3.2928 \(\int \frac{(2+3 x)^{9/2} \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=218 \[ \frac{\sqrt{5 x+3} (3 x+2)^{9/2}}{3 (1-2 x)^{3/2}}-\frac{166 \sqrt{5 x+3} (3 x+2)^{7/2}}{33 \sqrt{1-2 x}}-\frac{1327}{154} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{139163 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{3850}-\frac{6478333 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{38500}-\frac{6770629 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17500 \sqrt{33}}-\frac{112543103 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8750 \sqrt{33}} \]

[Out]

(-6478333*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/38500 - (139163*Sqrt[1 - 2*
x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/3850 - (1327*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqr
t[3 + 5*x])/154 - (166*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/(33*Sqrt[1 - 2*x]) + ((2 +
3*x)^(9/2)*Sqrt[3 + 5*x])/(3*(1 - 2*x)^(3/2)) - (112543103*EllipticE[ArcSin[Sqr
t[3/7]*Sqrt[1 - 2*x]], 35/33])/(8750*Sqrt[33]) - (6770629*EllipticF[ArcSin[Sqrt[
3/7]*Sqrt[1 - 2*x]], 35/33])/(17500*Sqrt[33])

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Rubi [A]  time = 0.491654, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{\sqrt{5 x+3} (3 x+2)^{9/2}}{3 (1-2 x)^{3/2}}-\frac{166 \sqrt{5 x+3} (3 x+2)^{7/2}}{33 \sqrt{1-2 x}}-\frac{1327}{154} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{139163 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{3850}-\frac{6478333 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{38500}-\frac{6770629 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17500 \sqrt{33}}-\frac{112543103 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8750 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[((2 + 3*x)^(9/2)*Sqrt[3 + 5*x])/(1 - 2*x)^(5/2),x]

[Out]

(-6478333*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/38500 - (139163*Sqrt[1 - 2*
x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/3850 - (1327*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqr
t[3 + 5*x])/154 - (166*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/(33*Sqrt[1 - 2*x]) + ((2 +
3*x)^(9/2)*Sqrt[3 + 5*x])/(3*(1 - 2*x)^(3/2)) - (112543103*EllipticE[ArcSin[Sqr
t[3/7]*Sqrt[1 - 2*x]], 35/33])/(8750*Sqrt[33]) - (6770629*EllipticF[ArcSin[Sqrt[
3/7]*Sqrt[1 - 2*x]], 35/33])/(17500*Sqrt[33])

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Rubi in Sympy [A]  time = 47.8461, size = 199, normalized size = 0.91 \[ - \frac{1327 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{154} - \frac{139163 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{3850} - \frac{6478333 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{38500} - \frac{112543103 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{288750} - \frac{6770629 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{612500} - \frac{166 \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{33 \sqrt{- 2 x + 1}} + \frac{\left (3 x + 2\right )^{\frac{9}{2}} \sqrt{5 x + 3}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**(9/2)*(3+5*x)**(1/2)/(1-2*x)**(5/2),x)

[Out]

-1327*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5*x + 3)/154 - 139163*sqrt(-2*x + 1)*
(3*x + 2)**(3/2)*sqrt(5*x + 3)/3850 - 6478333*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(
5*x + 3)/38500 - 112543103*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7),
35/33)/288750 - 6770629*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33
/35)/612500 - 166*(3*x + 2)**(7/2)*sqrt(5*x + 3)/(33*sqrt(-2*x + 1)) + (3*x + 2)
**(9/2)*sqrt(5*x + 3)/(3*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.361109, size = 130, normalized size = 0.6 \[ -\frac{10 \sqrt{3 x+2} \sqrt{5 x+3} \left (1336500 x^4+6664680 x^3+19375686 x^2-94671446 x+35797779\right )-226741655 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1155000 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((2 + 3*x)^(9/2)*Sqrt[3 + 5*x])/(1 - 2*x)^(5/2),x]

[Out]

-(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(35797779 - 94671446*x + 19375686*x^2 + 6664680
*x^3 + 1336500*x^4) + 450172412*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticE[ArcSin[Sqrt[2
/11]*Sqrt[3 + 5*x]], -33/2] - 226741655*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticF[ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(1155000*(1 - 2*x)^(3/2))

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Maple [C]  time = 0.053, size = 291, normalized size = 1.3 \[ -{\frac{1}{1155000\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 900344824\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-453483310\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,{x}^{6}-450172412\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}-10119455760\,{x}^{3}-11455366730\,{x}^{2}[phone]\,x+2147866740 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^(9/2)*(3+5*x)^(1/2)/(1-2*x)^(5/2),x)

[Out]

-1/1155000*(900344824*2^(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))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-45348331
0*2^(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))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*x^6-450172412*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))[phone]*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))[phone]*x^5+4252832100*x^4-10119455760*x^3-11455366730*
x^2+1121291250*x+2147866740)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)/(-1+2*x)^
2/(15*x^2+19*x+6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(3*x + 2)^(9/2)/(-2*x + 1)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(5*x + 3)*(3*x + 2)^(9/2)/(-2*x + 1)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(3*x + 2)^(9/2)/(-2*x + 1)^(5/2),x, algorithm="fricas")

[Out]

integral((81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*sqrt(5*x + 3)*sqrt(3*x + 2)/((
4*x^2 - 4*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)**(9/2)*(3+5*x)**(1/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(3*x + 2)^(9/2)/(-2*x + 1)^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(5*x + 3)*(3*x + 2)^(9/2)/(-2*x + 1)^(5/2), x)