3.2739 \(\int (1-2 x)^{5/2} \sqrt{2+3 x} \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=191 \[ \frac{2}{45} \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac{326 \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{4725}+\frac{10214 \sqrt{3 x+2} (5 x+3)^{3/2} \sqrt{1-2 x}}{118125}-\frac{110717 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{1063125}-\frac{110717 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5315625}-\frac{6799613 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5315625} \]

[Out]

(-110717*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1063125 + (10214*Sqrt[1 - 2*
x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/118125 + (326*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3
+ 5*x)^(3/2))/4725 + (2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/45 - (67
99613*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5315625 - (1
10717*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5315625

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Rubi [A]  time = 0.396527, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2}{45} \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac{326 \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{4725}+\frac{10214 \sqrt{3 x+2} (5 x+3)^{3/2} \sqrt{1-2 x}}{118125}-\frac{110717 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{1063125}-\frac{110717 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5315625}-\frac{6799613 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5315625} \]

Antiderivative was successfully verified.

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

[Out]

(-110717*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1063125 + (10214*Sqrt[1 - 2*
x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/118125 + (326*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3
+ 5*x)^(3/2))/4725 + (2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/45 - (67
99613*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5315625 - (1
10717*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5315625

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Rubi in Sympy [A]  time = 39.6121, size = 172, normalized size = 0.9 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{27} - \frac{59 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{315} + \frac{1286 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{7875} + \frac{394876 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{1063125} - \frac{6799613 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{15946875} - \frac{110717 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{15946875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(5/2)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/27 - 59*(-2*x + 1)**(5/2)*sqr
t(3*x + 2)*sqrt(5*x + 3)/315 + 1286*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*sqrt(5*x + 3
)/7875 + 394876*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/1063125 - 6799613*sqr
t(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/15946875 - 110717*sqrt(
33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/15946875

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Mathematica [A]  time = 0.341894, size = 102, normalized size = 0.53 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (945000 x^3-1111500 x^2+55530 x+526861\right )-9945565 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{15946875 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(526861 + 55530*x - 1111500*x^2 +
945000*x^3) + 13599226*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 9945
565*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(15946875*Sqrt[2])

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Maple [C]  time = 0.014, size = 179, normalized size = 0.9 \[{\frac{1}{956812500\,{x}^{3}[phone]\,{x}^{2}-223256250\,x-191362500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 850500000\,{x}^{6}+9945565\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -13599226\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -348300000\,{x}^{5}-915408000\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-120636210\,x-94834980 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/31893750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(850500000*x^6+9945565*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))-13599226*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))-348300000*x^5-915408000*x^4+575805600*x^3+551942790*x^2-120
636210*x-94834980)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((4*x^2 - 4*x + 1)*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)**(1/2)*(3+5*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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