3.2727 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{36968 \sqrt{1-2 x} \sqrt{3 x+2}}{21 \sqrt{5 x+3}}+\frac{6116 \sqrt{1-2 x}}{35 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{44 \sqrt{1-2 x}}{5 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{14 \sqrt{1-2 x}}{15 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{1112}{35} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{36968}{35} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(14*Sqrt[1 - 2*x])/(15*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (44*Sqrt[1 - 2*x])/(5*(2
+ 3*x)^(3/2)*Sqrt[3 + 5*x]) + (6116*Sqrt[1 - 2*x])/(35*Sqrt[2 + 3*x]*Sqrt[3 + 5
*x]) - (36968*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(21*Sqrt[3 + 5*x]) + (36968*Sqrt[11/3
]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/35 + (1112*Sqrt[11/3]*Ellip
ticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/35

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Rubi [A]  time = 0.435007, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{36968 \sqrt{1-2 x} \sqrt{3 x+2}}{21 \sqrt{5 x+3}}+\frac{6116 \sqrt{1-2 x}}{35 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{44 \sqrt{1-2 x}}{5 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{14 \sqrt{1-2 x}}{15 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{1112}{35} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{36968}{35} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(14*Sqrt[1 - 2*x])/(15*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (44*Sqrt[1 - 2*x])/(5*(2
+ 3*x)^(3/2)*Sqrt[3 + 5*x]) + (6116*Sqrt[1 - 2*x])/(35*Sqrt[2 + 3*x]*Sqrt[3 + 5
*x]) - (36968*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(21*Sqrt[3 + 5*x]) + (36968*Sqrt[11/3
]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/35 + (1112*Sqrt[11/3]*Ellip
ticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/35

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Rubi in Sympy [A]  time = 40.3079, size = 172, normalized size = 0.9 \[ - \frac{36968 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{35 \sqrt{3 x + 2}} - \frac{532 \sqrt{- 2 x + 1}}{3 \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{44 \sqrt{- 2 x + 1}}{5 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}} + \frac{14 \sqrt{- 2 x + 1}}{15 \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}} + \frac{36968 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{105} + \frac{12232 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1225} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-36968*sqrt(-2*x + 1)*sqrt(5*x + 3)/(35*sqrt(3*x + 2)) - 532*sqrt(-2*x + 1)/(3*s
qrt(3*x + 2)*sqrt(5*x + 3)) + 44*sqrt(-2*x + 1)/(5*(3*x + 2)**(3/2)*sqrt(5*x + 3
)) + 14*sqrt(-2*x + 1)/(15*(3*x + 2)**(5/2)*sqrt(5*x + 3)) + 36968*sqrt(33)*elli
ptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/105 + 12232*sqrt(35)*elliptic_f(a
sin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/1225

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Mathematica [A]  time = 0.235731, size = 105, normalized size = 0.55 \[ \frac{2}{105} \left (-\frac{3 \sqrt{1-2 x} \left (831780 x^3+1636038 x^2+1071882 x+233897\right )}{(3 x+2)^{5/2} \sqrt{5 x+3}}-2 \sqrt{2} \left (9242 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-4655 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((-3*Sqrt[1 - 2*x]*(233897 + 1071882*x + 1636038*x^2 + 831780*x^3))/((2 + 3*x
)^(5/2)*Sqrt[3 + 5*x]) - 2*Sqrt[2]*(9242*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]], -33/2] - 4655*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/105

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Maple [C]  time = 0.034, size = 386, normalized size = 2. \[ -{\frac{2}{1050\,{x}^{2}+105\,x-315}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 83790\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-166356\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+111720\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-221808\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+37240\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -73936\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +4990680\,{x}^{4}+7320888\,{x}^{3}+1523178\,{x}^{2}-1812264\,x-701691 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(83790*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)-166356*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)+111720*
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)-221808*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)+37240*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))
-73936*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))+4990680*x^4+7320888*x^3+1
523178*x^2-1812264*x-701691)/(2+3*x)^(5/2)/(10*x^2+x-3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-2*x + 1)^(3/2)/((135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*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)**(3/2)/(2+3*x)**(7/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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