3.2654 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac{118 \sqrt{1-2 x} (5 x+3)^{3/2}}{2079 (3 x+2)^{9/2}}+\frac{1305025844 \sqrt{1-2 x} \sqrt{5 x+3}}{524126295 \sqrt{3 x+2}}+\frac{19417096 \sqrt{1-2 x} \sqrt{5 x+3}}{74875185 (3 x+2)^{3/2}}+\frac{627806 \sqrt{1-2 x} \sqrt{5 x+3}}{10696455 (3 x+2)^{5/2}}-\frac{13022 \sqrt{1-2 x} \sqrt{5 x+3}}{305613 (3 x+2)^{7/2}}-\frac{37904696 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845 \sqrt{33}}-\frac{1305025844 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845 \sqrt{33}} \]

[Out]

(-13022*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(305613*(2 + 3*x)^(7/2)) + (627806*Sqrt[1 -
2*x]*Sqrt[3 + 5*x])/(10696455*(2 + 3*x)^(5/2)) + (19417096*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(74875185*(2 + 3*x)^(3/2)) + (1305025844*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(
524126295*Sqrt[2 + 3*x]) - (118*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2079*(2 + 3*x)^(
9/2)) - (2*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(33*(2 + 3*x)^(11/2)) - (1305025844*El
lipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(47647845*Sqrt[33]) - (37904696
*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(47647845*Sqrt[33])

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Rubi [A]  time = 0.593664, 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 \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac{118 \sqrt{1-2 x} (5 x+3)^{3/2}}{2079 (3 x+2)^{9/2}}+\frac{1305025844 \sqrt{1-2 x} \sqrt{5 x+3}}{524126295 \sqrt{3 x+2}}+\frac{19417096 \sqrt{1-2 x} \sqrt{5 x+3}}{74875185 (3 x+2)^{3/2}}+\frac{627806 \sqrt{1-2 x} \sqrt{5 x+3}}{10696455 (3 x+2)^{5/2}}-\frac{13022 \sqrt{1-2 x} \sqrt{5 x+3}}{305613 (3 x+2)^{7/2}}-\frac{37904696 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845 \sqrt{33}}-\frac{1305025844 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-13022*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(305613*(2 + 3*x)^(7/2)) + (627806*Sqrt[1 -
2*x]*Sqrt[3 + 5*x])/(10696455*(2 + 3*x)^(5/2)) + (19417096*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(74875185*(2 + 3*x)^(3/2)) + (1305025844*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(
524126295*Sqrt[2 + 3*x]) - (118*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2079*(2 + 3*x)^(
9/2)) - (2*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(33*(2 + 3*x)^(11/2)) - (1305025844*El
lipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(47647845*Sqrt[33]) - (37904696
*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(47647845*Sqrt[33])

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Rubi in Sympy [A]  time = 53.0374, size = 230, normalized size = 0.92 \[ \frac{1305025844 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{524126295 \sqrt{3 x + 2}} + \frac{19417096 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{74875185 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{627806 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{10696455 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{13022 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{305613 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{118 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{2079 \left (3 x + 2\right )^{\frac{9}{2}}} - \frac{2 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{33 \left (3 x + 2\right )^{\frac{11}{2}}} - \frac{1305025844 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1572378885} - \frac{37904696 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1572378885} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1305025844*sqrt(-2*x + 1)*sqrt(5*x + 3)/(524126295*sqrt(3*x + 2)) + 19417096*sqr
t(-2*x + 1)*sqrt(5*x + 3)/(74875185*(3*x + 2)**(3/2)) + 627806*sqrt(-2*x + 1)*sq
rt(5*x + 3)/(10696455*(3*x + 2)**(5/2)) - 13022*sqrt(-2*x + 1)*sqrt(5*x + 3)/(30
5613*(3*x + 2)**(7/2)) - 118*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/(2079*(3*x + 2)**(9
/2)) - 2*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/(33*(3*x + 2)**(11/2)) - 1305025844*sqr
t(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1572378885 - 37904696*s
qrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1572378885

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Mathematica [A]  time = 0.485797, size = 112, normalized size = 0.45 \[ \frac{\frac{48 \sqrt{2-4 x} \sqrt{5 x+3} \left (158560640046 x^5+534040213536 x^4+719808574005 x^3+484598540169 x^2+162787885893 x+21813966691\right )}{(3 x+2)^{11/2}}-10873573760 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{12579031080 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(21813966691 + 162787885893*x + 484598540169*x^
2 + 719808574005*x^3 + 534040213536*x^4 + 158560640046*x^5))/(2 + 3*x)^(11/2) +
20880413504*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 10873573760*Ell
ipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(12579031080*Sqrt[2])

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Maple [C]  time = 0.058, 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)*(1-2*x)^(1/2)/(2+3*x)^(13/2),x)

[Out]

2/1572378885*(82571200740*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)-15
8560640046*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)-528535466820*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)-704713955760*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)-469809303840
*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+815518032
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)-156603101280*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+10873573760*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))-20880413504*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+11891020005261*x^4-140844968748*x^3-3218604203
112*x^2-1399649072964*x-196325700219)*(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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((25*x^2 + 30*x + 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((729*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((3+5*x)**(5/2)*(1-2*x)**(1/2)/(2+3*x)**(13/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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