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

Optimal. Leaf size=195 \[ \frac{2 \sqrt{\frac{3}{143}} (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{31 \sqrt{2 x-5} \sqrt{4 x+1}}+\frac{10 \sqrt{\frac{11}{39}} \sqrt{2-3 x} \sqrt{\frac{5-2 x}{5 x+7}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{22}} \sqrt{4 x+1}}{\sqrt{5 x+7}}\right )|\frac{62}{39}\right )}{713 \sqrt{2 x-5} \sqrt{\frac{2-3 x}{5 x+7}}} \]

[Out]

(10*Sqrt[11/39]*Sqrt[2 - 3*x]*Sqrt[(5 - 2*x)/(7 + 5*x)]*EllipticE[ArcSin[(Sqrt[3
9/22]*Sqrt[1 + 4*x])/Sqrt[7 + 5*x]], 62/39])/(713*Sqrt[-5 + 2*x]*Sqrt[(2 - 3*x)/
(7 + 5*x)]) + (2*Sqrt[3/143]*(2 - 3*x)*Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[-((1 + 4*x
)/(2 - 3*x))]*EllipticF[ArcSin[(Sqrt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/3
9])/(31*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])

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Rubi [A]  time = 0.661174, antiderivative size = 270, normalized size of antiderivative = 1.38, number of steps used = 8, number of rules used = 7, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.189 \[ -\frac{10 \sqrt{2 x-5} \sqrt{4 x+1}}{897 \sqrt{2-3 x} \sqrt{5 x+7}}+\frac{6 \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{31 \sqrt{253} \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}-\frac{5 \sqrt{\frac{22}{31}} \sqrt{4 x+1} F\left (\tan ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{2 x-5}}{\sqrt{5 x+7}}\right )|\frac{39}{62}\right )}{1209 \sqrt{2-3 x} \sqrt{-\frac{4 x+1}{2-3 x}}}+\frac{10 \sqrt{\frac{22}{31}} \sqrt{4 x+1} E\left (\tan ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{2 x-5}}{\sqrt{5 x+7}}\right )|\frac{39}{62}\right )}{897 \sqrt{2-3 x} \sqrt{-\frac{4 x+1}{2-3 x}}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(897*Sqrt[2 - 3*x]*Sqrt[7 + 5*x]) + (10*Sqrt[
22/31]*Sqrt[1 + 4*x]*EllipticE[ArcTan[(Sqrt[31/11]*Sqrt[-5 + 2*x])/Sqrt[7 + 5*x]
], 39/62])/(897*Sqrt[2 - 3*x]*Sqrt[-((1 + 4*x)/(2 - 3*x))]) + (6*Sqrt[7 + 5*x]*E
llipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(31*Sqrt[253]*S
qrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) - (5*Sqrt[22/31]*Sqrt[1 + 4*x]*Elliptic
F[ArcTan[(Sqrt[31/11]*Sqrt[-5 + 2*x])/Sqrt[7 + 5*x]], 39/62])/(1209*Sqrt[2 - 3*x
]*Sqrt[-((1 + 4*x)/(2 - 3*x))])

