3.2794 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=158 \[ \frac{14 (1-2 x)^{3/2}}{3 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{6388 \sqrt{3 x+2} \sqrt{1-2 x}}{15 \sqrt{5 x+3}}-\frac{1012 \sqrt{3 x+2} \sqrt{1-2 x}}{15 (5 x+3)^{3/2}}-\frac{64}{25} \sqrt{33} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{6388}{25} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(14*(1 - 2*x)^(3/2))/(3*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (1012*Sqrt[1 - 2*x]*Sqr
t[2 + 3*x])/(15*(3 + 5*x)^(3/2)) + (6388*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(15*Sqrt[3
+ 5*x]) - (6388*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2
5 - (64*Sqrt[33]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/25

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Rubi [A]  time = 0.33598, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{14 (1-2 x)^{3/2}}{3 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{6388 \sqrt{3 x+2} \sqrt{1-2 x}}{15 \sqrt{5 x+3}}-\frac{1012 \sqrt{3 x+2} \sqrt{1-2 x}}{15 (5 x+3)^{3/2}}-\frac{64}{25} \sqrt{33} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{6388}{25} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(14*(1 - 2*x)^(3/2))/(3*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (1012*Sqrt[1 - 2*x]*Sqr
t[2 + 3*x])/(15*(3 + 5*x)^(3/2)) + (6388*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(15*Sqrt[3
+ 5*x]) - (6388*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2
5 - (64*Sqrt[33]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/25

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Rubi in Sympy [A]  time = 32.2947, size = 143, normalized size = 0.91 \[ \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}}}{3 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{6388 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{15 \sqrt{5 x + 3}} - \frac{1012 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{6388 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{75} - \frac{2112 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14*(-2*x + 1)**(3/2)/(3*sqrt(3*x + 2)*(5*x + 3)**(3/2)) + 6388*sqrt(-2*x + 1)*sq
rt(3*x + 2)/(15*sqrt(5*x + 3)) - 1012*sqrt(-2*x + 1)*sqrt(3*x + 2)/(15*(5*x + 3)
**(3/2)) - 6388*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/75 -
2112*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/875

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Mathematica [A]  time = 0.218298, size = 100, normalized size = 0.63 \[ \frac{2}{75} \left (\frac{5 \sqrt{1-2 x} \left (47910 x^2+59098 x+18187\right )}{\sqrt{3 x+2} (5 x+3)^{3/2}}+2 \sqrt{2} \left (1597 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-805 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((5*Sqrt[1 - 2*x]*(18187 + 59098*x + 47910*x^2))/(Sqrt[2 + 3*x]*(3 + 5*x)^(3/
2)) + 2*Sqrt[2]*(1597*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 805*E
llipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/75

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Maple [C]  time = 0.033, size = 267, normalized size = 1.7 \[{\frac{2}{450\,{x}^{2}+75\,x-150}\sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 8050\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-15970\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+4830\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -9582\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +479100\,{x}^{3}+351430\,{x}^{2}-113620\,x-90935 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/75*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(8050*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)-15970*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)+4830*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))-9582*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))+479100*x^3+351430*x^2-113620*x-90935)/(3+5*x)^(3/2)/(6*x^2+x-2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}{{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(-2*x + 1)/((75*x^3 + 140*x^2 + 87*x + 18)*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)**(3/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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