3.2648 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{\sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=160 \[ \frac{2}{21} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{1}{7} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{131}{189} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{131}{945} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{9013 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1890} \]

[Out]

(-131*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/189 - (Sqrt[1 - 2*x]*Sqrt[2 + 3
*x]*(3 + 5*x)^(3/2))/7 + (2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/21 - (9
013*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1890 - (131*Sq
rt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/945

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Rubi [A]  time = 0.336018, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2}{21} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{1}{7} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{131}{189} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{131}{945} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{9013 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1890} \]

Antiderivative was successfully verified.

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

[Out]

(-131*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/189 - (Sqrt[1 - 2*x]*Sqrt[2 + 3
*x]*(3 + 5*x)^(3/2))/7 + (2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/21 - (9
013*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1890 - (131*Sq
rt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/945

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Rubi in Sympy [A]  time = 31.0322, size = 141, normalized size = 0.88 \[ \frac{2 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{5}{2}}}{21} - \frac{\sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{7} - \frac{131 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{189} - \frac{9013 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{5670} - \frac{1441 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{33075} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(5/2)/21 - sqrt(-2*x + 1)*sqrt(3*x + 2
)*(5*x + 3)**(3/2)/7 - 131*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/189 - 9013
*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/5670 - 1441*sqrt(35
)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/33075

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Mathematica [A]  time = 0.354525, size = 97, normalized size = 0.61 \[ \frac{75 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (90 x^2+81 x-10\right )-4690 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+9013 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2835 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(75*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-10 + 81*x + 90*x^2) + 9013*Ellip
ticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 4690*EllipticF[ArcSin[Sqrt[2/11]
*Sqrt[3 + 5*x]], -33/2])/(2835*Sqrt[2])

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Maple [C]  time = 0.017, size = 174, normalized size = 1.1 \[{\frac{1}{170100\,{x}^{3}+130410\,{x}^{2}-39690\,x-34020}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 4690\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -9013\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +405000\,{x}^{5}+675000\,{x}^{4}+139950\,{x}^{3}-200550\,{x}^{2}-62400\,x+9000 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5670*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(4690*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))-9013*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*El
lipticE(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))+4050
00*x^5+675000*x^4+139950*x^3-200550*x^2-62400*x+9000)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{\sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((25*x^2 + 30*x + 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)/sqrt(3*x + 2), 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((3+5*x)**(5/2)*(1-2*x)**(1/2)/(2+3*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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