3.2644 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}+\frac{22738708 \sqrt{1-2 x} \sqrt{5 x+3}}{6806835 \sqrt{3 x+2}}+\frac{332372 \sqrt{1-2 x} \sqrt{5 x+3}}{972405 (3 x+2)^{3/2}}+\frac{8842 \sqrt{1-2 x} \sqrt{5 x+3}}{138915 (3 x+2)^{5/2}}-\frac{214 \sqrt{1-2 x} \sqrt{5 x+3}}{3969 (3 x+2)^{7/2}}-\frac{673072 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835}-\frac{22738708 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835} \]

[Out]

(-214*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3969*(2 + 3*x)^(7/2)) + (8842*Sqrt[1 - 2*x]*
Sqrt[3 + 5*x])/(138915*(2 + 3*x)^(5/2)) + (332372*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(
972405*(2 + 3*x)^(3/2)) + (22738708*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6806835*Sqrt[2
+ 3*x]) - (2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(27*(2 + 3*x)^(9/2)) - (22738708*Sq
rt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/6806835 - (673072*Sq
rt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/6806835

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Rubi [A]  time = 0.504916, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}+\frac{22738708 \sqrt{1-2 x} \sqrt{5 x+3}}{6806835 \sqrt{3 x+2}}+\frac{332372 \sqrt{1-2 x} \sqrt{5 x+3}}{972405 (3 x+2)^{3/2}}+\frac{8842 \sqrt{1-2 x} \sqrt{5 x+3}}{138915 (3 x+2)^{5/2}}-\frac{214 \sqrt{1-2 x} \sqrt{5 x+3}}{3969 (3 x+2)^{7/2}}-\frac{673072 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835}-\frac{22738708 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2 + 3*x)^(11/2),x]

[Out]

(-214*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3969*(2 + 3*x)^(7/2)) + (8842*Sqrt[1 - 2*x]*
Sqrt[3 + 5*x])/(138915*(2 + 3*x)^(5/2)) + (332372*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(
972405*(2 + 3*x)^(3/2)) + (22738708*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6806835*Sqrt[2
+ 3*x]) - (2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(27*(2 + 3*x)^(9/2)) - (22738708*Sq
rt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/6806835 - (673072*Sq
rt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/6806835

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Rubi in Sympy [A]  time = 45.3699, size = 201, normalized size = 0.91 \[ \frac{22738708 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{6806835 \sqrt{3 x + 2}} + \frac{332372 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{972405 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{8842 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{138915 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{214 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3969 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{2 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{27 \left (3 x + 2\right )^{\frac{9}{2}}} - \frac{22738708 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{20420505} - \frac{7403792 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{238239225} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)**(3/2)*(1-2*x)**(1/2)/(2+3*x)**(11/2),x)

[Out]

22738708*sqrt(-2*x + 1)*sqrt(5*x + 3)/(6806835*sqrt(3*x + 2)) + 332372*sqrt(-2*x
+ 1)*sqrt(5*x + 3)/(972405*(3*x + 2)**(3/2)) + 8842*sqrt(-2*x + 1)*sqrt(5*x + 3
)/(138915*(3*x + 2)**(5/2)) - 214*sqrt(-2*x + 1)*sqrt(5*x + 3)/(3969*(3*x + 2)**
(7/2)) - 2*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/(27*(3*x + 2)**(9/2)) - 22738708*sqrt
(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/20420505 - 7403792*sqrt(
35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/238239225

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Mathematica [A]  time = 0.405756, size = 107, normalized size = 0.48 \[ \frac{\frac{24 \sqrt{2-4 x} \sqrt{5 x+3} \left (920917674 x^4+2487189618 x^3+2520548433 x^2+1134125364 x+190959271\right )}{(3 x+2)^{9/2}}-93064160 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{81682020 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2 + 3*x)^(11/2),x]

[Out]

((24*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(190959271 + 1134125364*x + 2520548433*x^2 + 24
87189618*x^3 + 920917674*x^4))/(2 + 3*x)^(9/2) + 181909664*EllipticE[ArcSin[Sqrt
[2/11]*Sqrt[3 + 5*x]], -33/2] - 93064160*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]], -33/2])/(81682020*Sqrt[2])

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Maple [C]  time = 0.055, size = 624, normalized size = 2.8 \[{\frac{2}{204205050\,{x}^{2}[phone]\,x-61261515} \left ( 471137310\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-920917674\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-2455780464\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2455780464\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1091457984\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -181909664\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-13553781675\,{x}^{2}-9634250463\,x-1718633439 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(11/2),x)

[Out]

2/20420505*(471137310*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)-920917
674*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)*Ellip
ticF(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)-2455780464*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)-2455780464*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)-1091457984*2^(1/2)*EllipticE(1/1
1*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+93064160*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))-181909664*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))+
77378441562*x^5+74789762778*x^4+19200699657*x^3-13553781675*x^2-9634250463*x-171
8633439)*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(9/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*sqrt(-2*x + 1)/(3*x + 2)^(11/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(3/2)*sqrt(-2*x + 1)/(3*x + 2)^(11/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}{{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*sqrt(-2*x + 1)/(3*x + 2)^(11/2),x, algorithm="fricas")

[Out]

integral((5*x + 3)^(3/2)*sqrt(-2*x + 1)/((243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2
+ 240*x + 32)*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)**(3/2)*(1-2*x)**(1/2)/(2+3*x)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*sqrt(-2*x + 1)/(3*x + 2)^(11/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(3/2)*sqrt(-2*x + 1)/(3*x + 2)^(11/2), x)