3.2768 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx\)

Optimal. Leaf size=311 \[ \frac{16636 \sqrt{1-2 x} (5 x+3)^{5/2}}{11583 (3 x+2)^{11/2}}+\frac{74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{351 (3 x+2)^{13/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}-\frac{1085156 \sqrt{1-2 x} (5 x+3)^{3/2}}{729729 (3 x+2)^{9/2}}+\frac{12641611554328 \sqrt{1-2 x} \sqrt{5 x+3}}{183968329545 \sqrt{3 x+2}}+\frac{181941877952 \sqrt{1-2 x} \sqrt{5 x+3}}{26281189935 (3 x+2)^{3/2}}+\frac{3914701972 \sqrt{1-2 x} \sqrt{5 x+3}}{3754455705 (3 x+2)^{5/2}}-\frac{112817764 \sqrt{1-2 x} \sqrt{5 x+3}}{107270163 (3 x+2)^{7/2}}-\frac{380220959152 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16724393595 \sqrt{33}}-\frac{12641611554328 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16724393595 \sqrt{33}} \]

[Out]

(-112817764*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(107270163*(2 + 3*x)^(7/2)) + (39147019
72*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3754455705*(2 + 3*x)^(5/2)) + (181941877952*Sqr
t[1 - 2*x]*Sqrt[3 + 5*x])/(26281189935*(2 + 3*x)^(3/2)) + (12641611554328*Sqrt[1
- 2*x]*Sqrt[3 + 5*x])/(183968329545*Sqrt[2 + 3*x]) - (1085156*Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/(729729*(2 + 3*x)^(9/2)) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(45
*(2 + 3*x)^(15/2)) + (74*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(351*(2 + 3*x)^(13/2))
+ (16636*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(11583*(2 + 3*x)^(11/2)) - (12641611554
328*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(16724393595*Sqrt[33]) -
(380220959152*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(16724393595*Sq
rt[33])

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Rubi [A]  time = 0.78065, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{16636 \sqrt{1-2 x} (5 x+3)^{5/2}}{11583 (3 x+2)^{11/2}}+\frac{74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{351 (3 x+2)^{13/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}-\frac{1085156 \sqrt{1-2 x} (5 x+3)^{3/2}}{729729 (3 x+2)^{9/2}}+\frac{12641611554328 \sqrt{1-2 x} \sqrt{5 x+3}}{183968329545 \sqrt{3 x+2}}+\frac{181941877952 \sqrt{1-2 x} \sqrt{5 x+3}}{26281189935 (3 x+2)^{3/2}}+\frac{3914701972 \sqrt{1-2 x} \sqrt{5 x+3}}{3754455705 (3 x+2)^{5/2}}-\frac{112817764 \sqrt{1-2 x} \sqrt{5 x+3}}{107270163 (3 x+2)^{7/2}}-\frac{380220959152 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16724393595 \sqrt{33}}-\frac{12641611554328 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16724393595 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(17/2),x]

[Out]

(-112817764*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(107270163*(2 + 3*x)^(7/2)) + (39147019
72*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3754455705*(2 + 3*x)^(5/2)) + (181941877952*Sqr
t[1 - 2*x]*Sqrt[3 + 5*x])/(26281189935*(2 + 3*x)^(3/2)) + (12641611554328*Sqrt[1
- 2*x]*Sqrt[3 + 5*x])/(183968329545*Sqrt[2 + 3*x]) - (1085156*Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/(729729*(2 + 3*x)^(9/2)) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(45
*(2 + 3*x)^(15/2)) + (74*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(351*(2 + 3*x)^(13/2))
+ (16636*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(11583*(2 + 3*x)^(11/2)) - (12641611554
328*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(16724393595*Sqrt[33]) -
(380220959152*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(16724393595*Sq
rt[33])

