3.2737 \(\int (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=249 \[ \frac{2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}+\frac{62 (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}}{2145}+\frac{34 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}}{2475}+\frac{32717 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}}{1126125}-\frac{445024 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{9384375}-\frac{69808931 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{168918750}-\frac{69808931 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{76781250 \sqrt{33}}-\frac{1163388067 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{38390625 \sqrt{33}} \]

[Out]

(-69808931*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/168918750 - (445024*Sqrt[1
- 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/9384375 + (32717*Sqrt[1 - 2*x]*(2 + 3*x)^
(3/2)*(3 + 5*x)^(3/2))/1126125 + (34*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/
2))/2475 + (62*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/2145 + (2*(1 - 2
*x)^(5/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/65 - (1163388067*EllipticE[ArcSin[Sqr
t[3/7]*Sqrt[1 - 2*x]], 35/33])/(38390625*Sqrt[33]) - (69808931*EllipticF[ArcSin[
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(76781250*Sqrt[33])

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Rubi [A]  time = 0.546473, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}+\frac{62 (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}}{2145}+\frac{34 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}}{2475}+\frac{32717 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}}{1126125}-\frac{445024 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{9384375}-\frac{69808931 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{168918750}-\frac{69808931 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{76781250 \sqrt{33}}-\frac{1163388067 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{38390625 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(5/2)*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x],x]

[Out]

(-69808931*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/168918750 - (445024*Sqrt[1
- 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/9384375 + (32717*Sqrt[1 - 2*x]*(2 + 3*x)^
(3/2)*(3 + 5*x)^(3/2))/1126125 + (34*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/
2))/2475 + (62*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/2145 + (2*(1 - 2
*x)^(5/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/65 - (1163388067*EllipticE[ArcSin[Sqr
t[3/7]*Sqrt[1 - 2*x]], 35/33])/(38390625*Sqrt[33]) - (69808931*EllipticF[ArcSin[
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(76781250*Sqrt[33])

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Rubi in Sympy [A]  time = 58.0211, size = 230, normalized size = 0.92 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{39} - \frac{59 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{715} - \frac{401 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{2145} - \frac{18787 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{50050} + \frac{51694 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{144375} + \frac{68473859 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{84459375} - \frac{1163388067 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1266890625} - \frac{69808931 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2533781250} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(2+3*x)**(5/2)*(3+5*x)**(1/2),x)

[Out]

2*(-2*x + 1)**(5/2)*(3*x + 2)**(7/2)*sqrt(5*x + 3)/39 - 59*(-2*x + 1)**(5/2)*(3*
x + 2)**(5/2)*sqrt(5*x + 3)/715 - 401*(-2*x + 1)**(5/2)*(3*x + 2)**(3/2)*sqrt(5*
x + 3)/2145 - 18787*(-2*x + 1)**(5/2)*sqrt(3*x + 2)*sqrt(5*x + 3)/50050 + 51694*
(-2*x + 1)**(3/2)*sqrt(3*x + 2)*sqrt(5*x + 3)/144375 + 68473859*sqrt(-2*x + 1)*s
qrt(3*x + 2)*sqrt(5*x + 3)/84459375 - 1163388067*sqrt(33)*elliptic_e(asin(sqrt(2
1)*sqrt(-2*x + 1)/7), 35/33)/1266890625 - 69808931*sqrt(33)*elliptic_f(asin(sqrt
(21)*sqrt(-2*x + 1)/7), 35/33)/2533781250

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Mathematica [A]  time = 0.43827, size = 112, normalized size = 0.45 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (935550000 x^5+433755000 x^4-936022500 x^3-309143250 x^2+380959290 x+84411073\right )-2349857545 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2533781250 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(5/2)*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x],x]

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(84411073 + 380959290*x - 30914325
0*x^2 - 936022500*x^3 + 433755000*x^4 + 935550000*x^5) + 4653552268*EllipticE[Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2349857545*EllipticF[ArcSin[Sqrt[2/11]*
Sqrt[3 + 5*x]], -33/2])/(2533781250*Sqrt[2])

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Maple [C]  time = 0.016, size = 189, normalized size = 0.8 \[{\frac{1}{152026875000\,{x}^{3}[phone]\,{x}^{2}-35472937500\,x-30405375000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 841995000000\,{x}^{8}[phone]\,{x}^{7}-739594800000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -4653552268\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1183572000000\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-86298997530\,x-15193993140 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(2+3*x)^(5/2)*(3+5*x)^(1/2),x)

[Out]

1/5067562500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(841995000000*x^8+1035909
000000*x^7-739594800000*x^6+2349857545*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*
x)^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^
(1/2))-4653552268*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/
11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))-1183572000000*
x^5+248043343500*x^4+572236008300*x^3+33887974470*x^2-86298997530*x-15193993140)
/(30*x^3+23*x^2-7*x-6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(3*x + 2)^(5/2)*(-2*x + 1)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(5*x + 3)*(3*x + 2)^(5/2)*(-2*x + 1)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(3*x + 2)^(5/2)*(-2*x + 1)^(5/2),x, algorithm="fricas")

[Out]

integral((36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-
2*x + 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)**(5/2)*(2+3*x)**(5/2)*(3+5*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(3*x + 2)^(5/2)*(-2*x + 1)^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(5*x + 3)*(3*x + 2)^(5/2)*(-2*x + 1)^(5/2), x)