3.2708 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{315 (3 x+2)^{5/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}+\frac{2108 \sqrt{1-2 x} (5 x+3)^{3/2}}{6615 (3 x+2)^{3/2}}+\frac{249448 \sqrt{1-2 x} \sqrt{5 x+3}}{138915 \sqrt{3 x+2}}+\frac{249448 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{138915}-\frac{962678 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{138915} \]

[Out]

(249448*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(138915*Sqrt[2 + 3*x]) + (2108*Sqrt[1 - 2*x
]*(3 + 5*x)^(3/2))/(6615*(2 + 3*x)^(3/2)) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/
(21*(2 + 3*x)^(7/2)) + (362*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(315*(2 + 3*x)^(5/2))
- (962678*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/138915
+ (249448*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/138915

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Rubi [A]  time = 0.415628, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{315 (3 x+2)^{5/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}+\frac{2108 \sqrt{1-2 x} (5 x+3)^{3/2}}{6615 (3 x+2)^{3/2}}+\frac{249448 \sqrt{1-2 x} \sqrt{5 x+3}}{138915 \sqrt{3 x+2}}+\frac{249448 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{138915}-\frac{962678 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{138915} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(9/2),x]

[Out]

(249448*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(138915*Sqrt[2 + 3*x]) + (2108*Sqrt[1 - 2*x
]*(3 + 5*x)^(3/2))/(6615*(2 + 3*x)^(3/2)) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/
(21*(2 + 3*x)^(7/2)) + (362*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(315*(2 + 3*x)^(5/2))
- (962678*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/138915
+ (249448*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/138915

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Rubi in Sympy [A]  time = 41.1549, size = 172, normalized size = 0.9 \[ - \frac{14054 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{46305 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{362 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{2205 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{21 \left (3 x + 2\right )^{\frac{7}{2}}} + \frac{20378 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3969 \sqrt{3 x + 2}} - \frac{962678 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{416745} + \frac{2743928 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{4862025} \]

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)**(9/2),x)

[Out]

-14054*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(46305*(3*x + 2)**(3/2)) - 362*(-2*x + 1)
**(3/2)*(5*x + 3)**(3/2)/(2205*(3*x + 2)**(5/2)) - 2*(-2*x + 1)**(3/2)*(5*x + 3)
**(5/2)/(21*(3*x + 2)**(7/2)) + 20378*sqrt(-2*x + 1)*sqrt(5*x + 3)/(3969*sqrt(3*
x + 2)) - 962678*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/416
745 + 2743928*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/48620
25

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Mathematica [A]  time = 0.353925, size = 104, normalized size = 0.54 \[ \frac{2 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (10680903 x^3+20067219 x^2+12594615 x+2640643\right )}{(3 x+2)^{7/2}}+\sqrt{2} \left (481339 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2539285 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{416745} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(9/2),x]

[Out]

(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2640643 + 12594615*x + 20067219*x^2 + 106809
03*x^3))/(2 + 3*x)^(7/2) + Sqrt[2]*(481339*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]], -33/2] - 2539285*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/41
6745

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Maple [C]  time = 0.028, size = 505, normalized size = 2.6 \[{\frac{2}{4167450\,{x}^{2}+416745\,x-1250235} \left ( 68560695\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-12996153\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-25992306\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-17328204\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -3850712\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-63601836\,{x}^{2}-105429606\,x-23765787 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}} \]

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)^(9/2),x)

[Out]

2/416745*(68560695*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^3*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)-12996153*
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)-25992306*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]*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)-173282
04*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]*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))-3850712*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+634059279*x^4+341911980*x^3-63601836*x^2-105429606*x-2376578
7)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \]

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)^(9/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{3 \, x + 2}}, x\right ) \]

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)^(9/2),x, algorithm="fricas")

[Out]

integral(-(50*x^3 + 35*x^2 - 12*x - 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((81*x^4 + 2
16*x^3 + 216*x^2 + 96*x + 16)*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)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \]

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)^(9/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(9/2), x)