3.2766 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx\)

Optimal. Leaf size=249 \[ \frac{36980 \sqrt{1-2 x} (5 x+3)^{5/2}}{18711 (3 x+2)^{7/2}}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{891 (3 x+2)^{9/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac{55772 \sqrt{1-2 x} (5 x+3)^{3/2}}{43659 (3 x+2)^{5/2}}+\frac{584888452 \sqrt{1-2 x} \sqrt{5 x+3}}{57760857 \sqrt{3 x+2}}-\frac{17089252 \sqrt{1-2 x} \sqrt{5 x+3}}{8251551 (3 x+2)^{3/2}}-\frac{13235368 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5250987 \sqrt{33}}-\frac{584888452 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5250987 \sqrt{33}} \]

[Out]

(-17089252*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8251551*(2 + 3*x)^(3/2)) + (584888452*S
qrt[1 - 2*x]*Sqrt[3 + 5*x])/(57760857*Sqrt[2 + 3*x]) - (55772*Sqrt[1 - 2*x]*(3 +
5*x)^(3/2))/(43659*(2 + 3*x)^(5/2)) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(33*(
2 + 3*x)^(11/2)) + (370*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(891*(2 + 3*x)^(9/2)) +
(36980*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(18711*(2 + 3*x)^(7/2)) - (584888452*Elli
pticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5250987*Sqrt[33]) - (13235368*El
lipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5250987*Sqrt[33])

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Rubi [A]  time = 0.588197, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{36980 \sqrt{1-2 x} (5 x+3)^{5/2}}{18711 (3 x+2)^{7/2}}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{891 (3 x+2)^{9/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac{55772 \sqrt{1-2 x} (5 x+3)^{3/2}}{43659 (3 x+2)^{5/2}}+\frac{584888452 \sqrt{1-2 x} \sqrt{5 x+3}}{57760857 \sqrt{3 x+2}}-\frac{17089252 \sqrt{1-2 x} \sqrt{5 x+3}}{8251551 (3 x+2)^{3/2}}-\frac{13235368 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5250987 \sqrt{33}}-\frac{584888452 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5250987 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(13/2),x]

[Out]

(-17089252*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8251551*(2 + 3*x)^(3/2)) + (584888452*S
qrt[1 - 2*x]*Sqrt[3 + 5*x])/(57760857*Sqrt[2 + 3*x]) - (55772*Sqrt[1 - 2*x]*(3 +
5*x)^(3/2))/(43659*(2 + 3*x)^(5/2)) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(33*(
2 + 3*x)^(11/2)) + (370*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(891*(2 + 3*x)^(9/2)) +
(36980*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(18711*(2 + 3*x)^(7/2)) - (584888452*Elli
pticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5250987*Sqrt[33]) - (13235368*El
lipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5250987*Sqrt[33])

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Rubi in Sympy [A]  time = 58.0702, size = 230, normalized size = 0.92 \[ - \frac{48490 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{916839 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{370 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{6237 \left (3 x + 2\right )^{\frac{9}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{33 \left (3 x + 2\right )^{\frac{11}{2}}} + \frac{293926 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{916839 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{584888452 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{57760857 \sqrt{3 x + 2}} + \frac{6617684 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{8251551 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{584888452 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{173282571} - \frac{13235368 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{183784545} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**(13/2),x)

[Out]

-48490*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(916839*(3*x + 2)**(7/2)) - 370*(-2*x + 1
)**(5/2)*(5*x + 3)**(3/2)/(6237*(3*x + 2)**(9/2)) - 2*(-2*x + 1)**(5/2)*(5*x + 3
)**(5/2)/(33*(3*x + 2)**(11/2)) + 293926*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(916839
*(3*x + 2)**(5/2)) + 584888452*sqrt(-2*x + 1)*sqrt(5*x + 3)/(57760857*sqrt(3*x +
2)) + 6617684*sqrt(-2*x + 1)*sqrt(5*x + 3)/(8251551*(3*x + 2)**(3/2)) - 5848884
52*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/173282571 - 13235
368*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/183784545

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Mathematica [A]  time = 0.558912, size = 112, normalized size = 0.45 \[ \frac{\frac{48 \sqrt{2-4 x} \sqrt{5 x+3} \left (71063946918 x^5+237923150688 x^4+320012032635 x^3+215597947743 x^2+72620507583 x+9770732477\right )}{(3 x+2)^{11/2}}-5864078080 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1386260568 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(13/2),x]

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(9770732477 + 72620507583*x + 215597947743*x^2
+ 320012032635*x^3 + 237923150688*x^4 + 71063946918*x^5))/(2 + 3*x)^(11/2) + 935
8215232*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5864078080*Elliptic
F[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(1386260568*Sqrt[2])

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Maple [C]  time = 0.031, size = 743, normalized size = 3. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(13/2),x)

[Out]

2/173282571*(44530342920*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^5*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-710
63946918*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^5*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*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^4*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-236879823060*2^(1/2)*EllipticE(1/1
1*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x^4*(3+5*x)^(1/
2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*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)-315839764080*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)
[phone]*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)-210559842720*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)[phone]*x^7+43980585600
*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)-70186614240*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)[phone]*x^6+5864078080*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))-9358215232*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+5286666174003*x^4-54699222996*x^3-1429398032628*x^2-6
24272370816*x-87936592293)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(11/
2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(13/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(13/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(13/2),x, algorithm="fricas")

[Out]

integral((100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((72
9*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*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)**(5/2)/(2+3*x)**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(13/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(13/2), x)