3.2796 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=222 \[ \frac{14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac{96808 \sqrt{3 x+2} \sqrt{1-2 x}}{3 \sqrt{5 x+3}}-\frac{16016 \sqrt{3 x+2} \sqrt{1-2 x}}{3 (5 x+3)^{3/2}}+\frac{35948 \sqrt{1-2 x}}{45 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{1232 \sqrt{1-2 x}}{45 (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{2912}{5} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{96808}{5} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(14*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (1232*Sqrt[1 - 2*x])
/(45*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (35948*Sqrt[1 - 2*x])/(45*Sqrt[2 + 3*x]*
(3 + 5*x)^(3/2)) - (16016*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3*(3 + 5*x)^(3/2)) + (96
808*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3*Sqrt[3 + 5*x]) - (96808*Sqrt[11/3]*EllipticE
[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (2912*Sqrt[11/3]*EllipticF[ArcSin[
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5

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Rubi [A]  time = 0.518457, 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 \[ \frac{14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac{96808 \sqrt{3 x+2} \sqrt{1-2 x}}{3 \sqrt{5 x+3}}-\frac{16016 \sqrt{3 x+2} \sqrt{1-2 x}}{3 (5 x+3)^{3/2}}+\frac{35948 \sqrt{1-2 x}}{45 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{1232 \sqrt{1-2 x}}{45 (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{2912}{5} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{96808}{5} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(14*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (1232*Sqrt[1 - 2*x])
/(45*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (35948*Sqrt[1 - 2*x])/(45*Sqrt[2 + 3*x]*
(3 + 5*x)^(3/2)) - (16016*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3*(3 + 5*x)^(3/2)) + (96
808*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3*Sqrt[3 + 5*x]) - (96808*Sqrt[11/3]*EllipticE
[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (2912*Sqrt[11/3]*EllipticF[ArcSin[
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5

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Rubi in Sympy [A]  time = 47.231, size = 201, normalized size = 0.91 \[ \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}}}{15 \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{96808 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{3 \sqrt{5 x + 3}} - \frac{16016 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{3 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{35948 \sqrt{- 2 x + 1}}{45 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{1232 \sqrt{- 2 x + 1}}{45 \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{96808 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{15} - \frac{4576 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14*(-2*x + 1)**(3/2)/(15*(3*x + 2)**(5/2)*(5*x + 3)**(3/2)) + 96808*sqrt(-2*x +
1)*sqrt(3*x + 2)/(3*sqrt(5*x + 3)) - 16016*sqrt(-2*x + 1)*sqrt(3*x + 2)/(3*(5*x
+ 3)**(3/2)) + 35948*sqrt(-2*x + 1)/(45*sqrt(3*x + 2)*(5*x + 3)**(3/2)) + 1232*s
qrt(-2*x + 1)/(45*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)) - 96808*sqrt(33)*elliptic_e
(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/15 - 4576*sqrt(35)*elliptic_f(asin(sqrt
(55)*sqrt(-2*x + 1)/11), 33/35)/25

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Mathematica [A]  time = 0.373151, size = 109, normalized size = 0.49 \[ \frac{2}{15} \left (\frac{\sqrt{1-2 x} \left (32672700 x^4+83867940 x^3+80662602 x^2+34450018 x+5512543\right )}{(3 x+2)^{5/2} (5 x+3)^{3/2}}+4 \sqrt{2} \left (12101 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-6095 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[1 - 2*x]*(5512543 + 34450018*x + 80662602*x^2 + 83867940*x^3 + 3267270
0*x^4))/((2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + 4*Sqrt[2]*(12101*EllipticE[ArcSin[Sq
rt[2/11]*Sqrt[3 + 5*x]], -33/2] - 6095*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]
], -33/2])))/15

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Maple [C]  time = 0.036, size = 502, normalized size = 2.3 \[{\frac{2}{-15+30\,x}\sqrt{1-2\,x} \left ( 1097100\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-2178180\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}+2121060\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-4211148\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1365280\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2710624\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+292560\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -580848\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-11762566\,{x}^{2}-23424932\,x-5512543 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(1-2*x)^(1/2)*(1097100*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)-
2178180*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)+2121060*2^(1/2)*Elli
pticF(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)-4211148*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^2*(3+5*x)^(1/2)*(2+3*x)^
(1/2)*(1-2*x)^(1/2)+1365280*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)-27
10624*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)+292560*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))-580848*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+135063180*x^4+77457264*x^3-11762566*x^2-23424932*x-5512543)/(
2+3*x)^(5/2)/(3+5*x)^(3/2)/(-1+2*x)

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

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)^(7/2)),x, algorithm="maxima")

[Out]

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

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

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)^(7/2)),x, algorithm="fricas")

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(-2*x + 1)/((675*x^5 + 2160*x^4 + 2763*x^3 + 1766
*x^2 + 564*x + 72)*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)**(7/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

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)^(7/2)),x, algorithm="giac")

[Out]

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