3.2823 \(\int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^{13/2}} \, dx\)

Optimal. Leaf size=249 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}+\frac{924247516 \sqrt{1-2 x} \sqrt{5 x+3}}{733776813 \sqrt{3 x+2}}+\frac{11460644 \sqrt{1-2 x} \sqrt{5 x+3}}{104825259 (3 x+2)^{3/2}}-\frac{362666 \sqrt{1-2 x} \sqrt{5 x+3}}{14975037 (3 x+2)^{5/2}}-\frac{251590 \sqrt{1-2 x} \sqrt{5 x+3}}{2139291 (3 x+2)^{7/2}}+\frac{940 \sqrt{1-2 x} \sqrt{5 x+3}}{43659 (3 x+2)^{9/2}}-\frac{31704544 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{66706983 \sqrt{33}}-\frac{924247516 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{66706983 \sqrt{33}} \]

[Out]

(940*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43659*(2 + 3*x)^(9/2)) - (251590*Sqrt[1 - 2*x
]*Sqrt[3 + 5*x])/(2139291*(2 + 3*x)^(7/2)) - (362666*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]
)/(14975037*(2 + 3*x)^(5/2)) + (11460644*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(104825259
*(2 + 3*x)^(3/2)) + (924247516*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(733776813*Sqrt[2 +
3*x]) + (2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(231*(2 + 3*x)^(11/2)) - (924247516*El
lipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(66706983*Sqrt[33]) - (31704544
*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(66706983*Sqrt[33])

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Rubi [A]  time = 0.587936, 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 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}+\frac{924247516 \sqrt{1-2 x} \sqrt{5 x+3}}{733776813 \sqrt{3 x+2}}+\frac{11460644 \sqrt{1-2 x} \sqrt{5 x+3}}{104825259 (3 x+2)^{3/2}}-\frac{362666 \sqrt{1-2 x} \sqrt{5 x+3}}{14975037 (3 x+2)^{5/2}}-\frac{251590 \sqrt{1-2 x} \sqrt{5 x+3}}{2139291 (3 x+2)^{7/2}}+\frac{940 \sqrt{1-2 x} \sqrt{5 x+3}}{43659 (3 x+2)^{9/2}}-\frac{31704544 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{66706983 \sqrt{33}}-\frac{924247516 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{66706983 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(940*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43659*(2 + 3*x)^(9/2)) - (251590*Sqrt[1 - 2*x
]*Sqrt[3 + 5*x])/(2139291*(2 + 3*x)^(7/2)) - (362666*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]
)/(14975037*(2 + 3*x)^(5/2)) + (11460644*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(104825259
*(2 + 3*x)^(3/2)) + (924247516*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(733776813*Sqrt[2 +
3*x]) + (2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(231*(2 + 3*x)^(11/2)) - (924247516*El
lipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(66706983*Sqrt[33]) - (31704544
*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(66706983*Sqrt[33])

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Rubi in Sympy [A]  time = 57.0011, size = 230, normalized size = 0.92 \[ \frac{924247516 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{733776813 \sqrt{3 x + 2}} + \frac{11460644 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{104825259 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{362666 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{14975037 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{251590 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2139291 \left (3 x + 2\right )^{\frac{7}{2}}} + \frac{940 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{43659 \left (3 x + 2\right )^{\frac{9}{2}}} + \frac{2 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{231 \left (3 x + 2\right )^{\frac{11}{2}}} - \frac{924247516 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2201330439} - \frac{31704544 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{2334744405} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

924247516*sqrt(-2*x + 1)*sqrt(5*x + 3)/(733776813*sqrt(3*x + 2)) + 11460644*sqrt
(-2*x + 1)*sqrt(5*x + 3)/(104825259*(3*x + 2)**(3/2)) - 362666*sqrt(-2*x + 1)*sq
rt(5*x + 3)/(14975037*(3*x + 2)**(5/2)) - 251590*sqrt(-2*x + 1)*sqrt(5*x + 3)/(2
139291*(3*x + 2)**(7/2)) + 940*sqrt(-2*x + 1)*sqrt(5*x + 3)/(43659*(3*x + 2)**(9
/2)) + 2*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/(231*(3*x + 2)**(11/2)) - 924247516*sqr
t(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/2201330439 - 31704544*s
qrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/2334744405

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Mathematica [A]  time = 0.519498, size = 112, normalized size = 0.45 \[ \frac{\frac{48 \sqrt{2-4 x} \sqrt{5 x+3} \left (112296073194 x^5+377569336554 x^4+507518001945 x^3+340525216341 x^2+113962415157 x+15211411193\right )}{(3 x+2)^{11/2}}-6417960640 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{17610643512 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(15211411193 + 113962415157*x + 340525216341*x^
2 + 507518001945*x^3 + 377569336554*x^4 + 112296073194*x^5))/(2 + 3*x)^(11/2) +
14787960256*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 6417960640*Elli
pticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(17610643512*Sqrt[2])

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Maple [C]  time = 0.034, size = 743, normalized size = 3. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/2201330439*(48736388610*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)-11
2296073194*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)-374320243980*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)-499093658640*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)-332729105760
*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+481347048
00*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)-110909701920*2^(1/2)*Ellipt
icE(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+6417960640*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))-14787960256*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+8340186467079*x^4-127213913772*x^3-226649736580
8*x^2-980027502834*x-136902700737)*(1-2*x)^(1/2)*(3+5*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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{13}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}{{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((25*x^2 + 30*x + 9)*sqrt(5*x + 3)/((729*x^6 + 2916*x^5 + 4860*x^4 + 432
0*x^3 + 2160*x^2 + 576*x + 64)*sqrt(3*x + 2)*sqrt(-2*x + 1)), 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((3+5*x)**(5/2)/(2+3*x)**(13/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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