3.2787 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=222 \[ \frac{2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} \sqrt{5 x+3}}-\frac{9795160 \sqrt{3 x+2} \sqrt{1-2 x}}{441 \sqrt{5 x+3}}+\frac{324104 \sqrt{1-2 x}}{147 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{2332 \sqrt{1-2 x}}{21 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{104 \sqrt{1-2 x}}{9 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{58928}{147} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{1959032}{147} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(2*(1 - 2*x)^(3/2))/(3*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x]) + (104*Sqrt[1 - 2*x])/(9*(
2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (2332*Sqrt[1 - 2*x])/(21*(2 + 3*x)^(3/2)*Sqrt[3
+ 5*x]) + (324104*Sqrt[1 - 2*x])/(147*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (9795160*Sq
rt[1 - 2*x]*Sqrt[2 + 3*x])/(441*Sqrt[3 + 5*x]) + (1959032*Sqrt[11/3]*EllipticE[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/147 + (58928*Sqrt[11/3]*EllipticF[ArcSin
[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/147

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Rubi [A]  time = 0.518641, antiderivative size = 222, 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 \[ \frac{2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} \sqrt{5 x+3}}-\frac{9795160 \sqrt{3 x+2} \sqrt{1-2 x}}{441 \sqrt{5 x+3}}+\frac{324104 \sqrt{1-2 x}}{147 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{2332 \sqrt{1-2 x}}{21 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{104 \sqrt{1-2 x}}{9 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{58928}{147} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{1959032}{147} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*(1 - 2*x)^(3/2))/(3*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x]) + (104*Sqrt[1 - 2*x])/(9*(
2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (2332*Sqrt[1 - 2*x])/(21*(2 + 3*x)^(3/2)*Sqrt[3
+ 5*x]) + (324104*Sqrt[1 - 2*x])/(147*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (9795160*Sq
rt[1 - 2*x]*Sqrt[2 + 3*x])/(441*Sqrt[3 + 5*x]) + (1959032*Sqrt[11/3]*EllipticE[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/147 + (58928*Sqrt[11/3]*EllipticF[ArcSin
[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/147

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Rubi in Sympy [A]  time = 48.5841, size = 201, normalized size = 0.91 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{3 \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}} - \frac{9795160 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{441 \sqrt{5 x + 3}} + \frac{324104 \sqrt{- 2 x + 1}}{147 \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{2332 \sqrt{- 2 x + 1}}{21 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}} + \frac{104 \sqrt{- 2 x + 1}}{9 \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}} + \frac{1959032 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{441} + \frac{648208 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{5145} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(3/2)/(3*(3*x + 2)**(7/2)*sqrt(5*x + 3)) - 9795160*sqrt(-2*x + 1)*
sqrt(3*x + 2)/(441*sqrt(5*x + 3)) + 324104*sqrt(-2*x + 1)/(147*sqrt(3*x + 2)*sqr
t(5*x + 3)) + 2332*sqrt(-2*x + 1)/(21*(3*x + 2)**(3/2)*sqrt(5*x + 3)) + 104*sqrt
(-2*x + 1)/(9*(3*x + 2)**(5/2)*sqrt(5*x + 3)) + 1959032*sqrt(33)*elliptic_e(asin
(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/441 + 648208*sqrt(35)*elliptic_f(asin(sqrt(5
5)*sqrt(-2*x + 1)/11), 33/35)/5145

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Mathematica [A]  time = 0.362829, size = 110, normalized size = 0.5 \[ \frac{2}{441} \left (-\frac{3 \sqrt{1-2 x} \left (132234660 x^4+348250356 x^3+343801494 x^2+150788294 x+24789615\right )}{(3 x+2)^{7/2} \sqrt{5 x+3}}-4 \sqrt{2} \left (244879 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-123340 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((-3*Sqrt[1 - 2*x]*(24789615 + 150788294*x + 343801494*x^2 + 348250356*x^3 +
132234660*x^4))/((2 + 3*x)^(7/2)*Sqrt[3 + 5*x]) - 4*Sqrt[2]*(244879*EllipticE[Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 123340*EllipticF[ArcSin[Sqrt[2/11]*Sqrt
[3 + 5*x]], -33/2])))/441

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Maple [C]  time = 0.036, size = 505, normalized size = 2.3 \[ -{\frac{2}{4410\,{x}^{2}+441\,x-1323}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 13320720\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-26446932\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-52893864\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-35262576\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+3946880\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -7836128\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-126674718\,{x}^{2}-303627192\,x-74368845 \right ) \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/441*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(13320720*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*(1-2*x)^(1/2)*(3+5*x)^(1/
2)*(2+3*x)^(1/2)-26446932*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)+26
641440*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^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-52893864*2^(1/2)*Elli
pticE(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^2*(3
+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*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)-35262576*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)+39
46880*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))-7836128*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]*x^5+1692798156*x^4+1018057896*x^3-126674
718*x^2-303627192*x-74368845)/(2+3*x)^(7/2)/(10*x^2+x-3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}{{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(-2*x + 1)/((405*x^5 + 1323*x^4 + 1728*x^3 + 1128
*x^2 + 368*x + 48)*sqrt(5*x + 3)*sqrt(3*x + 2)), 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((1-2*x)**(5/2)/(2+3*x)**(9/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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