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

Optimal. Leaf size=351 \[ \frac{1}{4} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7}+\frac{509 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{240 \sqrt{2 x-5}}+\frac{8959 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{720 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}-\frac{509 \sqrt{\frac{143}{3}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{160 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}+\frac{2198489 (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{3600 \sqrt{429} \sqrt{2 x-5} \sqrt{4 x+1}} \]

[Out]

(509*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(240*Sqrt[-5 + 2*x]) + (Sqrt[2 -
3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/4 - (509*Sqrt[143/3]*Sqrt[2 -
3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sqrt[1 + 4*x])/Sqrt
[-5 + 2*x]], -23/39])/(160*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) + (8959*Sqrt
[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -
39/23])/(720*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) + (2198489*(2 - 3*x)*Sqrt
[(5 - 2*x)/(2 - 3*x)]*Sqrt[-((1 + 4*x)/(2 - 3*x))]*EllipticPi[-69/55, ArcSin[(Sq
rt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/(3600*Sqrt[429]*Sqrt[-5 + 2*x]
*Sqrt[1 + 4*x])

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Rubi [A]  time = 1.12616, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.27 \[ \frac{1}{4} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7}+\frac{509 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{240 \sqrt{2 x-5}}+\frac{8959 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{720 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}-\frac{509 \sqrt{\frac{143}{3}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{160 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}+\frac{2198489 (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{3600 \sqrt{429} \sqrt{2 x-5} \sqrt{4 x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(509*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(240*Sqrt[-5 + 2*x]) + (Sqrt[2 -
3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/4 - (509*Sqrt[143/3]*Sqrt[2 -
3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sqrt[1 + 4*x])/Sqrt
[-5 + 2*x]], -23/39])/(160*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) + (8959*Sqrt
[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -
39/23])/(720*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) + (2198489*(2 - 3*x)*Sqrt
[(5 - 2*x)/(2 - 3*x)]*Sqrt[-((1 + 4*x)/(2 - 3*x))]*EllipticPi[-69/55, ArcSin[(Sq
rt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/(3600*Sqrt[429]*Sqrt[-5 + 2*x]
*Sqrt[1 + 4*x])

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

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)**(1/2)/(-5+2*x)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 4.34546, size = 347, normalized size = 0.99 \[ \frac{\sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7} \left (66960 (2-3 x)-\frac{3 \left (76756 \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )+94674 \sqrt{682} (2-3 x) (5 x+7) \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} E\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )+\sqrt{\frac{5 x+7}{3 x-2}} \left (70919 \sqrt{682} \sqrt{\frac{4 x+1}{3 x-2}} \sqrt{\frac{10 x^2-11 x-35}{(2-3 x)^2}} (2-3 x)^2 \Pi \left (\frac{117}{62};\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )+284022 \left (40 x^3-34 x^2-151 x-35\right )\right )\right )}{(2-3 x) \left (\frac{5 x+7}{3 x-2}\right )^{3/2} \left (-8 x^2+18 x+5\right )}\right )}{267840 \sqrt{2-3 x}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x]*(66960*(2 - 3*x) - (3*(94674*Sqrt[68
2]*(2 - 3*x)*(7 + 5*x)*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*EllipticE[ArcSin[Sq
rt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62] + 76756*Sqrt[682]*Sqrt[(-5 - 18*x
+ 8*x^2)/(2 - 3*x)^2]*(-14 + 11*x + 15*x^2)*EllipticF[ArcSin[Sqrt[31/39]*Sqrt[(
-5 + 2*x)/(-2 + 3*x)]], 39/62] + Sqrt[(7 + 5*x)/(-2 + 3*x)]*(284022*(-35 - 151*x
- 34*x^2 + 40*x^3) + 70919*Sqrt[682]*(2 - 3*x)^2*Sqrt[(1 + 4*x)/(-2 + 3*x)]*Sqr
t[(-35 - 11*x + 10*x^2)/(2 - 3*x)^2]*EllipticPi[117/62, ArcSin[Sqrt[31/39]*Sqrt[
(-5 + 2*x)/(-2 + 3*x)]], 39/62])))/((2 - 3*x)*((7 + 5*x)/(-2 + 3*x))^(3/2)*(5 +
18*x - 8*x^2))))/(267840*Sqrt[2 - 3*x])

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Maple [B]  time = 0.029, size = 934, normalized size = 2.7 \[ \text{result too large to display} \]

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

[Out]

-1/2059200*(2-3*x)^(1/2)*(1+4*x)^(1/2)*(7+5*x)^(1/2)*(-5+2*x)^(1/2)*(3622960*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))-26098192*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*EllipticP
i(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2),124/55,1/39*2^(1/2)*3^(1/2)*31^
(1/2)*13^(1/2))-34937760*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))+1811480*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))-13049096*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*EllipticPi(1/31*
31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2),124/55,1/39*2^(1/2)*3^(1/2)*31^(1/2)*1
3^(1/2))-17468880*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(1/31*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))+226435*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))-1631137*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)*EllipticPi(1/31*31^(1/2)*11^(1/
2)*((7+5*x)/(1+4*x))^(1/2),124/55,1/39*2^(1/2)*3^(1/2)*31^(1/2)*13^(1/2))-218361
0*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(1/31*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))-168339600*x^3-61776000*x^4+661123320*x^2+
623542920*x-647446800)/(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. \[ \int \frac{\sqrt{5 \, x + 7} \sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{\sqrt{2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- 3 x + 2} \sqrt{4 x + 1} \sqrt{5 x + 7}}{\sqrt{2 x - 5}}\, dx \]

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)**(1/2)/(-5+2*x)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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