3.84 \(\int \frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{(7+5 x)^{9/2}} \, dx\)

Optimal. Leaf size=370 \[ -\frac{65687975672 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{2257624501329015 \sqrt{2 x-5}}+\frac{32843987836 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{451524900265803 \sqrt{5 x+7}}+\frac{23758016 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{57992193675 (5 x+7)^{3/2}}+\frac{2558 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{695175 (5 x+7)^{5/2}}-\frac{2 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{35 (5 x+7)^{7/2}}-\frac{1212290288 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{1867348636335 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}+\frac{32843987836 \sqrt{\frac{11}{39}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{57887807726385 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}} \]

[Out]

(-2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(35*(7 + 5*x)^(7/2)) + (2558*Sqr
t[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(695175*(7 + 5*x)^(5/2)) + (23758016*Sq
rt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(57992193675*(7 + 5*x)^(3/2)) + (32843
987836*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(451524900265803*Sqrt[7 + 5*x
]) - (65687975672*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(2257624501329015*S
qrt[-5 + 2*x]) + (32843987836*Sqrt[11/39]*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])/(578878
07726385*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) - (1212290288*Sqrt[11/23]*Sqrt
[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(186
7348636335*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)])

_______________________________________________________________________________________

Rubi [A]  time = 1.24673, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.216 \[ -\frac{65687975672 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{2257624501329015 \sqrt{2 x-5}}+\frac{32843987836 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{451524900265803 \sqrt{5 x+7}}+\frac{23758016 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{57992193675 (5 x+7)^{3/2}}+\frac{2558 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{695175 (5 x+7)^{5/2}}-\frac{2 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{35 (5 x+7)^{7/2}}-\frac{1212290288 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{1867348636335 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}+\frac{32843987836 \sqrt{\frac{11}{39}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{57887807726385 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(7 + 5*x)^(9/2),x]

[Out]

(-2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(35*(7 + 5*x)^(7/2)) + (2558*Sqr
t[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(695175*(7 + 5*x)^(5/2)) + (23758016*Sq
rt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(57992193675*(7 + 5*x)^(3/2)) + (32843
987836*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(451524900265803*Sqrt[7 + 5*x
]) - (65687975672*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(2257624501329015*S
qrt[-5 + 2*x]) + (32843987836*Sqrt[11/39]*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])/(578878
07726385*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) - (1212290288*Sqrt[11/23]*Sqrt
[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(186
7348636335*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)])

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- 3 x + 2} \sqrt{2 x - 5} \sqrt{4 x + 1}}{\left (5 x + 7\right )^{\frac{9}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2-3*x)**(1/2)*(-5+2*x)**(1/2)*(1+4*x)**(1/2)/(7+5*x)**(9/2),x)

[Out]

Integral(sqrt(-3*x + 2)*sqrt(2*x - 5)*sqrt(4*x + 1)/(5*x + 7)**(9/2), x)

_______________________________________________________________________________________

Mathematica [A]  time = 2.71175, size = 259, normalized size = 0.7 \[ \frac{2 \sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7} \left (\frac{242 \left (203578437 \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )[phone] \sqrt{682} (3 x-2) \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} F\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )-67859479 \sqrt{682} (3 x-2) \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} E\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )\right )}{\sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )}-\frac{(3 x-2) \left (10263746198750 x^3+54668919175710 x^2+113490310442229 x+15395515423270\right )}{(5 x+7)^4}\right )}{2257624501329015 \sqrt{2-3 x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(7 + 5*x)^(9/2),x]

[Out]

(2*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x]*(-(((-2 + 3*x)*(15395515423270 + 1
13490310442229*x + 54668919175710*x^2 + 10263746198750*x^3))/(7 + 5*x)^4) + (242
*(203578437*Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-5 - 18*x + 8*x^2) - 67859479*Sqrt[682]*
(-2 + 3*x)*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*EllipticE[ArcSin[Sqrt[31/39]*Sq
rt[(-5 + 2*x)/(-2 + 3*x)]], 39/62] + 19017205*Sqrt[682]*(-2 + 3*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]))/(Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-5 - 18*x + 8*x^2))))/(225762450132901
5*Sqrt[2 - 3*x])

_______________________________________________________________________________________

Maple [B]  time = 0.061, size = 1232, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(9/2),x)

[Out]

-2/2257624501329015*(4737011092000*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))*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*x^5+3284
3987836000*3^(1/2)*13^(1/2)*((-5+2*x)/(1+4*x))^(1/2)*((-2+3*x)/(1+4*x))^(1/2)*El
lipticE(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))*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*x^5+22263952132400*((-5+2*x)/(1+
4*x))^(1/2)*3^(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))*13^(1/2)*11^(1/2)*
((7+5*x)/(1+4*x))^(1/2)*x^4+154366742829200*((-5+2*x)/(1+4*x))^(1/2)*3^(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))*13^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)
*x^4+38097411707410*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))*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*x^3+264147772171030*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*3
1^(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))
*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*x^3+28168636458578*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*Ell
ipticF(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^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))[phone]
4534*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
*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)*3
1^(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)*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]
4*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))[phone]*x^5-173342585590346*x^4+215
3615020704860*x^3-4639703191080657*x^2+51366440607272*x+1423213141652020)*(1+4*x
)^(1/2)*(-5+2*x)^(1/2)*(2-3*x)^(1/2)/(120*x^4-182*x^3-385*x^2+197*x+70)/(7+5*x)^
(5/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(5*x + 7)^(9/2),x, algorithm="maxima")

[Out]

integrate(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(5*x + 7)^(9/2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{{\left (625 \, x^{4} + 3500 \, x^{3} + 7350 \, x^{2} + 6860 \, x + 2401\right )} \sqrt{5 \, x + 7}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(5*x + 7)^(9/2),x, algorithm="fricas")

[Out]

integral(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/((625*x^4 + 3500*x^3 + 7350*
x^2 + 6860*x + 2401)*sqrt(5*x + 7)), x)

_______________________________________________________________________________________

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)*(-5+2*x)**(1/2)*(1+4*x)**(1/2)/(7+5*x)**(9/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(5*x + 7)^(9/2),x, algorithm="giac")

[Out]

integrate(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(5*x + 7)^(9/2), x)