (3 i x+4 x^2\right )^{5/2}}{2048}+\frac {1}{64} (3 i+8 x) \left (3 i x+4 x^2\right )^{7/2}+\frac {229635 i \arcsin \left (1-\frac {8 i x}{3}\right )}{16777216} \]
945/131072*(3*I+8*x)*(3*I*x+4*x^2)^(3/2)+21/2048*(3*I+8*x)*(3*I*x+4*x^2)^( 5/2)+1/64*(3*I+8*x)*(3*I*x+4*x^2)^(7/2)-229635/16777216*I*arcsin(-1+8/3*I* x)+25515/4194304*(3*I+8*x)*(3*I*x+4*x^2)^(1/2)
Time = 0.20 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.97 \[ \int \left (3 i x+4 x^2\right )^{7/2} \, dx=\frac {\sqrt {x} \sqrt {3 i+4 x} \left (2 \sqrt {x} \sqrt {3 i+4 x} \left (76545 i-68040 x-72576 i x^2+82944 x^3-25067520 i x^4-79429632 x^5+88080384 i x^6+33554432 x^7\right )-229635 \log \left (-2 \sqrt {x}+\sqrt {3 i+4 x}\right )\right )}{8388608 \sqrt {x (3 i+4 x)}} \]
(Sqrt[x]*Sqrt[3*I + 4*x]*(2*Sqrt[x]*Sqrt[3*I + 4*x]*(76545*I - 68040*x - ( 72576*I)*x^2 + 82944*x^3 - (25067520*I)*x^4 - 79429632*x^5 + (88080384*I)* x^6 + 33554432*x^7) - 229635*Log[-2*Sqrt[x] + Sqrt[3*I + 4*x]]))/(8388608* Sqrt[x*(3*I + 4*x)])
Time = 0.23 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1087, 1087, 1087, 1087, 1090, 222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (4 x^2+3 i x\right )^{7/2} \, dx\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {63}{128} \int \left (4 x^2+3 i x\right )^{5/2}dx+\frac {1}{64} (8 x+3 i) \left (4 x^2+3 i x\right )^{7/2}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {63}{128} \left (\frac {15}{32} \int \left (4 x^2+3 i x\right )^{3/2}dx+\frac {1}{48} (8 x+3 i) \left (4 x^2+3 i x\right )^{5/2}\right )+\frac {1}{64} (8 x+3 i) \left (4 x^2+3 i x\right )^{7/2}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {63}{128} \left (\frac {15}{32} \left (\frac {27}{64} \int \sqrt {4 x^2+3 i x}dx+\frac {1}{32} (8 x+3 i) \left (4 x^2+3 i x\right )^{3/2}\right )+\frac {1}{48} (8 x+3 i) \left (4 x^2+3 i x\right )^{5/2}\right )+\frac {1}{64} (8 x+3 i) \left (4 x^2+3 i x\right )^{7/2}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {63}{128} \left (\frac {15}{32} \left (\frac {27}{64} \left (\frac {9}{32} \int \frac {1}{\sqrt {4 x^2+3 i x}}dx+\frac {1}{16} \sqrt {4 x^2+3 i x} (8 x+3 i)\right )+\frac {1}{32} (8 x+3 i) \left (4 x^2+3 i x\right )^{3/2}\right )+\frac {1}{48} (8 x+3 i) \left (4 x^2+3 i x\right )^{5/2}\right )+\frac {1}{64} (8 x+3 i) \left (4 x^2+3 i x\right )^{7/2}\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {63}{128} \left (\frac {15}{32} \left (\frac {27}{64} \left (\frac {3}{64} \int \frac {1}{\sqrt {\frac {1}{9} (8 x+3 i)^2+1}}d(8 x+3 i)+\frac {1}{16} \sqrt {4 x^2+3 i x} (8 x+3 i)\right )+\frac {1}{32} (8 x+3 i) \left (4 x^2+3 i x\right )^{3/2}\right )+\frac {1}{48} (8 x+3 i) \left (4 x^2+3 i x\right )^{5/2}\right )+\frac {1}{64} (8 x+3 i) \left (4 x^2+3 i x\right )^{7/2}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {63}{128} \left (\frac {15}{32} \left (\frac {27}{64} \left (\frac {9}{64} \text {arcsinh}\left (\frac {1}{3} (8 x+3 i)\right )+\frac {1}{16} \sqrt {4 x^2+3 i x} (8 x+3 i)\right )+\frac {1}{32} (8 x+3 i) \left (4 x^2+3 i x\right )^{3/2}\right )+\frac {1}{48} (8 x+3 i) \left (4 x^2+3 i x\right )^{5/2}\right )+\frac {1}{64} (8 x+3 i) \left (4 x^2+3 i x\right )^{7/2}\) |
