3.2850 \(\int \frac{(2+3 x)^{7/2}}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{5/2}}{55 \sqrt{5 x+3}}-\frac{69 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{1375}-\frac{2577 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{6875}-\frac{942 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3125}-\frac{61151 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6250} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(55*Sqrt[3 + 5*x]) - (2577*Sqrt[1 - 2*x]*Sqrt
[2 + 3*x]*Sqrt[3 + 5*x])/6875 - (69*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])
/1375 - (61151*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/625
0 - (942*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3125

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Rubi [A]  time = 0.334696, 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 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{5/2}}{55 \sqrt{5 x+3}}-\frac{69 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{1375}-\frac{2577 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{6875}-\frac{942 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3125}-\frac{61151 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6250} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x)^(7/2)/(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)),x]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(55*Sqrt[3 + 5*x]) - (2577*Sqrt[1 - 2*x]*Sqrt
[2 + 3*x]*Sqrt[3 + 5*x])/6875 - (69*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])
/1375 - (61151*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/625
0 - (942*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3125

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Rubi in Sympy [A]  time = 34.3714, size = 144, normalized size = 0.9 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}}{55 \sqrt{5 x + 3}} - \frac{69 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{1375} - \frac{2577 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{6875} - \frac{61151 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{68750} - \frac{942 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{34375} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**(7/2)/(3+5*x)**(3/2)/(1-2*x)**(1/2),x)

[Out]

-2*sqrt(-2*x + 1)*(3*x + 2)**(5/2)/(55*sqrt(5*x + 3)) - 69*sqrt(-2*x + 1)*(3*x +
2)**(3/2)*sqrt(5*x + 3)/1375 - 2577*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/
6875 - 61151*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/68750 -
942*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/34375

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Mathematica [A]  time = 0.280916, size = 122, normalized size = 0.76 \[ \frac{61151 \sqrt{2} (5 x+3) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (2 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (7425 x^2+22440 x+10801\right )+6013 \sqrt{2} (5 x+3) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{68750 (5 x+3)} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x)^(7/2)/(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)),x]

[Out]

(61151*Sqrt[2]*(3 + 5*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5*
(2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(10801 + 22440*x + 7425*x^2) + 6013
*Sqrt[2]*(3 + 5*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/(68750*(
3 + 5*x))

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Maple [C]  time = 0.027, size = 169, normalized size = 1.1 \[{\frac{1}{2062500\,{x}^{3}+1581250\,{x}^{2}-481250\,x-412500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 30065\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -61151\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -445500\,{x}^{4}-1420650\,{x}^{3}-723960\,{x}^{2}+340790\,x+216020 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^(7/2)/(3+5*x)^(3/2)/(1-2*x)^(1/2),x)

[Out]

1/68750*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(30065*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*1
1^(1/2)*3^(1/2)*2^(1/2))-61151*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))-4
45500*x^4-1420650*x^3-723960*x^2+340790*x+216020)/(30*x^3+23*x^2-7*x-6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(7/2)/((5*x + 3)^(3/2)*sqrt(-2*x + 1)),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(7/2)/((5*x + 3)^(3/2)*sqrt(-2*x + 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{3 \, x + 2}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(7/2)/((5*x + 3)^(3/2)*sqrt(-2*x + 1)),x, algorithm="fricas")

[Out]

integral((27*x^3 + 54*x^2 + 36*x + 8)*sqrt(3*x + 2)/((5*x + 3)^(3/2)*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((2+3*x)**(7/2)/(3+5*x)**(3/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(7/2)/((5*x + 3)^(3/2)*sqrt(-2*x + 1)),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(7/2)/((5*x + 3)^(3/2)*sqrt(-2*x + 1)), x)