3.2679 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=150 \[ \frac{2660 \sqrt{1-2 x} \sqrt{3 x+2}}{33 \sqrt{5 x+3}}-\frac{40 \sqrt{1-2 x} \sqrt{3 x+2}}{3 (5 x+3)^{3/2}}+\frac{2 \sqrt{1-2 x}}{\sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{16 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}}-\frac{532 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}} \]

[Out]

(2*Sqrt[1 - 2*x])/(Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (40*Sqrt[1 - 2*x]*Sqrt[2 + 3
*x])/(3*(3 + 5*x)^(3/2)) + (2660*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(33*Sqrt[3 + 5*x])
- (532*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/Sqrt[33] - (16*Ellipt
icF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/Sqrt[33]

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Rubi [A]  time = 0.337069, antiderivative size = 150, 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 \[ \frac{2660 \sqrt{1-2 x} \sqrt{3 x+2}}{33 \sqrt{5 x+3}}-\frac{40 \sqrt{1-2 x} \sqrt{3 x+2}}{3 (5 x+3)^{3/2}}+\frac{2 \sqrt{1-2 x}}{\sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{16 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}}-\frac{532 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x])/(Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (40*Sqrt[1 - 2*x]*Sqrt[2 + 3
*x])/(3*(3 + 5*x)^(3/2)) + (2660*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(33*Sqrt[3 + 5*x])
- (532*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/Sqrt[33] - (16*Ellipt
icF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/Sqrt[33]

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Rubi in Sympy [A]  time = 30.4672, size = 141, normalized size = 0.94 \[ \frac{2660 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{33 \sqrt{5 x + 3}} - \frac{40 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{3 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{2 \sqrt{- 2 x + 1}}{\sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{532 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{33} - \frac{16 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{35} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2660*sqrt(-2*x + 1)*sqrt(3*x + 2)/(33*sqrt(5*x + 3)) - 40*sqrt(-2*x + 1)*sqrt(3*
x + 2)/(3*(5*x + 3)**(3/2)) + 2*sqrt(-2*x + 1)/(sqrt(3*x + 2)*(5*x + 3)**(3/2))
- 532*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/33 - 16*sqrt(3
5)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/35

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Mathematica [A]  time = 0.265202, size = 99, normalized size = 0.66 \[ \frac{2}{33} \left (\frac{\sqrt{1-2 x} \left (19950 x^2+24610 x+7573\right )}{\sqrt{3 x+2} (5 x+3)^{3/2}}+2 \sqrt{2} \left (133 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-67 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[1 - 2*x]*(7573 + 24610*x + 19950*x^2))/(Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))
+ 2*Sqrt[2]*(133*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 67*Ellipt
icF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/33

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Maple [C]  time = 0.033, size = 267, normalized size = 1.8 \[{\frac{2}{198\,{x}^{2}+33\,x-66}\sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 670\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1330\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+402\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -798\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +39900\,{x}^{3}+29270\,{x}^{2}-9464\,x-7573 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/33*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(670*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)-1330*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)+402*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))-798*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))+39900*x^3+29270*x^2-9464*x-7573)/(3+5*x)^(3/2)/(6*x^2+x-2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-2 \, x + 1}}{{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(-2*x + 1)/((75*x^3 + 140*x^2 + 87*x + 18)*sqrt(5*x + 3)*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)**(1/2)/(2+3*x)**(3/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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