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Rubi in Sympy [A]  time = 58.4603, size = 479, normalized size = 2.46 \[ - \frac{220 \sqrt{\frac{- 156 x - 39}{- 110 x - 154}} \sqrt{\frac{117 x - 78}{55 x + 77}} \sqrt{2 x - 5} \sqrt{5 x + 7} \sqrt{\frac{23 \left (2 x - 5\right )}{22 \left (5 x + 7\right )} + 1}}{34983 \sqrt{- 3 x + 2} \sqrt{4 x + 1} \sqrt{\frac{31 \left (2 x - 5\right )}{11 \left (5 x + 7\right )} + 1}} + \frac{220 \sqrt{341} \sqrt{\frac{- 156 x - 39}{- 110 x - 154}} \sqrt{\frac{117 x - 78}{55 x + 77}} \left (5 x + 7\right ) \sqrt{\frac{23 \left (2 x - 5\right )}{22 \left (5 x + 7\right )} + 1} E\left (\operatorname{atan}{\left (\frac{\sqrt{341} \sqrt{2 x - 5}}{11 \sqrt{5 x + 7}} \right )}\middle | \frac{39}{62}\right )}{1084473 \sqrt{\frac{\frac{23 \left (2 x - 5\right )}{22 \left (5 x + 7\right )} + 1}{\frac{31 \left (2 x - 5\right )}{11 \left (5 x + 7\right )} + 1}} \sqrt{- 3 x + 2} \sqrt{4 x + 1} \sqrt{\frac{31 \left (2 x - 5\right )}{11 \left (5 x + 7\right )} + 1}} - \frac{110 \sqrt{341} \sqrt{\frac{- 156 x - 39}{- 110 x - 154}} \sqrt{\frac{117 x - 78}{55 x + 77}} \left (5 x + 7\right ) \sqrt{\frac{23 \left (2 x - 5\right )}{22 \left (5 x + 7\right )} + 1} F\left (\operatorname{atan}{\left (\frac{\sqrt{341} \sqrt{2 x - 5}}{11 \sqrt{5 x + 7}} \right )}\middle | \frac{39}{62}\right )}{1461681 \sqrt{\frac{\frac{23 \left (2 x - 5\right )}{22 \left (5 x + 7\right )} + 1}{\frac{31 \left (2 x - 5\right )}{11 \left (5 x + 7\right )} + 1}} \sqrt{- 3 x + 2} \sqrt{4 x + 1} \sqrt{\frac{31 \left (2 x - 5\right )}{11 \left (5 x + 7\right )} + 1}} + \frac{\sqrt{897} \sqrt{\frac{- 66 x + 44}{- 22 x + 55}} \sqrt{\frac{110 x + 154}{- 46 x + 115}} \left (- 2 x + 5\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{897} \sqrt{4 x + 1}}{23 \sqrt{2 x - 5}} \right )}\middle | - \frac{23}{39}\right )}{4433 \sqrt{- 3 x + 2} \sqrt{5 x + 7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-220*sqrt((-156*x - 39)/(-110*x - 154))*sqrt((117*x - 78)/(55*x + 77))*sqrt(2*x
- 5)*sqrt(5*x + 7)*sqrt(23*(2*x - 5)/(22*(5*x + 7)) + 1)/(34983*sqrt(-3*x + 2)*s
qrt(4*x + 1)*sqrt(31*(2*x - 5)/(11*(5*x + 7)) + 1)) + 220*sqrt(341)*sqrt((-156*x
- 39)/(-110*x - 154))*sqrt((117*x - 78)/(55*x + 77))*(5*x + 7)*sqrt(23*(2*x - 5
)/(22*(5*x + 7)) + 1)*elliptic_e(atan(sqrt(341)*sqrt(2*x - 5)/(11*sqrt(5*x + 7))
), 39/62)/(1084473*sqrt((23*(2*x - 5)/(22*(5*x + 7)) + 1)/(31*(2*x - 5)/(11*(5*x
+ 7)) + 1))*sqrt(-3*x + 2)*sqrt(4*x + 1)*sqrt(31*(2*x - 5)/(11*(5*x + 7)) + 1))
- 110*sqrt(341)*sqrt((-156*x - 39)/(-110*x - 154))*sqrt((117*x - 78)/(55*x + 77
))*(5*x + 7)*sqrt(23*(2*x - 5)/(22*(5*x + 7)) + 1)*elliptic_f(atan(sqrt(341)*sqr
t(2*x - 5)/(11*sqrt(5*x + 7))), 39/62)/(1461681*sqrt((23*(2*x - 5)/(22*(5*x + 7)
) + 1)/(31*(2*x - 5)/(11*(5*x + 7)) + 1))*sqrt(-3*x + 2)*sqrt(4*x + 1)*sqrt(31*(
2*x - 5)/(11*(5*x + 7)) + 1)) + sqrt(897)*sqrt((-66*x + 44)/(-22*x + 55))*sqrt((
110*x + 154)/(-46*x + 115))*(-2*x + 5)*elliptic_f(asin(sqrt(897)*sqrt(4*x + 1)/(
23*sqrt(2*x - 5))), -23/39)/(4433*sqrt(-3*x + 2)*sqrt(5*x + 7))

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Mathematica [A]  time = 1.68367, size = 237, normalized size = 1.22 \[ -\frac{2 \sqrt{2 x-5} \sqrt{4 x+1} \left (1705 \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )-23 \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )-55 \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )\right )}{305877 \sqrt{2-3 x} \sqrt{5 x+7} \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(1705*Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-5 - 18*x + 8
*x^2) - 55*Sqrt[682]*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*(-14 + 11*x + 15*x^2)
*EllipticE[ArcSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62] - 23*Sqrt[682
]*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*(-14 + 11*x + 15*x^2)*EllipticF[ArcSin[S
qrt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62]))/(305877*Sqrt[2 - 3*x]*Sqrt[7 +
5*x]*Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-5 - 18*x + 8*x^2))

