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

Optimal. Leaf size=471 \[ \frac{1}{25} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^{7/2}-\frac{427 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^{5/2}}{2400}-\frac{83363 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^{3/2}}{34560}-\frac{70489981 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7}}{1658880}-\frac{1450582567 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{3686400 \sqrt{2 x-5}}-\frac{245264762213 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{99532800 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}+\frac{1450582567 \sqrt{\frac{143}{3}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{2457600 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}-\frac{57691792727443 (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{497664000 \sqrt{429} \sqrt{2 x-5} \sqrt{4 x+1}} \]

[Out]

(-1450582567*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(3686400*Sqrt[-5 + 2*x])
- (70489981*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/1658880 -
(83363*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(3/2))/34560 - (427
*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(5/2))/2400 + (Sqrt[2 - 3*
x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(7/2))/25 + (1450582567*Sqrt[143/3]*Sq
rt[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])/(2457600*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) -
(245264762213*Sqrt[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]
*Sqrt[2 - 3*x])], -39/23])/(99532800*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) -
(57691792727443*(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])
/(497664000*Sqrt[429]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])

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Rubi [A]  time = 1.62607, antiderivative size = 471, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.297 \[ \frac{1}{25} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^{7/2}-\frac{427 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^{5/2}}{2400}-\frac{83363 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^{3/2}}{34560}-\frac{70489981 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7}}{1658880}-\frac{1450582567 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{3686400 \sqrt{2 x-5}}-\frac{245264762213 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{99532800 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}+\frac{1450582567 \sqrt{\frac{143}{3}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{2457600 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}-\frac{57691792727443 (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{497664000 \sqrt{429} \sqrt{2 x-5} \sqrt{4 x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(-1450582567*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(3686400*Sqrt[-5 + 2*x])
- (70489981*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/1658880 -
(83363*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(3/2))/34560 - (427
*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(5/2))/2400 + (Sqrt[2 - 3*
x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(7/2))/25 + (1450582567*Sqrt[143/3]*Sq
rt[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])/(2457600*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) -
(245264762213*Sqrt[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]
*Sqrt[2 - 3*x])], -39/23])/(99532800*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) -
(57691792727443*(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])
/(497664000*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. \[ \int \sqrt{- 3 x + 2} \sqrt{2 x - 5} \sqrt{4 x + 1} \left (5 x + 7\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 3.74196, size = 350, normalized size = 0.74 \[ -\frac{\sqrt{2 x-5} \sqrt{4 x+1} \left (-62507925572 \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )[phone] \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )+\sqrt{\frac{5 x+7}{3 x-2}} \left (6 \left (39813120000 x^7+71414784000 x^6-288728294400 x^5-849459145920 x^4-795166559320 x^3+2861488598626 x^2+5225923788019 x+1118234665415\right )-60033082963 \sqrt{682} (2-3 x)^2 \sqrt{\frac{4 x+1}{3 x-2}} \sqrt{\frac{10 x^2-11 x-35}{(2-3 x)^2}} \Pi \left (\frac{117}{62};\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )\right )\right )}{398131200 \sqrt{2-3 x} \sqrt{5 x+7} \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(78331458618*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] - 62507925572*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)]*(6*(1118234665415 + 5225923788019*x + 286
1488598626*x^2 - 795166559320*x^3 - 849459145920*x^4 - 288728294400*x^5 + 714147
84000*x^6 + 39813120000*x^7) - 60033082963*Sqrt[682]*(2 - 3*x)^2*Sqrt[(1 + 4*x)/
(-2 + 3*x)]*Sqrt[(-35 - 11*x + 10*x^2)/(2 - 3*x)^2]*EllipticPi[117/62, ArcSin[Sq
rt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62])))/(398131200*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 = 0.148, size = 949, normalized size = 2. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/284663808000*(7+5*x)^(1/2)*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)*(6243373
1183120*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))-684857410441904*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))-896111886589920*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*Ellipt
icE(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*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))-342428705220952
*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))-448055943294960*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))[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))-428035
88152619*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))-56006992911870*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^7+306369423360000*x^6-12386443829760
00*x^5-3644179735996800*x^4-3411264539482800*x^3+18436555308411240*x^2+156423669
08265240*x-16765465556439600)/(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. \[ \int{\left (5 \, x + 7\right )}^{\frac{5}{2}} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((5*x + 7)^(5/2)*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. \[{\rm integral}\left ({\left (25 \, x^{2} + 70 \, x + 49\right )} \sqrt{5 \, x + 7} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((25*x^2 + 70*x + 49)*sqrt(5*x + 7)*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((7+5*x)**(5/2)*(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. \[ \int{\left (5 \, x + 7\right )}^{\frac{5}{2}} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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