((3*I + 8*x)*((3*I)*x + 4*x^2)^(7/2))/64 + (63*(((3*I + 8*x)*((3*I)*x + 4* x^2)^(5/2))/48 + (15*(((3*I + 8*x)*((3*I)*x + 4*x^2)^(3/2))/32 + (27*(((3* I + 8*x)*Sqrt[(3*I)*x + 4*x^2])/16 + (9*ArcSinh[(3*I + 8*x)/3])/64))/64))/ 32))/128
3.1.2.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
Time = 2.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.57
method | result | size |
risch | \(\frac {\left (33554432 x^{7}+88080384 i x^{6}-79429632 x^{5}-25067520 i x^{4}+82944 x^{3}-72576 i x^{2}-68040 x +76545 i\right ) \left (3 i+4 x \right ) x}{4194304 \sqrt {x \left (3 i+4 x \right )}}+\frac {229635 \,\operatorname {arcsinh}\left (i+\frac {8 x}{3}\right )}{16777216}\) | \(69\) |
default | \(\frac {\left (3 i+8 x \right ) \left (4 x^{2}+3 i x \right )^{\frac {7}{2}}}{64}+\frac {21 \left (3 i+8 x \right ) \left (4 x^{2}+3 i x \right )^{\frac {5}{2}}}{2048}+\frac {945 \left (3 i+8 x \right ) \left (4 x^{2}+3 i x \right )^{\frac {3}{2}}}{131072}+\frac {25515 \left (3 i+8 x \right ) \sqrt {4 x^{2}+3 i x}}{4194304}+\frac {229635 \,\operatorname {arcsinh}\left (i+\frac {8 x}{3}\right )}{16777216}\) | \(91\) |
trager | \(\left (21 i x^{6}+8 x^{7}-\frac {765}{128} i x^{4}-\frac {303}{16} x^{5}-\frac {567}{32768} i x^{2}+\frac {81}{4096} x^{3}+\frac {76545}{4194304} i-\frac {8505}{524288} x \right ) \sqrt {4 x^{2}+3 i x}+\frac {229635 \ln \left (440 x +144+165 i-192 i \sqrt {4 x^{2}+3 i x}-384 i x +220 \sqrt {4 x^{2}+3 i x}\right )}{16777216}\) | \(97\) |
1/4194304*(76545*I-68040*x-72576*I*x^2+82944*x^3-25067520*I*x^4-79429632*x ^5+88080384*I*x^6+33554432*x^7)*(3*I+4*x)*x/(x*(3*I+4*x))^(1/2)+229635/167 77216*arcsinh(I+8/3*x)
Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.57 \[ \int \left (3 i x+4 x^2\right )^{7/2} \, dx=\frac {1}{4194304} \, {\left (33554432 \, x^{7} + 88080384 i \, x^{6} - 79429632 \, x^{5} - 25067520 i \, x^{4} + 82944 \, x^{3} - 72576 i \, x^{2} - 68040 \, x + 76545 i\right )} \sqrt {4 \, x^{2} + 3 i \, x} - \frac {229635}{16777216} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 3 i \, x} - \frac {3}{4} i\right ) - \frac {1165671}{268435456} \]
1/4194304*(33554432*x^7 + 88080384*I*x^6 - 79429632*x^5 - 25067520*I*x^4 + 82944*x^3 - 72576*I*x^2 - 68040*x + 76545*I)*sqrt(4*x^2 + 3*I*x) - 229635 /16777216*log(-2*x + sqrt(4*x^2 + 3*I*x) - 3/4*I) - 1165671/268435456
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (107) = 214\).