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Maple [B]  time = 0.038, size = 635, normalized size = 3.3 \[{\frac{2}{36705240\,{x}^{4}-55669614\,{x}^{3}-117762645\,{x}^{2}[phone]\,x+21411390}\sqrt{2-3\,x}\sqrt{-5+2\,x}\sqrt{1+4\,x}\sqrt{7+5\,x} \left ( 1104\,\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{-5+2\,x}{1+4\,x}}}\sqrt{{\frac{-2+3\,x}{1+4\,x}}}{x}^{2}{\it EllipticF} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}},1/39\,\sqrt{2}\sqrt{3}\sqrt{31}\sqrt{13} \right ) +880\,\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{-5+2\,x}{1+4\,x}}}\sqrt{{\frac{-2+3\,x}{1+4\,x}}}{x}^{2}{\it EllipticE} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}},1/39\,\sqrt{2}\sqrt{3}\sqrt{31}\sqrt{13} \right ) +552\,\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{-5+2\,x}{1+4\,x}}}\sqrt{{\frac{-2+3\,x}{1+4\,x}}}x{\it EllipticF} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}},1/39\,\sqrt{2}\sqrt{3}\sqrt{31}\sqrt{13} \right ) +440\,\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{-5+2\,x}{1+4\,x}}}\sqrt{{\frac{-2+3\,x}{1+4\,x}}}x{\it EllipticE} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}},1/39\,\sqrt{2}\sqrt{3}\sqrt{31}\sqrt{13} \right ) +69\,\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{-5+2\,x}{1+4\,x}}}\sqrt{{\frac{-2+3\,x}{1+4\,x}}}{\it EllipticF} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}},1/39\,\sqrt{2}\sqrt{3}\sqrt{31}\sqrt{13} \right ) +55\,\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{-5+2\,x}{1+4\,x}}}\sqrt{{\frac{-2+3\,x}{1+4\,x}}}{\it EllipticE} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}},1/39\,\sqrt{2}\sqrt{3}\sqrt{31}\sqrt{13} \right ) +7590\,{x}^{2}-24035\,x+12650 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/305877*(7+5*x)^(1/2)*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)*(1104*11^(1/2)
*((7+5*x)/(1+4*x))^(1/2)*3^(1/2)*13^(1/2)*((-5+2*x)/(1+4*x))^(1/2)*((-2+3*x)/(1+
4*x))^(1/2)*x^2*EllipticF(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2),1/39*2^
(1/2)*3^(1/2)*31^(1/2)*13^(1/2))+880*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*3^(1/2)*13
^(1/2)*((-5+2*x)/(1+4*x))^(1/2)*((-2+3*x)/(1+4*x))^(1/2)*x^2*EllipticE(1/31*31^(
1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2),1/39*2^(1/2)*3^(1/2)*31^(1/2)*13^(1/2))+55
2*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*3^(1/2)*13^(1/2)*((-5+2*x)/(1+4*x))^(1/2)*((-
2+3*x)/(1+4*x))^(1/2)*x*EllipticF(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)
,1/39*2^(1/2)*3^(1/2)*31^(1/2)*13^(1/2))+440*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*3^
(1/2)*13^(1/2)*((-5+2*x)/(1+4*x))^(1/2)*((-2+3*x)/(1+4*x))^(1/2)*x*EllipticE(1/3
1*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2),1/39*2^(1/2)*3^(1/2)*31^(1/2)*13^(1/
2))+69*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*3^(1/2)*13^(1/2)*((-5+2*x)/(1+4*x))^(1/2
)*((-2+3*x)/(1+4*x))^(1/2)*EllipticF(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1
/2),1/39*2^(1/2)*3^(1/2)*31^(1/2)*13^(1/2))+55*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*
3^(1/2)*13^(1/2)*((-5+2*x)/(1+4*x))^(1/2)*((-2+3*x)/(1+4*x))^(1/2)*EllipticE(1/3
1*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2),1/39*2^(1/2)*3^(1/2)*31^(1/2)*13^(1/
2))+7590*x^2-24035*x+12650)/(120*x^4-182*x^3-385*x^2+197*x+70)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((5*x + 7)^(3/2)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)), 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/(7+5*x)**(3/2)/(2-3*x)**(1/2)/(-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. \[ \int \frac{1}{{\left (5 \, x + 7\right )}^{\frac{3}{2}} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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