3.2710 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx\)

Optimal. Leaf size=249 \[ \frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{891 (3 x+2)^{9/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac{13292 \sqrt{1-2 x} (5 x+3)^{3/2}}{43659 (3 x+2)^{7/2}}+\frac{3316711588 \sqrt{1-2 x} \sqrt{5 x+3}}{673876665 \sqrt{3 x+2}}+\frac{45748292 \sqrt{1-2 x} \sqrt{5 x+3}}{96268095 (3 x+2)^{3/2}}-\frac{1366496 \sqrt{1-2 x} \sqrt{5 x+3}}{4584195 (3 x+2)^{5/2}}-\frac{103970992 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{61261515 \sqrt{33}}-\frac{3316711588 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{61261515 \sqrt{33}} \]

[Out]

(-1366496*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(4584195*(2 + 3*x)^(5/2)) + (45748292*Sqr
t[1 - 2*x]*Sqrt[3 + 5*x])/(96268095*(2 + 3*x)^(3/2)) + (3316711588*Sqrt[1 - 2*x]
*Sqrt[3 + 5*x])/(673876665*Sqrt[2 + 3*x]) - (13292*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)
)/(43659*(2 + 3*x)^(7/2)) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(33*(2 + 3*x)^(1
1/2)) + (362*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(891*(2 + 3*x)^(9/2)) - (3316711588*
EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(61261515*Sqrt[33]) - (103970
992*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(61261515*Sqrt[33])

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Rubi [A]  time = 0.585511, 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 \[ \frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{891 (3 x+2)^{9/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac{13292 \sqrt{1-2 x} (5 x+3)^{3/2}}{43659 (3 x+2)^{7/2}}+\frac{3316711588 \sqrt{1-2 x} \sqrt{5 x+3}}{673876665 \sqrt{3 x+2}}+\frac{45748292 \sqrt{1-2 x} \sqrt{5 x+3}}{96268095 (3 x+2)^{3/2}}-\frac{1366496 \sqrt{1-2 x} \sqrt{5 x+3}}{4584195 (3 x+2)^{5/2}}-\frac{103970992 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{61261515 \sqrt{33}}-\frac{3316711588 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{61261515 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-1366496*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(4584195*(2 + 3*x)^(5/2)) + (45748292*Sqr
t[1 - 2*x]*Sqrt[3 + 5*x])/(96268095*(2 + 3*x)^(3/2)) + (3316711588*Sqrt[1 - 2*x]
*Sqrt[3 + 5*x])/(673876665*Sqrt[2 + 3*x]) - (13292*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)
)/(43659*(2 + 3*x)^(7/2)) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(33*(2 + 3*x)^(1
1/2)) + (362*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(891*(2 + 3*x)^(9/2)) - (3316711588*
EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(61261515*Sqrt[33]) - (103970
992*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(61261515*Sqrt[33])

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Rubi in Sympy [A]  time = 57.2645, size = 230, normalized size = 0.92 \[ - \frac{14582 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{305613 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{362 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{6237 \left (3 x + 2\right )^{\frac{9}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{33 \left (3 x + 2\right )^{\frac{11}{2}}} + \frac{3316711588 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{673876665 \sqrt{3 x + 2}} + \frac{45748292 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{96268095 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{1039534 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{4584195 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{3316711588 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2021629995} - \frac{103970992 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2021629995} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-14582*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(305613*(3*x + 2)**(7/2)) - 362*(-2*x + 1
)**(3/2)*(5*x + 3)**(3/2)/(6237*(3*x + 2)**(9/2)) - 2*(-2*x + 1)**(3/2)*(5*x + 3
)**(5/2)/(33*(3*x + 2)**(11/2)) + 3316711588*sqrt(-2*x + 1)*sqrt(5*x + 3)/(67387
6665*sqrt(3*x + 2)) + 45748292*sqrt(-2*x + 1)*sqrt(5*x + 3)/(96268095*(3*x + 2)*
*(3/2)) + 1039534*sqrt(-2*x + 1)*sqrt(5*x + 3)/(4584195*(3*x + 2)**(5/2)) - 3316
711588*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/2021629995 -
103970992*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/2021629995

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Mathematica [A]  time = 0.438001, size = 112, normalized size = 0.45 \[ \frac{\frac{48 \sqrt{2-4 x} \sqrt{5 x+3} \left (402980457942 x^5+1356237833922 x^4+1829570010885 x^3+1234133449713 x^2+415681177941 x+55875107717\right )}{(3 x+2)^{11/2}}-25619043520 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{16173039960 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(55875107717 + 415681177941*x + 1234133449713*x
^2 + 1829570010885*x^3 + 1356237833922*x^4 + 402980457942*x^5))/(2 + 3*x)^(11/2)
+ 53067385408*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 25619043520*
EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(16173039960*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)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(13/2),x)

[Out]

2/2021629995*(194544611730*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)-4
02980457942*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)-1343268193140*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)-1791024257520*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*1
1^(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)-119401617
1680*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+1921
42826400*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)-398005390560*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+25619043520*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))-53067385408*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+30306573018747*x^4-293294410596*x^3-8183
904282084*x^2-3573505278318*x-502875969453)*(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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(50*x^3 + 35*x^2 - 12*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((1-2*x)**(3/2)*(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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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