3.2780 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{8}{45} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{1972 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{4725}-\frac{2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{5 \sqrt{5 x+3}}+\frac{167228 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{118125}+\frac{196499 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{590625}-\frac{299863 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2953125}-\frac{1509007 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2953125} \]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(5/2))/(5*Sqrt[3 + 5*x]) + (196499*Sqrt[1 - 2*x]*S
qrt[2 + 3*x]*Sqrt[3 + 5*x])/590625 + (167228*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[
3 + 5*x])/118125 - (1972*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/4725 - (8*
(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/45 - (1509007*Sqrt[11/3]*Elliptic
E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2953125 - (299863*Sqrt[11/3]*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2953125

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Rubi [A]  time = 0.495084, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{8}{45} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{1972 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{4725}-\frac{2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{5 \sqrt{5 x+3}}+\frac{167228 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{118125}+\frac{196499 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{590625}-\frac{299863 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2953125}-\frac{1509007 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2953125} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(5/2))/(5*Sqrt[3 + 5*x]) + (196499*Sqrt[1 - 2*x]*S
qrt[2 + 3*x]*Sqrt[3 + 5*x])/590625 + (167228*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[
3 + 5*x])/118125 - (1972*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/4725 - (8*
(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/45 - (1509007*Sqrt[11/3]*Elliptic
E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2953125 - (299863*Sqrt[11/3]*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2953125

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Rubi in Sympy [A]  time = 51.1868, size = 201, normalized size = 0.91 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{5}{2}}}{5 \sqrt{5 x + 3}} - \frac{8 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{45} + \frac{986 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{1575} + \frac{887 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{13125} + \frac{103364 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{590625} - \frac{1509007 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{8859375} - \frac{3298493 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{103359375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(5/2)*(3*x + 2)**(5/2)/(5*sqrt(5*x + 3)) - 8*(-2*x + 1)**(3/2)*(3
*x + 2)**(5/2)*sqrt(5*x + 3)/45 + 986*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2)*sqrt(5*
x + 3)/1575 + 887*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*sqrt(5*x + 3)/13125 + 103364*s
qrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/590625 - 1509007*sqrt(33)*elliptic_e(a
sin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/8859375 - 3298493*sqrt(35)*elliptic_f(asi
n(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/103359375

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Mathematica [A]  time = 0.454914, size = 112, normalized size = 0.5 \[ \frac{\frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \left (945000 x^4-382500 x^3-844650 x^2+650155 x+443337\right )}{\sqrt{5 x+3}}+6877465 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+3018014 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{17718750} \]

Antiderivative was successfully verified.

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(443337 + 650155*x - 844650*x^2 - 382500*x^3 +
945000*x^4))/Sqrt[3 + 5*x] + 3018014*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]], -33/2] + 6877465*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -3
3/2])/17718750

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Maple [C]  time = 0.026, size = 179, normalized size = 0.8 \[ -{\frac{1}{531562500\,{x}^{3}[phone]\,{x}^{2}-124031250\,x-106312500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -170100000\,{x}^{6}+6877465\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +3018014\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}-114638400\,{x}^{3}-149984310\,{x}^{2}[phone]\,x+26600220 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/17718750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-170100000*x^6+6877465*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))+3018014*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+220212000*x^4-114638400*x^3-149984310*x^2+257
09190*x+26600220)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*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)**(5/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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