Time = 1.24 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.22 \[ \int \left (3 i x+4 x^2\right )^{7/2} \, dx=- 108 \sqrt {4 x^{2} + 3 i x} \left (\frac {x^{5}}{6} + \frac {i x^{4}}{80} + \frac {27 x^{3}}{2560} - \frac {189 i x^{2}}{20480} - \frac {567 x}{65536} + \frac {5103 i}{524288}\right ) + 64 \sqrt {4 x^{2} + 3 i x} \left (\frac {x^{7}}{8} + \frac {3 i x^{6}}{448} + \frac {39 x^{5}}{7168} - \frac {1287 i x^{4}}{286720} - \frac {34749 x^{3}}{9175040} + \frac {34749 i x^{2}}{10485760} + \frac {104247 x}{33554432} - \frac {938223 i}{268435456}\right ) - 27 i \left (\sqrt {4 x^{2} + 3 i x} \left (\frac {x^{4}}{5} + \frac {3 i x^{3}}{160} + \frac {21 x^{2}}{1280} - \frac {63 i x}{4096} - \frac {567}{32768}\right ) + \frac {1701 i \operatorname {asinh}{\left (\frac {8 x}{3} + i \right )}}{131072}\right ) + 144 i \left (\sqrt {4 x^{2} + 3 i x} \left (\frac {x^{6}}{7} + \frac {i x^{5}}{112} + \frac {33 x^{4}}{4480} - \frac {891 i x^{3}}{143360} - \frac {891 x^{2}}{163840} + \frac {2673 i x}{524288} + \frac {24057}{4194304}\right ) - \frac {72171 i \operatorname {asinh}{\left (\frac {8 x}{3} + i \right )}}{16777216}\right ) - \frac {16041645 \operatorname {asinh}{\left (\frac {8 x}{3} + i \right )}}{16777216} \]
-108*sqrt(4*x**2 + 3*I*x)*(x**5/6 + I*x**4/80 + 27*x**3/2560 - 189*I*x**2/ 20480 - 567*x/65536 + 5103*I/524288) + 64*sqrt(4*x**2 + 3*I*x)*(x**7/8 + 3 *I*x**6/448 + 39*x**5/7168 - 1287*I*x**4/286720 - 34749*x**3/9175040 + 347 49*I*x**2/10485760 + 104247*x/33554432 - 938223*I/268435456) - 27*I*(sqrt( 4*x**2 + 3*I*x)*(x**4/5 + 3*I*x**3/160 + 21*x**2/1280 - 63*I*x/4096 - 567/ 32768) + 1701*I*asinh(8*x/3 + I)/131072) + 144*I*(sqrt(4*x**2 + 3*I*x)*(x* *6/7 + I*x**5/112 + 33*x**4/4480 - 891*I*x**3/143360 - 891*x**2/163840 + 2 673*I*x/524288 + 24057/4194304) - 72171*I*asinh(8*x/3 + I)/16777216) - 160 41645*asinh(8*x/3 + I)/16777216
Time = 0.26 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.07 \[ \int \left (3 i x+4 x^2\right )^{7/2} \, dx=\frac {1}{8} \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {7}{2}} x + \frac {3}{64} i \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {7}{2}} + \frac {21}{256} \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {5}{2}} x + \frac {63}{2048} i \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {5}{2}} + \frac {945}{16384} \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {3}{2}} x + \frac {2835}{131072} i \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {3}{2}} + \frac {25515}{524288} \, \sqrt {4 \, x^{2} + 3 i \, x} x + \frac {76545}{4194304} i \, \sqrt {4 \, x^{2} + 3 i \, x} + \frac {229635}{16777216} \, \log \left (8 \, x + 4 \, \sqrt {4 \, x^{2} + 3 i \, x} + 3 i\right ) \]
