3.2882 \(\int \frac{(2+3 x)^{7/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=219 \[ \frac{(5 x+3)^{3/2} (3 x+2)^{7/2}}{\sqrt{1-2 x}}+\frac{5}{3} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}+\frac{1397}{210} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}+\frac{24358}{875} \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}+\frac{6770629 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{31500}+\frac{6770629 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{157500}+\frac{112543103 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{78750} \]

[Out]

(6770629*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/31500 + (24358*Sqrt[1 - 2*x]
*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/875 + (1397*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5
*x)^(3/2))/210 + (5*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/3 + ((2 + 3*x
)^(7/2)*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] + (112543103*Sqrt[11/3]*EllipticE[ArcSin[
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/78750 + (6770629*Sqrt[11/3]*EllipticF[ArcSin[S
qrt[3/7]*Sqrt[1 - 2*x]], 35/33])/157500

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Rubi [A]  time = 0.459204, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{(5 x+3)^{3/2} (3 x+2)^{7/2}}{\sqrt{1-2 x}}+\frac{5}{3} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}+\frac{1397}{210} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}+\frac{24358}{875} \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}+\frac{6770629 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{31500}+\frac{6770629 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{157500}+\frac{112543103 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{78750} \]

Antiderivative was successfully verified.

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

[Out]

(6770629*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/31500 + (24358*Sqrt[1 - 2*x]
*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/875 + (1397*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5
*x)^(3/2))/210 + (5*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/3 + ((2 + 3*x
)^(7/2)*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] + (112543103*Sqrt[11/3]*EllipticE[ArcSin[
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/78750 + (6770629*Sqrt[11/3]*EllipticF[ArcSin[S
qrt[3/7]*Sqrt[1 - 2*x]], 35/33])/157500

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Rubi in Sympy [A]  time = 46.7573, size = 197, normalized size = 0.9 \[ \frac{5 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3} + \frac{1397 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{126} + \frac{139163 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{3150} + \frac{6478333 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{31500} + \frac{112543103 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{236250} + \frac{74476919 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{5512500} + \frac{\left (3 x + 2\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*(5*x + 3)**(3/2)/3 + 1397*sqrt(-2*x + 1)*(3*x
+ 2)**(5/2)*sqrt(5*x + 3)/126 + 139163*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x
+ 3)/3150 + 6478333*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/31500 + 112543103
*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/236250 + 74476919*s
qrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/5512500 + (3*x + 2)*
*(7/2)*(5*x + 3)**(3/2)/sqrt(-2*x + 1)

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Mathematica [A]  time = 0.303697, size = 120, normalized size = 0.55 \[ \frac{-30 \sqrt{3 x+2} \sqrt{5 x+3} \left (472500 x^4+2002500 x^3+4128030 x^2+6609296 x-12044593\right )[phone] \sqrt{2-4 x} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-450172412 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{945000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-12044593 + 6609296*x + 4128030*x^2 + 2002500*
x^3 + 472500*x^4) - 450172412*Sqrt[2 - 4*x]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]], -33/2] + 226741655*Sqrt[2 - 4*x]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]], -33/2])/(945000*Sqrt[1 - 2*x])

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Maple [C]  time = 0.026, size = 179, normalized size = 0.8 \[ -{\frac{1}{28350000\,{x}^{3}[phone]\,{x}^{2}-6615000\,x-5670000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -212625000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -450172412\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1170450000\,{x}^{5}-3084088500\,{x}^{4}-5687610300\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+2168026740 \right ) } \]

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)^(3/2),x)

[Out]

-1/945000*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-212625000*x^6+226741655*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))-450172412*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))-1170450000*x^5-3084088500*x^4-5687610300*x^3+909722730*x
^2+5675744730*x+2168026740)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*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)**(3/2)/(1-2*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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