3.2980 \(\int \frac{1}{(1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=187 \[ \frac{598660 \sqrt{1-2 x} \sqrt{3 x+2}}{2152227 \sqrt{5 x+3}}-\frac{18470 \sqrt{1-2 x} \sqrt{3 x+2}}{195657 (5 x+3)^{3/2}}+\frac{368 \sqrt{3 x+2}}{5929 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{4 \sqrt{3 x+2}}{231 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{7388 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{65219 \sqrt{33}}-\frac{119732 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{65219 \sqrt{33}} \]

[Out]

(4*Sqrt[2 + 3*x])/(231*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (368*Sqrt[2 + 3*x])/(5
929*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (18470*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(195657
*(3 + 5*x)^(3/2)) + (598660*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(2152227*Sqrt[3 + 5*x])
- (119732*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(65219*Sqrt[33]) -
(7388*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(65219*Sqrt[33])

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Rubi [A]  time = 0.431198, antiderivative size = 187, 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 \[ \frac{598660 \sqrt{1-2 x} \sqrt{3 x+2}}{2152227 \sqrt{5 x+3}}-\frac{18470 \sqrt{1-2 x} \sqrt{3 x+2}}{195657 (5 x+3)^{3/2}}+\frac{368 \sqrt{3 x+2}}{5929 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{4 \sqrt{3 x+2}}{231 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{7388 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{65219 \sqrt{33}}-\frac{119732 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{65219 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[2 + 3*x])/(231*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (368*Sqrt[2 + 3*x])/(5
929*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (18470*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(195657
*(3 + 5*x)^(3/2)) + (598660*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(2152227*Sqrt[3 + 5*x])
- (119732*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(65219*Sqrt[33]) -
(7388*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(65219*Sqrt[33])

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Rubi in Sympy [A]  time = 41.6047, size = 172, normalized size = 0.92 \[ - \frac{119732 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2152227} - \frac{7388 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2152227} - \frac{239464 \sqrt{3 x + 2} \sqrt{5 x + 3}}{2152227 \sqrt{- 2 x + 1}} + \frac{18160 \sqrt{3 x + 2}}{27951 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} - \frac{370 \sqrt{3 x + 2}}{2541 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{4 \sqrt{3 x + 2}}{231 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-119732*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/2152227 - 73
88*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/2152227 - 239464*
sqrt(3*x + 2)*sqrt(5*x + 3)/(2152227*sqrt(-2*x + 1)) + 18160*sqrt(3*x + 2)/(2795
1*sqrt(-2*x + 1)*sqrt(5*x + 3)) - 370*sqrt(3*x + 2)/(2541*sqrt(-2*x + 1)*(5*x +
3)**(3/2)) + 4*sqrt(3*x + 2)/(231*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.257881, size = 103, normalized size = 0.55 \[ \frac{2 \left (\frac{\sqrt{3 x+2} \left (5986600 x^3-2800980 x^2-1822554 x+881831\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}+\sqrt{2} \left (1085 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+59866 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{2152227} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[2 + 3*x]*(881831 - 1822554*x - 2800980*x^2 + 5986600*x^3))/((1 - 2*x)^
(3/2)*(3 + 5*x)^(3/2)) + Sqrt[2]*(59866*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x
]], -33/2] + 1085*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/2152227

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Maple [C]  time = 0.038, size = 383, normalized size = 2.1 \[ -{\frac{2}{2152227\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 10850\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+598660\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1085\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+59866\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3255\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -179598\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -17959800\,{x}^{4}-3570260\,{x}^{3}[phone]\,{x}^{2}+999615\,x-1763662 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{2+3\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/2152227*(1-2*x)^(1/2)*(10850*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)+598660*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)+1085*2^(1/2)*Elli
pticF(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)+59866*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)-3255*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))-179598*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))-17959800*x^4-3570260*x^3+11069622*x^2+
999615*x-1763662)/(3+5*x)^(3/2)/(-1+2*x)^2/(2+3*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*sqrt(5*x + 3)*sqrt(3*x + 2)*sq
rt(-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)**(5/2)/(3+5*x)**(5/2)/(2+3*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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