3.2749 \(\int (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=218 \[ \frac{2}{55} (1-2 x)^{5/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{326 (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}}{7425}+\frac{30362 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{779625}-\frac{78797 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{3898125}-\frac{12996374 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{35083125}-\frac{12996374 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15946875 \sqrt{33}}-\frac{829177897 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{31893750 \sqrt{33}} \]

[Out]

(-12996374*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/35083125 - (78797*Sqrt[1 -
2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/3898125 + (30362*Sqrt[1 - 2*x]*Sqrt[2 + 3*x
]*(3 + 5*x)^(5/2))/779625 + (326*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/
7425 + (2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/55 - (829177897*Ellipti
cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(31893750*Sqrt[33]) - (12996374*Elli
pticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(15946875*Sqrt[33])

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Rubi [A]  time = 0.48088, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2}{55} (1-2 x)^{5/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{326 (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}}{7425}+\frac{30362 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{779625}-\frac{78797 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{3898125}-\frac{12996374 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{35083125}-\frac{12996374 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15946875 \sqrt{33}}-\frac{829177897 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{31893750 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-12996374*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/35083125 - (78797*Sqrt[1 -
2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/3898125 + (30362*Sqrt[1 - 2*x]*Sqrt[2 + 3*x
]*(3 + 5*x)^(5/2))/779625 + (326*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/
7425 + (2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/55 - (829177897*Ellipti
cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(31893750*Sqrt[33]) - (12996374*Elli
pticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(15946875*Sqrt[33])

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Rubi in Sympy [A]  time = 46.9914, size = 201, normalized size = 0.92 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{33} - \frac{181 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{891} + \frac{6646 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{31185} + \frac{683248 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{2338875} - \frac{11437073 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{35083125} - \frac{829177897 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1052493750} - \frac{12996374 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{558140625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(5/2)*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)/33 - 181*(-2*x + 1)**(5/2)
*(3*x + 2)**(3/2)*sqrt(5*x + 3)/891 + 6646*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2)*sq
rt(5*x + 3)/31185 + 683248*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/2338875
- 11437073*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/35083125 - 829177897*sqrt
(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1052493750 - 12996374*sq
rt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/558140625

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Mathematica [A]  time = 0.419843, size = 107, normalized size = 0.49 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (127575000 x^4-51502500 x^3-95024250 x^2+48272535 x+22517617\right )-400297555 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{526246875 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(22517617 + 48272535*x - 95024250*
x^2 - 51502500*x^3 + 127575000*x^4) + 829177897*EllipticE[ArcSin[Sqrt[2/11]*Sqrt
[3 + 5*x]], -33/2] - 400297555*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2
])/(526246875*Sqrt[2])

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Maple [C]  time = 0.016, size = 184, normalized size = 0.8 \[{\frac{1}{31574812500\,{x}^{3}[phone]\,{x}^{2}-7367456250\,x-6314962500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 114817500000\,{x}^{7}[phone]\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -829177897\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -147849300000\,{x}^{5}-34269426000\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-13417755870\,x-4053171060 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1052493750*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(114817500000*x^7+4167450
0000*x^6+400297555*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))-829177897*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))-147849300000*x^5-34269426000*x^4+82
799446950*x^3+22504288380*x^2-13417755870*x-4053171060)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((20*x^3 - 8*x^2 - 7*x + 3)*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)*(3+5*x)**(3/2)*(2+3*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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