3.2 \(\int \left (3 i x+4 x^2\right )^{7/2} \, dx\)

Optimal. Leaf size=121 \[ \frac{1}{64} (8 x+3 i) \left (4 x^2+3 i x\right )^{7/2}+\frac{21 (8 x+3 i) \left (4 x^2+3 i x\right )^{5/2}}{2048}+\frac{945 (8 x+3 i) \left (4 x^2+3 i x\right )^{3/2}}{131072}+\frac{25515 (8 x+3 i) \sqrt{4 x^2+3 i x}}{4194304}+\frac{229635 i \sin ^{-1}\left (1-\frac{8 i x}{3}\right )}{16777216} \]

[Out]

(25515*(3*I + 8*x)*Sqrt[(3*I)*x + 4*x^2])/4194304 + (945*(3*I + 8*x)*((3*I)*x +
4*x^2)^(3/2))/131072 + (21*(3*I + 8*x)*((3*I)*x + 4*x^2)^(5/2))/2048 + ((3*I + 8
*x)*((3*I)*x + 4*x^2)^(7/2))/64 + ((229635*I)/16777216)*ArcSin[1 - ((8*I)/3)*x]

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Rubi [A]  time = 0.072225, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{64} (8 x+3 i) \left (4 x^2+3 i x\right )^{7/2}+\frac{21 (8 x+3 i) \left (4 x^2+3 i x\right )^{5/2}}{2048}+\frac{945 (8 x+3 i) \left (4 x^2+3 i x\right )^{3/2}}{131072}+\frac{25515 (8 x+3 i) \sqrt{4 x^2+3 i x}}{4194304}+\frac{229635 i \sin ^{-1}\left (1-\frac{8 i x}{3}\right )}{16777216} \]

Antiderivative was successfully verified.

[In]  Int[((3*I)*x + 4*x^2)^(7/2),x]

[Out]

(25515*(3*I + 8*x)*Sqrt[(3*I)*x + 4*x^2])/4194304 + (945*(3*I + 8*x)*((3*I)*x +
4*x^2)^(3/2))/131072 + (21*(3*I + 8*x)*((3*I)*x + 4*x^2)^(5/2))/2048 + ((3*I + 8
*x)*((3*I)*x + 4*x^2)^(7/2))/64 + ((229635*I)/16777216)*ArcSin[1 - ((8*I)/3)*x]

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Rubi in Sympy [A]  time = 4.32622, size = 104, normalized size = 0.86 \[ \frac{\left (8 x + 3 i\right ) \left (4 x^{2} + 3 i x\right )^{\frac{7}{2}}}{64} + \frac{21 \left (8 x + 3 i\right ) \left (4 x^{2} + 3 i x\right )^{\frac{5}{2}}}{2048} + \frac{945 \left (8 x + 3 i\right ) \left (4 x^{2} + 3 i x\right )^{\frac{3}{2}}}{131072} + \frac{25515 \left (8 x + 3 i\right ) \sqrt{4 x^{2} + 3 i x}}{4194304} + \frac{229635 \operatorname{asinh}{\left (\frac{8 x}{3} + i \right )}}{16777216} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*I*x+4*x**2)**(7/2),x)

[Out]

(8*x + 3*I)*(4*x**2 + 3*I*x)**(7/2)/64 + 21*(8*x + 3*I)*(4*x**2 + 3*I*x)**(5/2)/
2048 + 945*(8*x + 3*I)*(4*x**2 + 3*I*x)**(3/2)/131072 + 25515*(8*x + 3*I)*sqrt(4
*x**2 + 3*I*x)/4194304 + 229635*asinh(8*x/3 + I)/16777216

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Mathematica [A]  time = 0.116105, size = 117, normalized size = 0.97 \[ \frac{\sqrt{x} \sqrt{4 x+3 i} \left (2 \sqrt{x} \sqrt{4 x+3 i} \left (33554432 x^7+88080384 i x^6-79429632 x^5-25067520 i x^4+82944 x^3-72576 i x^2-68040 x+76545 i\right )+229635 \log \left (2 \sqrt{x}+\sqrt{4 x+3 i}\right )\right )}{8388608 \sqrt{x (4 x+3 i)}} \]

Antiderivative was successfully verified.

