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

Optimal. Leaf size=351 \[ \frac{5}{16} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7}+\frac{785 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{192 \sqrt{2 x-5}}+\frac{17515 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{576 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}-\frac{785 \sqrt{\frac{143}{3}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{128 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}+\frac{3730013 (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{2880 \sqrt{429} \sqrt{2 x-5} \sqrt{4 x+1}} \]

[Out]

(785*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(192*Sqrt[-5 + 2*x]) + (5*Sqrt[2
- 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/16 - (785*Sqrt[143/3]*Sqrt[2
- 3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sqrt[1 + 4*x])/S
qrt[-5 + 2*x]], -23/39])/(128*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) + (17515*
Sqrt[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])
], -39/23])/(576*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) + (3730013*(2 - 3*x)*
Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[-((1 + 4*x)/(2 - 3*x))]*EllipticPi[-69/55, ArcSin
[(Sqrt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/(2880*Sqrt[429]*Sqrt[-5 +
2*x]*Sqrt[1 + 4*x])

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Rubi [A]  time = 1.13837, 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 \[ \frac{5}{16} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7}+\frac{785 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{192 \sqrt{2 x-5}}+\frac{17515 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{576 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}-\frac{785 \sqrt{\frac{143}{3}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{128 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}+\frac{3730013 (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{2880 \sqrt{429} \sqrt{2 x-5} \sqrt{4 x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(785*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(192*Sqrt[-5 + 2*x]) + (5*Sqrt[2
- 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/16 - (785*Sqrt[143/3]*Sqrt[2
- 3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sqrt[1 + 4*x])/S
qrt[-5 + 2*x]], -23/39])/(128*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) + (17515*
Sqrt[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])
], -39/23])/(576*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) + (3730013*(2 - 3*x)*
Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[-((1 + 4*x)/(2 - 3*x))]*EllipticPi[-69/55, ArcSin
[(Sqrt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/(2880*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. \[ \int \frac{\sqrt{- 3 x + 2} \left (5 x + 7\right )^{\frac{3}{2}}}{\sqrt{2 x - 5} \sqrt{4 x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 5.13135, size = 349, normalized size = 0.99 \[ \frac{\sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7} \left (\frac{(2-3 x) \left (\frac{998820 \sqrt{682} (5 x+7) \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} F\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )}{(2-3 x)^2}-\frac{1314090 \sqrt{682} (5 x+7) \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} E\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )}{(2-3 x)^2}+\sqrt{\frac{5 x+7}{3 x-2}} \left (\frac{1082907 \sqrt{682} \sqrt{\frac{10 x^2-11 x-35}{(2-3 x)^2}} \left (\frac{4 x+1}{3 x-2}\right )^{3/2} \Pi \left (\frac{117}{62};\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )}{4 x+1}+\frac{3942270 \left (40 x^3-34 x^2-151 x-35\right )}{(3 x-2)^3}\right )\right )}{\left (\frac{5 x+7}{3 x-2}\right )^{3/2} \left (-8 x^2+18 x+5\right )}+200880\right )}{642816} \]

Warning: Unable to verify antiderivative.

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

[Out]

(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x]*(200880 + ((2 - 3*x)*(
(-1314090*Sqrt[682]*(7 + 5*x)*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*EllipticE[Ar
cSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62])/(2 - 3*x)^2 + (998820*Sqr
t[682]*(7 + 5*x)*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*EllipticF[ArcSin[Sqrt[31/
39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62])/(2 - 3*x)^2 + Sqrt[(7 + 5*x)/(-2 + 3*x
)]*((3942270*(-35 - 151*x - 34*x^2 + 40*x^3))/(-2 + 3*x)^3 + (1082907*Sqrt[682]*
((1 + 4*x)/(-2 + 3*x))^(3/2)*Sqrt[(-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])/(1 + 4*x))))/((
(7 + 5*x)/(-2 + 3*x))^(3/2)*(5 + 18*x - 8*x^2))))/642816

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Maple [B]  time = 0.033, size = 934, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1647360*(7+5*x)^(1/2)*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)*(1963280*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),1/3
9*2^(1/2)*3^(1/2)*31^(1/2)*13^(1/2))[phone]*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*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))[phone]*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))+981640*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))[phone]*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)*13^
(1/2))[phone]*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))+122705*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))+2767429*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))+3367650*
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),1/3
9*2^(1/2)*3^(1/2)*31^(1/2)*13^(1/2))[phone]*x^3+61776000*x^4-912139800*x^2-10
16644200*x+978978000)/(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. \[ \int \frac{{\left (5 \, x + 7\right )}^{\frac{3}{2}} \sqrt{-3 \, x + 2}}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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