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

Optimal. Leaf size=225 \[ -\frac{361 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{481988 (5 x+7)}+\frac{\sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{78 (5 x+7)^2}-\frac{6101 \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{231725 \sqrt{66} \sqrt{2 x-5}}+\frac{361 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{1204970 \sqrt{5-2 x}}-\frac{6655867 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{747081400 \sqrt{11} \sqrt{2 x-5}} \]

[Out]

(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(78*(7 + 5*x)^2) - (361*Sqrt[2 - 3*
x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(481988*(7 + 5*x)) + (361*Sqrt[11]*Sqrt[-5 + 2*
x]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(1204970*Sqrt[5 - 2*x])
- (6101*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(231725*
Sqrt[66]*Sqrt[-5 + 2*x]) - (6655867*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*S
qrt[2 - 3*x])/Sqrt[11]], -1/2])/(747081400*Sqrt[11]*Sqrt[-5 + 2*x])

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Rubi [A]  time = 0.961919, 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 \[ -\frac{361 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{481988 (5 x+7)}+\frac{\sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{78 (5 x+7)^2}-\frac{6101 \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{231725 \sqrt{66} \sqrt{2 x-5}}+\frac{361 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{1204970 \sqrt{5-2 x}}-\frac{6655867 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{747081400 \sqrt{11} \sqrt{2 x-5}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(78*(7 + 5*x)^2) - (361*Sqrt[2 - 3*
x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(481988*(7 + 5*x)) + (361*Sqrt[11]*Sqrt[-5 + 2*
x]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(1204970*Sqrt[5 - 2*x])
- (6101*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(231725*
Sqrt[66]*Sqrt[-5 + 2*x]) - (6655867*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*S
qrt[2 - 3*x])/Sqrt[11]], -1/2])/(747081400*Sqrt[11]*Sqrt[-5 + 2*x])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.66411, size = 137, normalized size = 0.61 \[ \frac{-\frac{17050 \sqrt{2-3 x} (2 x-5) \sqrt{4 x+1} (5415 x-10957)}{(5 x+7)^2}-3 \sqrt{55-22 x} \left (-9834812 F\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )+2462020 E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )-6655867 \Pi \left (\frac{55}{124};-\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )\right )}{24653686200 \sqrt{2 x-5}} \]

Antiderivative was successfully verified.

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

[Out]

((-17050*Sqrt[2 - 3*x]*(-5 + 2*x)*Sqrt[1 + 4*x]*(-10957 + 5415*x))/(7 + 5*x)^2 -
3*Sqrt[55 - 22*x]*(2462020*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2]
- 9834812*EllipticF[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] - 6655867*Elliptic
Pi[55/124, -ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2]))/(24653686200*Sqrt[-5 + 2
*x])

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Maple [B]  time = 0.025, size = 488, normalized size = 2.2 \[ -{\frac{1}{ \left ( 591688468800\,{x}^{3}-1725758034000\,{x}^{2}[phone]\,x+246536862000 \right ) \left ( 7+5\,x \right ) ^{2}}\sqrt{2-3\,x}\sqrt{-5+2\,x}\sqrt{1+4\,x} \left ( 737610900\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticF} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ){x}^{2}-184651500\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticE} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ){x}^{2}-499190025\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticPi} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},{\frac{55}{124}},i/2\sqrt{2} \right ){x}^{2}[phone]\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticF} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) x-517024200\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticE} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) x-1397732070\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticPi} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},{\frac{55}{124}},i/2\sqrt{2} \right ) x+1445717364\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticF} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -361916940\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticE} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -978412449\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticPi} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},{\frac{55}{124}},i/2\sqrt{2} \right ) [phone]\,{x}^{4}-10946406900\,{x}^{3}[phone]\,{x}^{2}-2999896350\,x-1868168500 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24653686200*(2-3*x)^(1/2)*(1+4*x)^(1/2)*(-5+2*x)^(1/2)*(737610900*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^2-184651500*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(1+4*x)^(1/2)*El
lipticE(2/11*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))*x^2-499190025*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),55/12
4,1/2*I*2^(1/2))*x^2+2065310520*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-517024200*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-1397732070*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(1+4*x)^(1/2)*Elli
pticPi(2/11*(2-3*x)^(1/2)*11^(1/2),55/124,1/2*I*2^(1/2))*x+1445717364*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))-361916940*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(1+4*x)^(1/2)*Ellip
ticE(2/11*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))-978412449*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^4-10946406900*x^3+15016020250*x^2-2999896350*x-1868168500
)/(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. \[ \int \frac{\sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3} \sqrt{2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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