[In]  Integrate[((3*I)*x + 4*x^2)^(7/2),x]

[Out]

(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 + 335544
32*x^7) + 229635*Log[2*Sqrt[x] + Sqrt[3*I + 4*x]]))/(8388608*Sqrt[x*(3*I + 4*x)]
)

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Maple [A]  time = 0.029, size = 91, normalized size = 0.8 \[{\frac{3\,i+8\,x}{64} \left ( 3\,ix+4\,{x}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{63\,i+168\,x}{2048} \left ( 3\,ix+4\,{x}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{2835\,i+7560\,x}{131072} \left ( 3\,ix+4\,{x}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{76545\,i+204120\,x}{4194304}\sqrt{3\,ix+4\,{x}^{2}}}+{\frac{229635}{16777216}{\it Arcsinh} \left ({\frac{8\,x}{3}}+i \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*I*x+4*x^2)^(7/2),x)

[Out]

1/64*(3*I+8*x)*(3*I*x+4*x^2)^(7/2)+21/2048*(3*I+8*x)*(3*I*x+4*x^2)^(5/2)+945/131
072*(3*I+8*x)*(3*I*x+4*x^2)^(3/2)+25515/4194304*(3*I+8*x)*(3*I*x+4*x^2)^(1/2)+22
9635/16777216*arcsinh(8/3*x+I)

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Maxima [A]  time = 0.789948, size = 176, normalized size = 1.45 \[ \frac{1}{8} \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{7}{2}} x + \frac{3}{64} i \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{7}{2}} + \frac{21}{256} \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{5}{2}} x + \frac{63}{2048} i \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{5}{2}} + \frac{945}{16384} \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{3}{2}} x + \frac{2835}{131072} i \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{3}{2}} + \frac{25515}{524288} \, \sqrt{4 \, x^{2} + 3 i \, x} x + \frac{76545}{4194304} i \, \sqrt{4 \, x^{2} + 3 i \, x} + \frac{229635}{16777216} \, \log \left (8 \, x + 4 \, \sqrt{4 \, x^{2} + 3 i \, x} + 3 i\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 + 3*I*x)^(7/2),x, algorithm="maxima")

[Out]

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 + 7
6545/4194304*I*sqrt(4*x^2 + 3*I*x) + 229635/16777216*log(8*x + 4*sqrt(4*x^2 + 3*
I*x) + 3*I)

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Fricas [A]  time = 0.226256, size = 494, normalized size = 4.08 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 + 3*I*x)^(7/2),x, algorithm="fricas")

[Out]

-(9223372036854775808*x^16 + 55340232221128654848*I*x^15 - 146997491837372989440
*x^14 - 226981421219472998400*I*x^13 + 224833204197217271808*x^12 + 148652564700
431646720*I*x^11 - 66132721700029071360*x^10 - 19461487753030533120*I*x^9 + 3598
235800719851520*x^8 + 341772402683805696*I*x^7 + 22469894033375232*x^6 + 1939053
7119301632*I*x^5 - 4790430374952960*x^4 - 557082135429120*I*x^3 + 21507697870848
*x^2 + (7890198520135680*x^8 + 23670595560407040*I*x^7 - 28848538339246080*x^6 -
18307726253752320*I*x^5 + 6436310011084800*x^4 + 1228750093025280*I*x^3 - 11519
5321221120*x^2 - (3945099260067840*x^7 + 10355885557678080*I*x^6 - 1081820187721
7280*x^5 - 5721164454297600*I*x^4 + 1609077502771200*x^3 + 230390642442240*I*x^2
- 14399415152640*x - 257132413440*I)*sqrt(4*x^2 + 3*I*x) - 4114118615040*I*x +
24106163760)*log(-2*x + sqrt(4*x^2 + 3*I*x) - 3/4*I) - (4611686018427387904*x^15
+ 25940733853654056960*I*x^14 - 64095229896736899072*x^13 - 9115736005760620953
6*I*x^12 + 82112302418497634304*x^11 + 48544353159984709632*I*x^10 - 18872887292
844834816*x^9 - 4697147896158486528*I*x^8 + 692278034027249664*x^7 + 36186599083
474944*I*x^6 + 12907046076678144*x^5 + 5439502364442624*I*x^4 - 996472905891840*
x^3 - 75654070087680*I*x^2 + 888535339776*x - 49436767584*I)*sqrt(4*x^2 + 3*I*x)
- 276723454464*I*x + 7647967431)/(576460752303423488*x^8 + 1729382256910270464*
I*x^7 - 2107684625609392128*x^6 - 1337569089329037312*I*x^5 + 470239132967239680
*x^4 + 89772925384654848*I*x^3 - 8416211754811392*x^2 - (288230376151711744*x^7
+ 756604737398243328*I*x^6 - 790381734603522048*x^5 - 417990340415324160*I*x^4 +
117559783241809920*x^3 + 16832423509622784*I*x^2 - 1052026469351424*x - 1878618
6952704*I)*sqrt(4*x^2 + 3*I*x) - 300578991243264*I*x + 1761205026816)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (4 x^{2} + 3 i x\right )^{\frac{7}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*I*x+4*x**2)**(7/2),x)

[Out]

Integral((4*x**2 + 3*I*x)**(7/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{7}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 + 3*I*x)^(7/2),x, algorithm="giac")

[Out]

integrate((4*x^2 + 3*I*x)^(7/2), x)