3.2757 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{15/2}} \, dx\)

Optimal. Leaf size=280 \[ -\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}+\frac{230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{1287 (3 x+2)^{11/2}}+\frac{1300 (5 x+3)^{3/2} \sqrt{1-2 x}}{891 (3 x+2)^{9/2}}+\frac{75041008472 \sqrt{5 x+3} \sqrt{1-2 x}}{584026443 \sqrt{3 x+2}}+\frac{1079936248 \sqrt{5 x+3} \sqrt{1-2 x}}{83432349 (3 x+2)^{3/2}}+\frac{23210828 \sqrt{5 x+3} \sqrt{1-2 x}}{11918907 (3 x+2)^{5/2}}-\frac{3347620 \sqrt{5 x+3} \sqrt{1-2 x}}{1702701 (3 x+2)^{7/2}}-\frac{2257166048 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{53093313 \sqrt{33}}-\frac{75041008472 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{53093313 \sqrt{33}} \]

[Out]

(-3347620*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1702701*(2 + 3*x)^(7/2)) + (23210828*Sqr
t[1 - 2*x]*Sqrt[3 + 5*x])/(11918907*(2 + 3*x)^(5/2)) + (1079936248*Sqrt[1 - 2*x]
*Sqrt[3 + 5*x])/(83432349*(2 + 3*x)^(3/2)) + (75041008472*Sqrt[1 - 2*x]*Sqrt[3 +
5*x])/(584026443*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(39*(2 +
3*x)^(13/2)) + (230*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(1287*(2 + 3*x)^(11/2)) + (
1300*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(891*(2 + 3*x)^(9/2)) - (75041008472*Ellipti
cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(53093313*Sqrt[33]) - (2257166048*El
lipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(53093313*Sqrt[33])

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Rubi [A]  time = 0.671497, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}+\frac{230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{1287 (3 x+2)^{11/2}}+\frac{1300 (5 x+3)^{3/2} \sqrt{1-2 x}}{891 (3 x+2)^{9/2}}+\frac{75041008472 \sqrt{5 x+3} \sqrt{1-2 x}}{584026443 \sqrt{3 x+2}}+\frac{1079936248 \sqrt{5 x+3} \sqrt{1-2 x}}{83432349 (3 x+2)^{3/2}}+\frac{23210828 \sqrt{5 x+3} \sqrt{1-2 x}}{11918907 (3 x+2)^{5/2}}-\frac{3347620 \sqrt{5 x+3} \sqrt{1-2 x}}{1702701 (3 x+2)^{7/2}}-\frac{2257166048 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{53093313 \sqrt{33}}-\frac{75041008472 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{53093313 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^(15/2),x]

[Out]

(-3347620*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1702701*(2 + 3*x)^(7/2)) + (23210828*Sqr
t[1 - 2*x]*Sqrt[3 + 5*x])/(11918907*(2 + 3*x)^(5/2)) + (1079936248*Sqrt[1 - 2*x]
*Sqrt[3 + 5*x])/(83432349*(2 + 3*x)^(3/2)) + (75041008472*Sqrt[1 - 2*x]*Sqrt[3 +
5*x])/(584026443*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(39*(2 +
3*x)^(13/2)) + (230*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(1287*(2 + 3*x)^(11/2)) + (
1300*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(891*(2 + 3*x)^(9/2)) - (75041008472*Ellipti
cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(53093313*Sqrt[33]) - (2257166048*El
lipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(53093313*Sqrt[33])

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Rubi in Sympy [A]  time = 66.4313, size = 258, normalized size = 0.92 \[ - \frac{230 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{9009 \left (3 x + 2\right )^{\frac{11}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{39 \left (3 x + 2\right )^{\frac{13}{2}}} + \frac{5870 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{81081 \left (3 x + 2\right )^{\frac{9}{2}}} + \frac{75041008472 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{584026443 \sqrt{3 x + 2}} + \frac{1079936248 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{83432349 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{23210828 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{11918907 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{556280 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1702701 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{75041008472 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1752079329} - \frac{2257166048 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1858265955} \]

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)**(15/2),x)

[Out]

-230*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(9009*(3*x + 2)**(11/2)) - 2*(-2*x + 1)**(5
/2)*(5*x + 3)**(3/2)/(39*(3*x + 2)**(13/2)) + 5870*(-2*x + 1)**(3/2)*sqrt(5*x +
3)/(81081*(3*x + 2)**(9/2)) + 75041008472*sqrt(-2*x + 1)*sqrt(5*x + 3)/(58402644
3*sqrt(3*x + 2)) + 1079936248*sqrt(-2*x + 1)*sqrt(5*x + 3)/(83432349*(3*x + 2)**
(3/2)) + 23210828*sqrt(-2*x + 1)*sqrt(5*x + 3)/(11918907*(3*x + 2)**(5/2)) + 556
280*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1702701*(3*x + 2)**(7/2)) - 75041008472*sqrt(3
3)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1752079329 - 2257166048*sq
rt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/1858265955

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Mathematica [A]  time = 0.445073, size = 117, normalized size = 0.42 \[ \frac{\frac{48 \sqrt{2-4 x} \sqrt{5 x+3} \left (27352447588044 x^6+110328276131100 x^5+185457331738206 x^4+166295375376786 x^3+83893544414217 x^2+22577209892436 x+2532151719515\right )}{(3 x+2)^{13/2}}-604764298880 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{14016634632 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^(15/2),x]

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(2532151719515 + 22577209892436*x + 83893544414
217*x^2 + 166295375376786*x^3 + 185457331738206*x^4 + 110328276131100*x^5 + 2735
2447588044*x^6))/(2 + 3*x)^(13/2) + 1200656135552*EllipticE[ArcSin[Sqrt[2/11]*Sq
rt[3 + 5*x]], -33/2] - 604764298880*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]],
-33/2])/(14016634632*Sqrt[2])

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Maple [C]  time = 0.032, size = 862, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^(15/2),x)

[Out]

2/1752079329*(13777286683860*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^6*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)
-27352447588044*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^6*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]
40*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)-109409790352176*2^(1/2)*E
llipticE(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)-182349650586960*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^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)-162088578299520*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)+8205734
27641320*x^8+40821590174400*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)-
81044289149760*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]
132*x^7+10885757379840*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)-2161181
0439936*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)+5648532752247084*x^6+1
209528597760*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))-2401312271104*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))+4552278771338298*x^5+1346576472913014
*x^4-567661448375343*x^3-611345718465195*x^2-195598433873379*x-22789365475635)*(
3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(13/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{15}{2}}}\,{d x} \]

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)^(15/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(15/2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \sqrt{3 \, x + 2}}, x\right ) \]

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)^(15/2),x, algorithm="fricas")

[Out]

integral((20*x^3 - 8*x^2 - 7*x + 3)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((2187*x^7 + 10
206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*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)**(15/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{15}{2}}}\,{d x} \]

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)^(15/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(15/2), x)