3.2779 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac{48}{275} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{7/2}-\frac{2972 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{7/2}}{7425}-\frac{2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt{5 x+3}}+\frac{346636 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{259875}+\frac{2020841 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{6496875}-\frac{703672 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{32484375}-\frac{7261561 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{14765625 \sqrt{33}}-\frac{264260033 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{29531250 \sqrt{33}} \]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(7/2))/(5*Sqrt[3 + 5*x]) - (703672*Sqrt[1 - 2*x]*S
qrt[2 + 3*x]*Sqrt[3 + 5*x])/32484375 + (2020841*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sq
rt[3 + 5*x])/6496875 + (346636*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/2598
75 - (2972*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/7425 - (48*(1 - 2*x)^(3/
2)*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/275 - (264260033*EllipticE[ArcSin[Sqrt[3/7]*Sq
rt[1 - 2*x]], 35/33])/(29531250*Sqrt[33]) - (7261561*EllipticF[ArcSin[Sqrt[3/7]*
Sqrt[1 - 2*x]], 35/33])/(14765625*Sqrt[33])

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Rubi [A]  time = 0.57652, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{48}{275} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{7/2}-\frac{2972 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{7/2}}{7425}-\frac{2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt{5 x+3}}+\frac{346636 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{259875}+\frac{2020841 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{6496875}-\frac{703672 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{32484375}-\frac{7261561 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{14765625 \sqrt{33}}-\frac{264260033 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{29531250 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(7/2))/(5*Sqrt[3 + 5*x]) - (703672*Sqrt[1 - 2*x]*S
qrt[2 + 3*x]*Sqrt[3 + 5*x])/32484375 + (2020841*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sq
rt[3 + 5*x])/6496875 + (346636*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/2598
75 - (2972*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/7425 - (48*(1 - 2*x)^(3/
2)*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/275 - (264260033*EllipticE[ArcSin[Sqrt[3/7]*Sq
rt[1 - 2*x]], 35/33])/(29531250*Sqrt[33]) - (7261561*EllipticF[ArcSin[Sqrt[3/7]*
Sqrt[1 - 2*x]], 35/33])/(14765625*Sqrt[33])

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Rubi in Sympy [A]  time = 58.9925, size = 230, normalized size = 0.92 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{7}{2}}}{5 \sqrt{5 x + 3}} - \frac{48 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{275} + \frac{1486 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{2475} + \frac{8717 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{86625} - \frac{82561 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{721875} + \frac{7965233 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{32484375} - \frac{264260033 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{974531250} - \frac{7261561 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{487265625} \]

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)**(3/2),x)

[Out]

-2*(-2*x + 1)**(5/2)*(3*x + 2)**(7/2)/(5*sqrt(5*x + 3)) - 48*(-2*x + 1)**(3/2)*(
3*x + 2)**(7/2)*sqrt(5*x + 3)/275 + 1486*(-2*x + 1)**(3/2)*(3*x + 2)**(5/2)*sqrt
(5*x + 3)/2475 + 8717*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/86625 - 8
2561*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*sqrt(5*x + 3)/721875 + 7965233*sqrt(-2*x +
1)*sqrt(3*x + 2)*sqrt(5*x + 3)/32484375 - 264260033*sqrt(33)*elliptic_e(asin(sqr
t(21)*sqrt(-2*x + 1)/7), 35/33)/974531250 - 7261561*sqrt(33)*elliptic_f(asin(sqr
t(21)*sqrt(-2*x + 1)/7), 35/33)/487265625

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Mathematica [A]  time = 0.44615, size = 125, normalized size = 0.5 \[ \frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \left (127575000 x^5+56227500 x^4-141221250 x^3-32807925 x^2+71568535 x+26378214\right )-24628520 \sqrt{10 x+6} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] \sqrt{10 x+6} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{974531250 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(26378214 + 71568535*x - 32807925*x^2 - 14122125
0*x^3 + 56227500*x^4 + 127575000*x^5) + 264260033*Sqrt[6 + 10*x]*EllipticE[ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 24628520*Sqrt[6 + 10*x]*EllipticF[ArcSin[S
qrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(974531250*Sqrt[3 + 5*x])

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Maple [C]  time = 0.027, size = 184, normalized size = 0.7 \[{\frac{1}{29235937500\,{x}^{3}[phone]\,{x}^{2}-6821718750\,x-5847187500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 22963500000\,{x}^{7}[phone]\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -264260033\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -31387500000\,{x}^{5}-13515714000\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-3502765680\,x-1582692840 \right ) } \]

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)^(3/2),x)

[Out]

1/974531250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(22963500000*x^7+139482000
00*x^6+24628520*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))-264260033*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))-31387500000*x^5-13515714000*x^4+203713
73550*x^3+8863610070*x^2-3502765680*x-1582692840)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*sqrt(3*x + 2)*sqrt(-2*x
+ 1)/(5*x + 3)^(3/2), 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)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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