3.2806 \(\int \frac{(2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{1}{7} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}-\frac{102}{175} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{4721 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{1050}-\frac{4721 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5250}-\frac{78472 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2625} \]

[Out]

(-4721*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1050 - (102*Sqrt[1 - 2*x]*Sqrt
[2 + 3*x]*(3 + 5*x)^(3/2))/175 - (Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))
/7 - (78472*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2625 -
(4721*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5250

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Rubi [A]  time = 0.318257, 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 \[ -\frac{1}{7} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}-\frac{102}{175} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{4721 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{1050}-\frac{4721 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5250}-\frac{78472 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2625} \]

Antiderivative was successfully verified.

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

[Out]

(-4721*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1050 - (102*Sqrt[1 - 2*x]*Sqrt
[2 + 3*x]*(3 + 5*x)^(3/2))/175 - (Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))
/7 - (78472*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2625 -
(4721*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5250

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Rubi in Sympy [A]  time = 33.4819, size = 143, normalized size = 0.89 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{7} - \frac{34 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{35} - \frac{4517 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{1050} - \frac{78472 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{7875} - \frac{4721 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{15750} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)/7 - 34*sqrt(-2*x + 1)*(3*x + 2
)**(3/2)*sqrt(5*x + 3)/35 - 4517*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/1050
- 78472*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/7875 - 4721
*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/15750

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Mathematica [A]  time = 0.302452, size = 100, normalized size = 0.62 \[ \frac{313888 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (2250 x^2+5910 x+7457\right )+31619 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{15750 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(313888*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5*(3*Sqrt[2 - 4*x]*
Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(7457 + 5910*x + 2250*x^2) + 31619*EllipticF[ArcSin[
Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/(15750*Sqrt[2])

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Maple [C]  time = 0.016, size = 174, normalized size = 1.1 \[{\frac{1}{945000\,{x}^{3}+724500\,{x}^{2}-220500\,x-189000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 158095\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -313888\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2025000\,{x}^{5}-6871500\,{x}^{4}-10316700\,{x}^{3}-3499230\,{x}^{2}+2629770\,x+1342260 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/31500*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(158095*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))-313888*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))
-2025000*x^5-6871500*x^4-10316700*x^3-3499230*x^2+2629770*x+1342260)/(30*x^3+23*
x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (15 \, x^{2} + 19 \, x + 6\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{\sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((15*x^2 + 19*x + 6)*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((2+3*x)**(3/2)*(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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}}{\sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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