3.2754 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{9/2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^{7/2}}+\frac{46 (5 x+3)^{3/2} (1-2 x)^{3/2}}{63 (3 x+2)^{5/2}}+\frac{608 (5 x+3)^{3/2} \sqrt{1-2 x}}{189 (3 x+2)^{3/2}}-\frac{4244 \sqrt{5 x+3} \sqrt{1-2 x}}{3969 \sqrt{3 x+2}}-\frac{4244 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3969}-\frac{11576 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3969} \]

[Out]

(-4244*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3969*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(5/2)*(3
+ 5*x)^(3/2))/(21*(2 + 3*x)^(7/2)) + (46*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(63*(
2 + 3*x)^(5/2)) + (608*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(189*(2 + 3*x)^(3/2)) - (1
1576*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3969 - (4244*
Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3969

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Rubi [A]  time = 0.41118, 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 \[ -\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^{7/2}}+\frac{46 (5 x+3)^{3/2} (1-2 x)^{3/2}}{63 (3 x+2)^{5/2}}+\frac{608 (5 x+3)^{3/2} \sqrt{1-2 x}}{189 (3 x+2)^{3/2}}-\frac{4244 \sqrt{5 x+3} \sqrt{1-2 x}}{3969 \sqrt{3 x+2}}-\frac{4244 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3969}-\frac{11576 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3969} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^(9/2),x]

[Out]

(-4244*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3969*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(5/2)*(3
+ 5*x)^(3/2))/(21*(2 + 3*x)^(7/2)) + (46*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(63*(
2 + 3*x)^(5/2)) + (608*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(189*(2 + 3*x)^(3/2)) - (1
1576*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3969 - (4244*
Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3969

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Rubi in Sympy [A]  time = 38.6479, size = 172, normalized size = 0.9 \[ - \frac{46 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{441 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{21 \left (3 x + 2\right )^{\frac{7}{2}}} + \frac{130 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{189 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{2260 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{567 \sqrt{3 x + 2}} - \frac{11576 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{11907} - \frac{46684 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{138915} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(3+5*x)**(3/2)/(2+3*x)**(9/2),x)

[Out]

-46*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(441*(3*x + 2)**(5/2)) - 2*(-2*x + 1)**(5/2)
*(5*x + 3)**(3/2)/(21*(3*x + 2)**(7/2)) + 130*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(1
89*(3*x + 2)**(3/2)) + 2260*sqrt(-2*x + 1)*sqrt(5*x + 3)/(567*sqrt(3*x + 2)) - 1
1576*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/11907 - 46684*s
qrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/138915

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Mathematica [A]  time = 0.286258, size = 104, normalized size = 0.54 \[ \frac{2 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (182736 x^3+409005 x^2+292578 x+67759\right )}{(3 x+2)^{7/2}}+\sqrt{2} \left (29225 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+5788 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{11907} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^(9/2),x]

[Out]

(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(67759 + 292578*x + 409005*x^2 + 182736*x^3))
/(2 + 3*x)^(7/2) + Sqrt[2]*(5788*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33
/2] + 29225*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/11907

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Maple [C]  time = 0.029, size = 505, normalized size = 2.6 \[ -{\frac{2}{119070\,{x}^{2}+11907\,x-35721} \left ( 789075\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}+156276\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}+1578150\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+312552\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1052100\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+208368\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+233800\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +46304\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -5482080\,{x}^{5}-12818358\,{x}^{4}-8359731\,{x}^{3}+770541\,{x}^{2}+2429925\,x+609831 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^(9/2),x)

[Out]

-2/11907*(789075*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)+156276*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)+1578150*2^(1/2)*EllipticF(1/11*1
1^(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)+312552*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)+1052100*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)+208368*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)+233800*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))+46304*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellipti
cE(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))-5482080*x
^5-12818358*x^4-8359731*x^3+770541*x^2+2429925*x+609831)*(3+5*x)^(1/2)*(1-2*x)^(
1/2)/(10*x^2+x-3)/(2+3*x)^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(9/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(9/2),x, algorithm="fricas")

[Out]

integral((20*x^3 - 8*x^2 - 7*x + 3)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((81*x^4 + 216*
x^3 + 216*x^2 + 96*x + 16)*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)*(3+5*x)**(3/2)/(2+3*x)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(9/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(9/2), x)