3.2891 \(\int \frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=246 \[ \frac{(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt{1-2 x}}+\frac{18}{11} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{5/2}+\frac{419}{66} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}+\frac{9741}{385} \sqrt{1-2 x} (5 x+3)^{5/2} \sqrt{3 x+2}+\frac{4066493 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{23100}+\frac{269045681 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{207900}+\frac{269045681 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{94500 \sqrt{33}}+\frac{17888580643 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{189000 \sqrt{33}} \]

[Out]

(269045681*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/207900 + (4066493*Sqrt[1 -
2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/23100 + (9741*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(
3 + 5*x)^(5/2))/385 + (419*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/66 + (
18*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/11 + ((2 + 3*x)^(7/2)*(3 + 5*x
)^(5/2))/Sqrt[1 - 2*x] + (17888580643*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/(189000*Sqrt[33]) + (269045681*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/(94500*Sqrt[33])

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Rubi [A]  time = 0.553857, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt{1-2 x}}+\frac{18}{11} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{5/2}+\frac{419}{66} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}+\frac{9741}{385} \sqrt{1-2 x} (5 x+3)^{5/2} \sqrt{3 x+2}+\frac{4066493 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{23100}+\frac{269045681 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{207900}+\frac{269045681 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{94500 \sqrt{33}}+\frac{17888580643 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{189000 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(269045681*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/207900 + (4066493*Sqrt[1 -
2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/23100 + (9741*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(
3 + 5*x)^(5/2))/385 + (419*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/66 + (
18*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/11 + ((2 + 3*x)^(7/2)*(3 + 5*x
)^(5/2))/Sqrt[1 - 2*x] + (17888580643*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/(189000*Sqrt[33]) + (269045681*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/(94500*Sqrt[33])

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Rubi in Sympy [A]  time = 54.7932, size = 226, normalized size = 0.92 \[ \frac{18 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{11} + \frac{2095 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{198} + \frac{277565 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{4158} + \frac{11059889 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{41580} + \frac{128715331 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{103950} + \frac{17888580643 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{6237000} + \frac{269045681 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3118500} + \frac{\left (3 x + 2\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

18*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*(5*x + 3)**(5/2)/11 + 2095*sqrt(-2*x + 1)*(3*
x + 2)**(5/2)*(5*x + 3)**(3/2)/198 + 277565*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt
(5*x + 3)/4158 + 11059889*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/41580 +
128715331*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/103950 + 17888580643*sqrt(3
3)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/6237000 + 269045681*sqrt(3
3)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/3118500 + (3*x + 2)**(7/2)
*(5*x + 3)**(5/2)/sqrt(-2*x + 1)

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Mathematica [A]  time = 0.342632, size = 125, normalized size = 0.51 \[ \frac{-30 \sqrt{3 x+2} \sqrt{5 x+3} \left (12757500 x^5+60196500 x^4+133330950 x^3+198895770 x^2+273928969 x-477155552\right )[phone] \sqrt{2-4 x} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-17888580643 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{6237000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-477155552 + 273928969*x + 198895770*x^2 + 133
330950*x^3 + 60196500*x^4 + 12757500*x^5) - 17888580643*Sqrt[2 - 4*x]*EllipticE[
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 9010073170*Sqrt[2 - 4*x]*EllipticF[Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(6237000*Sqrt[1 - 2*x])

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Maple [C]  time = 0.025, size = 184, normalized size = 0.8 \[ -{\frac{1}{187110000\,{x}^{3}[phone]\,{x}^{2}-43659000\,x-37422000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -5740875000\,{x}^{7}-34360200000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -17888580643\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -96607282500\,{x}^{5}-176337108000\,{x}^{4}-260638195950\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+85887999360 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6237000*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-5740875000*x^7-3436020000
0*x^6+9010073170*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/1
1*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))-17888580643*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))-96607282500*x^5-176337108000*x^4-26
0638195950*x^3+22779247470*x^2+222671450220*x+85887999360)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*sqrt(5*x + 3)*
sqrt(3*x + 2)/((2*x - 1)*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)**(5/2)/(1-2*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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