3.2816 \(\int \frac{\sqrt{2+3 x} (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{1}{7} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{20}{21} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{2645}{378} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{529}{378} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{17587}{378} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-2645*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/378 - (20*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*(3 + 5*x)^(3/2))/21 - (Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/7 -
(17587*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/378 - (529*
Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/378

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Rubi [A]  time = 0.325957, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{1}{7} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{20}{21} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{2645}{378} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{529}{378} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{17587}{378} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2645*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/378 - (20*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*(3 + 5*x)^(3/2))/21 - (Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/7 -
(17587*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/378 - (529*
Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/378

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Rubi in Sympy [A]  time = 32.37, size = 143, normalized size = 0.89 \[ - \frac{\sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{5}{2}}}{7} - \frac{20 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{21} - \frac{2645 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{378} - \frac{17587 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1134} - \frac{5819 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{13230} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(5/2)/7 - 20*sqrt(-2*x + 1)*sqrt(3*x +
2)*(5*x + 3)**(3/2)/21 - 2645*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/378 - 1
7587*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1134 - 5819*sqr
t(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/13230

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Mathematica [A]  time = 0.252413, size = 97, normalized size = 0.61 \[ \frac{-3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (1350 x^2+3420 x+4211\right )-17717 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+35174 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1134 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(4211 + 3420*x + 1350*x^2) + 35174
*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 17717*EllipticF[ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]], -33/2])/(1134*Sqrt[2])

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Maple [C]  time = 0.018, size = 174, normalized size = 1.1 \[{\frac{1}{68040\,{x}^{3}+52164\,{x}^{2}-15876\,x-13608}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 17717\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -35174\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -243000\,{x}^{5}-801900\,{x}^{4}-1173240\,{x}^{3}-388878\,{x}^{2}+299982\,x+151596 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2268*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(17717*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))-35174*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))-24
3000*x^5-801900*x^4-1173240*x^3-388878*x^2+299982*x+151596)/(30*x^3+23*x^2-7*x-6
)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{\sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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