3.57 \(\int \frac{\sqrt{2-3 x}}{\sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3} \, dx\)

Optimal. Leaf size=225 \[ -\frac{26825 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{33257172 (5 x+7)}-\frac{5 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{1794 (5 x+7)^2}-\frac{13243 \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{1065935 \sqrt{66} \sqrt{2 x-5}}+\frac{5365 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{16628586 \sqrt{5-2 x}}-\frac{16369941 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{3436574440 \sqrt{11} \sqrt{2 x-5}} \]

[Out]

(-5*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(1794*(7 + 5*x)^2) - (26825*Sqrt
[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(33257172*(7 + 5*x)) + (5365*Sqrt[11]*Sq
rt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(16628586*Sqrt
[5 - 2*x]) - (13243*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/
3])/(1065935*Sqrt[66]*Sqrt[-5 + 2*x]) - (16369941*Sqrt[5 - 2*x]*EllipticPi[55/12
4, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(3436574440*Sqrt[11]*Sqrt[-5 + 2*x
])

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Rubi [A]  time = 0.974281, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.314 \[ -\frac{26825 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{33257172 (5 x+7)}-\frac{5 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{1794 (5 x+7)^2}-\frac{13243 \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{1065935 \sqrt{66} \sqrt{2 x-5}}+\frac{5365 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{16628586 \sqrt{5-2 x}}-\frac{16369941 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{3436574440 \sqrt{11} \sqrt{2 x-5}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 - 3*x]/(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^3),x]

[Out]

(-5*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(1794*(7 + 5*x)^2) - (26825*Sqrt
[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(33257172*(7 + 5*x)) + (5365*Sqrt[11]*Sq
rt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(16628586*Sqrt
[5 - 2*x]) - (13243*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/
3])/(1065935*Sqrt[66]*Sqrt[-5 + 2*x]) - (16369941*Sqrt[5 - 2*x]*EllipticPi[55/12
4, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(3436574440*Sqrt[11]*Sqrt[-5 + 2*x
])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- 3 x + 2}}{\sqrt{2 x - 5} \sqrt{4 x + 1} \left (5 x + 7\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2-3*x)**(1/2)/(7+5*x)**3/(-5+2*x)**(1/2)/(1+4*x)**(1/2),x)

[Out]

Integral(sqrt(-3*x + 2)/(sqrt(2*x - 5)*sqrt(4*x + 1)*(5*x + 7)**3), x)

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Mathematica [A]  time = 0.409225, size = 144, normalized size = 0.64 \[ \frac{-17050 \sqrt{2-3 x} (2 x-5) \sqrt{4 x+1} (26825 x+56093)-\sqrt{55-22 x} (5 x+7)^2 \left (-64043148 F\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )[phone] E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )-49109823 \Pi \left (\frac{55}{124};-\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )\right )}{113406956520 \sqrt{2 x-5} (5 x+7)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[2 - 3*x]/(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^3),x]

[Out]

(-17050*Sqrt[2 - 3*x]*(-5 + 2*x)*Sqrt[1 + 4*x]*(56093 + 26825*x) - Sqrt[55 - 22*
x]*(7 + 5*x)^2*(36589300*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] - 6
4043148*EllipticF[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] - 49109823*EllipticP
i[55/124, -ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2]))/(113406956520*Sqrt[-5 + 2
*x]*(7 + 5*x)^2)

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Maple [B]  time = 0.027, size = 488, normalized size = 2.2 \[ -{\frac{1}{ \left ( 2721766956480\,{x}^{3}-7938486956400\,{x}^{2}[phone]\,x+1134069565200 \right ) \left ( 7+5\,x \right ) ^{2}}\sqrt{2-3\,x}\sqrt{-5+2\,x}\sqrt{1+4\,x} \left ( 1601078700\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticF} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ){x}^{2}-914732500\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticE} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ){x}^{2}-1227745575\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticPi} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},{\frac{55}{124}},i/2\sqrt{2} \right ){x}^{2}[phone]\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticF} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) x-2561251000\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticE} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) x-3437687610\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticPi} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},{\frac{55}{124}},i/2\sqrt{2} \right ) x+3138114252\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticF} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -1792875700\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticE} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -2406381327\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticPi} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},{\frac{55}{124}},i/2\sqrt{2} \right ) [phone]\,{x}^{4}-9062381900\,{x}^{3}-57342304250\,{x}^{2}[phone]\,x+9563856500 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2-3*x)^(1/2)/(7+5*x)^3/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x)

[Out]

-1/113406956520*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)*(1601078700*11^(1/2)*
(2-3*x)^(1/2)*(5-2*x)^(1/2)*(1+4*x)^(1/2)*EllipticF(2/11*(2-3*x)^(1/2)*11^(1/2),
1/2*I*2^(1/2))*x^2-914732500*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(1+4*x)^(1/2)*
EllipticE(2/11*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))*x^2-1227745575*11^(1/2)*(2-
3*x)^(1/2)*(5-2*x)^(1/2)*(1+4*x)^(1/2)*EllipticPi(2/11*(2-3*x)^(1/2)*11^(1/2),55
/124,1/2*I*2^(1/2))*x^2+4483020360*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(1+4*x)^
(1/2)*EllipticF(2/11*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))*x-2561251000*11^(1/2)
*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(1+4*x)^(1/2)*EllipticE(2/11*(2-3*x)^(1/2)*11^(1/2)
,1/2*I*2^(1/2))*x-3437687610*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(1+4*x)^(1/2)*
EllipticPi(2/11*(2-3*x)^(1/2)*11^(1/2),55/124,1/2*I*2^(1/2))*x+3138114252*11^(1/
2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(1+4*x)^(1/2)*EllipticF(2/11*(2-3*x)^(1/2)*11^(1/
2),1/2*I*2^(1/2))-1792875700*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(1+4*x)^(1/2)*
EllipticE(2/11*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))-2406381327*11^(1/2)*(2-3*x)
^(1/2)*(5-2*x)^(1/2)*(1+4*x)^(1/2)*EllipticPi(2/11*(2-3*x)^(1/2)*11^(1/2),55/124
,1/2*I*2^(1/2))[phone]*x^4-9062381900*x^3-57342304250*x^2+24657761150*x+956
3856500)/(24*x^3-70*x^2+21*x+10)/(7+5*x)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*x + 2)/((5*x + 7)^3*sqrt(4*x + 1)*sqrt(2*x - 5)),x, algorithm="maxima")

[Out]

integrate(sqrt(-3*x + 2)/((5*x + 7)^3*sqrt(4*x + 1)*sqrt(2*x - 5)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-3 \, x + 2}}{{\left (125 \, x^{3} + 525 \, x^{2} + 735 \, x + 343\right )} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*x + 2)/((5*x + 7)^3*sqrt(4*x + 1)*sqrt(2*x - 5)),x, algorithm="fricas")

[Out]

integral(sqrt(-3*x + 2)/((125*x^3 + 525*x^2 + 735*x + 343)*sqrt(4*x + 1)*sqrt(2*
x - 5)), 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((2-3*x)**(1/2)/(7+5*x)**3/(-5+2*x)**(1/2)/(1+4*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*x + 2)/((5*x + 7)^3*sqrt(4*x + 1)*sqrt(2*x - 5)),x, algorithm="giac")

[Out]

integrate(sqrt(-3*x + 2)/((5*x + 7)^3*sqrt(4*x + 1)*sqrt(2*x - 5)), x)