1/8*(4*x^2 + 3*I*x)^(7/2)*x + 3/64*I*(4*x^2 + 3*I*x)^(7/2) + 21/256*(4*x^2 + 3*I*x)^(5/2)*x + 63/2048*I*(4*x^2 + 3*I*x)^(5/2) + 945/16384*(4*x^2 + 3 *I*x)^(3/2)*x + 2835/131072*I*(4*x^2 + 3*I*x)^(3/2) + 25515/524288*sqrt(4* x^2 + 3*I*x)*x + 76545/4194304*I*sqrt(4*x^2 + 3*I*x) + 229635/16777216*log (8*x + 4*sqrt(4*x^2 + 3*I*x) + 3*I)
Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.16 \[ \int \left (3 i x+4 x^2\right )^{7/2} \, dx=\frac {1}{8388608} \, {\left (8 \, {\left (16 \, {\left (8 \, {\left (32 \, {\left (8 \, {\left (16 \, {\left (8 \, x + 21 i\right )} x - 303\right )} x - 765 i\right )} x + 81\right )} x - 567 i\right )} x - 8505\right )} x + 76545 i\right )} \sqrt {8 \, x^{2} + 2 \, \sqrt {16 \, x^{2} + 9} x} {\left (\frac {3 i \, x}{4 \, x^{2} + \sqrt {16 \, x^{4} + 9 \, x^{2}}} + 1\right )} - \frac {229635}{16777216} \, \log \left (2 \, \sqrt {8 \, x^{2} + 2 \, \sqrt {16 \, x^{2} + 9} x} {\left (\frac {3 i \, x}{4 \, x^{2} + \sqrt {16 \, x^{4} + 9 \, x^{2}}} + 1\right )} - 8 \, x - 3 i\right ) \]
1/8388608*(8*(16*(8*(32*(8*(16*(8*x + 21*I)*x - 303)*x - 765*I)*x + 81)*x - 567*I)*x - 8505)*x + 76545*I)*sqrt(8*x^2 + 2*sqrt(16*x^2 + 9)*x)*(3*I*x/ (4*x^2 + sqrt(16*x^4 + 9*x^2)) + 1) - 229635/16777216*log(2*sqrt(8*x^2 + 2 *sqrt(16*x^2 + 9)*x)*(3*I*x/(4*x^2 + sqrt(16*x^4 + 9*x^2)) + 1) - 8*x - 3* I)
Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.83 \[ \int \left (3 i x+4 x^2\right )^{7/2} \, dx=\frac {229635\,\ln \left (x+\frac {\sqrt {x\,\left (4\,x+3{}\mathrm {i}\right )}}{2}+\frac {3}{8}{}\mathrm {i}\right )}{16777216}+\frac {945\,\left (4\,x+\frac {3}{2}{}\mathrm {i}\right )\,{\left (4\,x^2+x\,3{}\mathrm {i}\right )}^{3/2}}{65536}+\frac {21\,\left (4\,x+\frac {3}{2}{}\mathrm {i}\right )\,{\left (4\,x^2+x\,3{}\mathrm {i}\right )}^{5/2}}{1024}+\frac {\left (4\,x+\frac {3}{2}{}\mathrm {i}\right )\,{\left (4\,x^2+x\,3{}\mathrm {i}\right )}^{7/2}}{32}+\frac {25515\,\left (\frac {x}{2}+\frac {3}{16}{}\mathrm {i}\right )\,\sqrt {4\,x^2+x\,3{}\mathrm {i}}}{262144} \]