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Rubi in Sympy [A]  time = 74.9808, size = 287, normalized size = 0.92 \[ - \frac{10226 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{567567 \left (3 x + 2\right )^{\frac{11}{2}}} - \frac{74 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{2457 \left (3 x + 2\right )^{\frac{13}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{45 \left (3 x + 2\right )^{\frac{15}{2}}} + \frac{450566 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{5108103 \left (3 x + 2\right )^{\frac{9}{2}}} + \frac{12641611554328 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{183968329545 \sqrt{3 x + 2}} + \frac{181941877952 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{26281189935 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{3914701972 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3754455705 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{16959884 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{107270163 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{12641611554328 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{551904988635} - \frac{380220959152 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{551904988635} \]

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)**(17/2),x)

[Out]

-10226*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(567567*(3*x + 2)**(11/2)) - 74*(-2*x + 1
)**(5/2)*(5*x + 3)**(3/2)/(2457*(3*x + 2)**(13/2)) - 2*(-2*x + 1)**(5/2)*(5*x +
3)**(5/2)/(45*(3*x + 2)**(15/2)) + 450566*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(51081
03*(3*x + 2)**(9/2)) + 12641611554328*sqrt(-2*x + 1)*sqrt(5*x + 3)/(183968329545
*sqrt(3*x + 2)) + 181941877952*sqrt(-2*x + 1)*sqrt(5*x + 3)/(26281189935*(3*x +
2)**(3/2)) + 3914701972*sqrt(-2*x + 1)*sqrt(5*x + 3)/(3754455705*(3*x + 2)**(5/2
)) + 16959884*sqrt(-2*x + 1)*sqrt(5*x + 3)/(107270163*(3*x + 2)**(7/2)) - 126416
11554328*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/55190498863
5 - 380220959152*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/551
904988635

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Mathematica [A]  time = 0.501348, size = 122, normalized size = 0.39 \[ \frac{\frac{96 \sqrt{2-4 x} \sqrt{5 x+3} \left (13823602234657668 x^7+64974368463330312 x^6+130900492508039982 x^5+146528498784887100 x^4+98427465692862075 x^3+39676146370896231 x^2+8886579657279639 x+853124799464729\right )}{(3 x+2)^{15/2}}-203774903306240 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{8830479818160 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(17/2),x]

[Out]

((96*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(853124799464729 + 8886579657279639*x + 3967614
6370896231*x^2 + 98427465692862075*x^3 + 146528498784887100*x^4 + 13090049250803
9982*x^5 + 64974368463330312*x^6 + 13823602234657668*x^7))/(2 + 3*x)^(15/2) + 40
4531569738496*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2037749033062
40*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(8830479818160*Sqrt[2])

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Maple [C]  time = 0.065, size = 981, normalized size = 3.2 \[ \text{result too large to display} \]

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)^(17/2),x)

[Out]

-2/551904988635*(77419842517122564*x-32495729111616960*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)+95570583350719680*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)-19256728362439680*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)-48141820906099200*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)-427
9272969431040*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)-7221273135914880
0*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)-64991458223233920*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)-2214305034568163712*x^5-420378712490
0760138*x^6-3997525460519271384*x^7+500221362404680812*x^3-166810299141489255*x^
4+304831834382285292*x^2-414708067039730040*x^9-1990701860603882364*x^8+84951629
64508416*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)+64510143761735784*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)+13823602234657668*2^(1/2)*Ellip
ticE(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^7*(3+
5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+143355875026079520*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)+38228233340287872*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)-6963370523917920*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^7*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*
x)^(1/2)+129020287523471568*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)-
407549806612480*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))[phone]*
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))+7678123195182561)*(3+5*x)^(1/2)*
(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(15/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{17}{2}}}\,{d x} \]

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)^(17/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(17/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256\right )} \sqrt{3 \, x + 2}}, x\right ) \]

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)^(17/2),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)/((65
61*x^8 + 34992*x^7 + 81648*x^6 + 108864*x^5 + 90720*x^4 + 48384*x^3 + 16128*x^2
+ 3072*x + 256)*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)**(17/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{17}{2}}}\,{d x} \]

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)^(17/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(17/2), x)