3.2916 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{1100380 \sqrt{1-2 x} \sqrt{3 x+2}}{41503 \sqrt{5 x+3}}+\frac{9876 \sqrt{1-2 x}}{3773 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{54 \sqrt{1-2 x}}{539 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{4}{77 \sqrt{1-2 x} (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{6584 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3773}+\frac{220076 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3773} \]

[Out]

4/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (54*Sqrt[1 - 2*x])/(539*(2
+ 3*x)^(3/2)*Sqrt[3 + 5*x]) + (9876*Sqrt[1 - 2*x])/(3773*Sqrt[2 + 3*x]*Sqrt[3 +
5*x]) - (1100380*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(41503*Sqrt[3 + 5*x]) + (220076*Sq
rt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3773 + (6584*Sqrt[3/
11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3773

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Rubi [A]  time = 0.434406, 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 \[ -\frac{1100380 \sqrt{1-2 x} \sqrt{3 x+2}}{41503 \sqrt{5 x+3}}+\frac{9876 \sqrt{1-2 x}}{3773 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{54 \sqrt{1-2 x}}{539 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{4}{77 \sqrt{1-2 x} (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{6584 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3773}+\frac{220076 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3773} \]

Antiderivative was successfully verified.

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

[Out]

4/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (54*Sqrt[1 - 2*x])/(539*(2
+ 3*x)^(3/2)*Sqrt[3 + 5*x]) + (9876*Sqrt[1 - 2*x])/(3773*Sqrt[2 + 3*x]*Sqrt[3 +
5*x]) - (1100380*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(41503*Sqrt[3 + 5*x]) + (220076*Sq
rt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3773 + (6584*Sqrt[3/
11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3773

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Rubi in Sympy [A]  time = 39.0499, size = 172, normalized size = 0.9 \[ - \frac{1100380 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{41503 \sqrt{5 x + 3}} + \frac{9876 \sqrt{- 2 x + 1}}{3773 \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{54 \sqrt{- 2 x + 1}}{539 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}} + \frac{220076 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{41503} + \frac{6584 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{41503} + \frac{4}{77 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1100380*sqrt(-2*x + 1)*sqrt(3*x + 2)/(41503*sqrt(5*x + 3)) + 9876*sqrt(-2*x + 1
)/(3773*sqrt(3*x + 2)*sqrt(5*x + 3)) + 54*sqrt(-2*x + 1)/(539*(3*x + 2)**(3/2)*s
qrt(5*x + 3)) + 220076*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/3
3)/41503 + 6584*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/4150
3 + 4/(77*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.226827, size = 104, normalized size = 0.54 \[ \frac{2 \left (\frac{9903420 x^3+7926942 x^2-2259236 x-2088967}{\sqrt{1-2 x} (3 x+2)^{3/2} \sqrt{5 x+3}}-2 \sqrt{2} \left (55019 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-27860 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{41503} \]

Antiderivative was successfully verified.

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

[Out]

(2*((-2088967 - 2259236*x + 7926942*x^2 + 9903420*x^3)/(Sqrt[1 - 2*x]*(2 + 3*x)^
(3/2)*Sqrt[3 + 5*x]) - 2*Sqrt[2]*(55019*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x
]], -33/2] - 27860*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/41503

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Maple [C]  time = 0.037, size = 267, normalized size = 1.4 \[ -{\frac{2}{415030\,{x}^{2}+41503\,x-124509}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 167160\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-330114\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+111440\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -220076\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +9903420\,{x}^{3}+7926942\,{x}^{2}-2259236\,x-2088967 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/41503*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(167160*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)-330114*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)+111440*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))-220076*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))+9903420*x^3+7926942*x^2-2259236*x-2088967)/(2+3*x)^(3/2)/(
10*x^2+x-3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-1/((90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)*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(1/(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. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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