3.2733 \(\int \frac{(1-2 x)^{3/2}}{\sqrt{2+3 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=127 \[ \frac{148 \sqrt{1-2 x} \sqrt{3 x+2}}{15 \sqrt{5 x+3}}-\frac{22 \sqrt{1-2 x} \sqrt{3 x+2}}{15 (5 x+3)^{3/2}}-\frac{52 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25 \sqrt{33}}-\frac{148}{25} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-22*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(15*(3 + 5*x)^(3/2)) + (148*Sqrt[1 - 2*x]*Sqrt
[2 + 3*x])/(15*Sqrt[3 + 5*x]) - (148*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[
1 - 2*x]], 35/33])/25 - (52*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(
25*Sqrt[33])

_______________________________________________________________________________________

Rubi [A]  time = 0.263025, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{148 \sqrt{1-2 x} \sqrt{3 x+2}}{15 \sqrt{5 x+3}}-\frac{22 \sqrt{1-2 x} \sqrt{3 x+2}}{15 (5 x+3)^{3/2}}-\frac{52 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25 \sqrt{33}}-\frac{148}{25} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-22*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(15*(3 + 5*x)^(3/2)) + (148*Sqrt[1 - 2*x]*Sqrt
[2 + 3*x])/(15*Sqrt[3 + 5*x]) - (148*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[
1 - 2*x]], 35/33])/25 - (52*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(
25*Sqrt[33])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 26.0014, size = 114, normalized size = 0.9 \[ \frac{148 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{15 \sqrt{5 x + 3}} - \frac{22 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{148 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{75} - \frac{52 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

148*sqrt(-2*x + 1)*sqrt(3*x + 2)/(15*sqrt(5*x + 3)) - 22*sqrt(-2*x + 1)*sqrt(3*x
+ 2)/(15*(5*x + 3)**(3/2)) - 148*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x +
1)/7), 35/33)/75 - 52*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/3
5)/875

_______________________________________________________________________________________

Mathematica [A]  time = 0.368846, size = 97, normalized size = 0.76 \[ \frac{2}{75} \left (\frac{5 \sqrt{1-2 x} \sqrt{3 x+2} (370 x+211)}{(5 x+3)^{3/2}}-35 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+74 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((5*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(211 + 370*x))/(3 + 5*x)^(3/2) + 74*Sqrt[2]*E
llipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 35*Sqrt[2]*EllipticF[ArcSin[
Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/75

_______________________________________________________________________________________

Maple [C]  time = 0.03, size = 267, normalized size = 2.1 \[{\frac{2}{450\,{x}^{2}+75\,x-150} \left ( 175\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-370\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+105\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -222\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +11100\,{x}^{3}+8180\,{x}^{2}-2645\,x-2110 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/75*(175*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)-370*2^(1/2)*Elliptic
E(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)+105*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))-222*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))+11100*x^3+8180*x^2-264
5*x-2110)*(2+3*x)^(1/2)*(1-2*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-2*x + 1)^(3/2)/((25*x^2 + 30*x + 9)*sqrt(5*x + 3)*sqrt(3*x + 2)), x)

_______________________________________________________________________________________

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)/(3+5*x)**(5/2)/(2+3*x)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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