3.2726 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{2660 \sqrt{1-2 x} \sqrt{3 x+2}}{9 \sqrt{5 x+3}}+\frac{88 \sqrt{1-2 x}}{3 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{14 \sqrt{1-2 x}}{9 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{16}{3} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{532}{3} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(14*Sqrt[1 - 2*x])/(9*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (88*Sqrt[1 - 2*x])/(3*Sqr
t[2 + 3*x]*Sqrt[3 + 5*x]) - (2660*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(9*Sqrt[3 + 5*x])
+ (532*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3 + (16*Sq
rt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3

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Rubi [A]  time = 0.344304, 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 \[ -\frac{2660 \sqrt{1-2 x} \sqrt{3 x+2}}{9 \sqrt{5 x+3}}+\frac{88 \sqrt{1-2 x}}{3 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{14 \sqrt{1-2 x}}{9 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{16}{3} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{532}{3} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(14*Sqrt[1 - 2*x])/(9*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (88*Sqrt[1 - 2*x])/(3*Sqr
t[2 + 3*x]*Sqrt[3 + 5*x]) - (2660*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(9*Sqrt[3 + 5*x])
+ (532*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3 + (16*Sq
rt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3

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Rubi in Sympy [A]  time = 33.1996, size = 143, normalized size = 0.89 \[ - \frac{2660 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{9 \sqrt{5 x + 3}} + \frac{88 \sqrt{- 2 x + 1}}{3 \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{14 \sqrt{- 2 x + 1}}{9 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}} + \frac{532 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{9} + \frac{176 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{105} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2660*sqrt(-2*x + 1)*sqrt(3*x + 2)/(9*sqrt(5*x + 3)) + 88*sqrt(-2*x + 1)/(3*sqrt
(3*x + 2)*sqrt(5*x + 3)) + 14*sqrt(-2*x + 1)/(9*(3*x + 2)**(3/2)*sqrt(5*x + 3))
+ 532*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/9 + 176*sqrt(3
5)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/105

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Mathematica [A]  time = 0.195298, size = 100, normalized size = 0.62 \[ -\frac{2 \sqrt{1-2 x} \left (3990 x^2+5188 x+1683\right )}{3 (3 x+2)^{3/2} \sqrt{5 x+3}}-\frac{4}{9} \sqrt{2} \left (133 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-67 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(1683 + 5188*x + 3990*x^2))/(3*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])
- (4*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]))/9

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Maple [C]  time = 0.034, size = 267, normalized size = 1.7 \[ -{\frac{2}{90\,{x}^{2}+9\,x-27}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 402\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-798\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+268\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -532\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +23940\,{x}^{3}+19158\,{x}^{2}-5466\,x-5049 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(402*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)-798*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)+268*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))-532*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))+23940*x^3+19158*x^2-5466*x-5049)/(2+3*x)^(3/2)/(10*x^2+x-3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-2*x + 1)^(3/2)/((45*x^3 + 87*x^2 + 56*x + 12)*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)**(3/2)/(2+3*x)**(5/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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