3.2908 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=191 \[ \frac{733812 \sqrt{1-2 x} \sqrt{5 x+3}}{132055 \sqrt{3 x+2}}+\frac{10308 \sqrt{1-2 x} \sqrt{5 x+3}}{18865 (3 x+2)^{3/2}}+\frac{138 \sqrt{1-2 x} \sqrt{5 x+3}}{2695 (3 x+2)^{5/2}}+\frac{4 \sqrt{5 x+3}}{77 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{7536 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005}-\frac{244604 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005} \]

[Out]

(4*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) + (138*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(2695*(2 + 3*x)^(5/2)) + (10308*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(18865*(2
+ 3*x)^(3/2)) + (733812*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(132055*Sqrt[2 + 3*x]) - (2
44604*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/12005 - (753
6*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/12005

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Rubi [A]  time = 0.422974, 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 \[ \frac{733812 \sqrt{1-2 x} \sqrt{5 x+3}}{132055 \sqrt{3 x+2}}+\frac{10308 \sqrt{1-2 x} \sqrt{5 x+3}}{18865 (3 x+2)^{3/2}}+\frac{138 \sqrt{1-2 x} \sqrt{5 x+3}}{2695 (3 x+2)^{5/2}}+\frac{4 \sqrt{5 x+3}}{77 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{7536 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005}-\frac{244604 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 - 2*x)^(3/2)*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x]),x]

[Out]

(4*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) + (138*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(2695*(2 + 3*x)^(5/2)) + (10308*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(18865*(2
+ 3*x)^(3/2)) + (733812*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(132055*Sqrt[2 + 3*x]) - (2
44604*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/12005 - (753
6*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/12005

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Rubi in Sympy [A]  time = 39.2183, size = 172, normalized size = 0.9 \[ \frac{733812 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{132055 \sqrt{3 x + 2}} + \frac{10308 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{18865 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{138 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2695 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{244604 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{132055} - \frac{22608 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{420175} + \frac{4 \sqrt{5 x + 3}}{77 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-2*x)**(3/2)/(2+3*x)**(7/2)/(3+5*x)**(1/2),x)

[Out]

733812*sqrt(-2*x + 1)*sqrt(5*x + 3)/(132055*sqrt(3*x + 2)) + 10308*sqrt(-2*x + 1
)*sqrt(5*x + 3)/(18865*(3*x + 2)**(3/2)) + 138*sqrt(-2*x + 1)*sqrt(5*x + 3)/(269
5*(3*x + 2)**(5/2)) - 244604*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7)
, 35/33)/132055 - 22608*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33
/35)/420175 + 4*sqrt(5*x + 3)/(77*sqrt(-2*x + 1)*(3*x + 2)**(5/2))

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Mathematica [A]  time = 0.257628, size = 106, normalized size = 0.55 \[ \frac{4 \left (\frac{\sqrt{5 x+3} \left (-6604308 x^3-5720058 x^2+1424784 x+1546591\right )}{2 \sqrt{1-2 x} (3 x+2)^{5/2}}+\sqrt{2} \left (61151 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-30065 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{132055} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 - 2*x)^(3/2)*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x]),x]

[Out]

(4*((Sqrt[3 + 5*x]*(1546591 + 1424784*x - 5720058*x^2 - 6604308*x^3))/(2*Sqrt[1
- 2*x]*(2 + 3*x)^(5/2)) + Sqrt[2]*(61151*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]], -33/2] - 30065*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/132055

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Maple [C]  time = 0.039, size = 386, normalized size = 2. \[{\frac{2}{1320550\,{x}^{2}+132055\,x-396165}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 541170\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1100718\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+721560\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1467624\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+240520\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -489208\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-12007307\,x-4639773 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-2*x)^(3/2)/(2+3*x)^(7/2)/(3+5*x)^(1/2),x)

[Out]

2/132055*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(541170*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)-1100718*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)+721
560*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)-1467624*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)+240520*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))-489208*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/11*1
1^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))[phone]*x^4+48413
214*x^3+10036254*x^2-12007307*x-4639773)/(2+3*x)^(5/2)/(10*x^2+x-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(5*x + 3)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(5*x + 3)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(5*x + 3)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)),x, algorithm="fricas")

[Out]

integral(-1/((54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*sqrt(5*x + 3)*sqrt(3*x + 2)*s
qrt(-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)**(7/2)/(3+5*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(5*x + 3)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(5*x + 3)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)), x)