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

Optimal. Leaf size=243 \[ \frac{2}{45} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^3-\frac{61}{270} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^2-\frac{8141 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)}{2700}-\frac{5256763 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{97200}+\frac{5592499 \sqrt{\frac{11}{6}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{3888 \sqrt{2 x-5}}-\frac{17746949 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{29160 \sqrt{5-2 x}} \]

[Out]

(-5256763*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/97200 - (8141*Sqrt[2 - 3*x
]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x))/2700 - (61*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x
]*Sqrt[1 + 4*x]*(7 + 5*x)^2)/270 + (2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]
*(7 + 5*x)^3)/45 - (17746949*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2
- 3*x])/Sqrt[11]], -1/2])/(29160*Sqrt[5 - 2*x]) + (5592499*Sqrt[11/6]*Sqrt[5 - 2
*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(3888*Sqrt[-5 + 2*x])

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Rubi [A]  time = 0.720587, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229 \[ \frac{2}{45} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^3-\frac{61}{270} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^2-\frac{8141 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)}{2700}-\frac{5256763 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{97200}+\frac{5592499 \sqrt{\frac{11}{6}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{3888 \sqrt{2 x-5}}-\frac{17746949 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{29160 \sqrt{5-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-5256763*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/97200 - (8141*Sqrt[2 - 3*x
]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x))/2700 - (61*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x
]*Sqrt[1 + 4*x]*(7 + 5*x)^2)/270 + (2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]
*(7 + 5*x)^3)/45 - (17746949*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2
- 3*x])/Sqrt[11]], -1/2])/(29160*Sqrt[5 - 2*x]) + (5592499*Sqrt[11/6]*Sqrt[5 - 2
*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(3888*Sqrt[-5 + 2*x])

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Rubi in Sympy [A]  time = 94.6192, size = 286, normalized size = 1.18 \[ - \frac{25 \left (- 3 x + 2\right )^{\frac{3}{2}} \sqrt{2 x - 5} \left (4 x + 1\right )^{\frac{5}{2}}}{216} - \frac{655 \left (- 3 x + 2\right )^{\frac{3}{2}} \sqrt{2 x - 5} \left (4 x + 1\right )^{\frac{3}{2}}}{756} + \frac{115 \sqrt{- 3 x + 2} \left (2 x - 5\right )^{\frac{3}{2}} \left (4 x + 1\right )^{\frac{3}{2}}}{112} + \frac{49319 \sqrt{- 3 x + 2} \sqrt{2 x - 5} \left (4 x + 1\right )^{\frac{3}{2}}}{7560} - \frac{684673 \sqrt{- 3 x + 2} \sqrt{2 x - 5} \sqrt{4 x + 1}}{9720} - \frac{17746949 \sqrt{11} \sqrt{\frac{12 x}{11} + \frac{3}{11}} \sqrt{2 x - 5} E\left (\operatorname{asin}{\left (\frac{2 \sqrt{11} \sqrt{- 3 x + 2}}{11} \right )}\middle | - \frac{1}{2}\right )}{29160 \sqrt{- \frac{6 x}{11} + \frac{15}{11}} \sqrt{4 x + 1}} + \frac{61517489 \sqrt{33} \sqrt{- \frac{12 x}{11} + \frac{8}{11}} \sqrt{- \frac{4 x}{11} + \frac{10}{11}} F\left (\operatorname{asin}{\left (\frac{\sqrt{33} \sqrt{4 x + 1}}{11} \right )}\middle | \frac{1}{3}\right )}{46656 \sqrt{- 3 x + 2} \sqrt{2 x - 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25*(-3*x + 2)**(3/2)*sqrt(2*x - 5)*(4*x + 1)**(5/2)/216 - 655*(-3*x + 2)**(3/2)
*sqrt(2*x - 5)*(4*x + 1)**(3/2)/756 + 115*sqrt(-3*x + 2)*(2*x - 5)**(3/2)*(4*x +
1)**(3/2)/112 + 49319*sqrt(-3*x + 2)*sqrt(2*x - 5)*(4*x + 1)**(3/2)/7560 - 6846
73*sqrt(-3*x + 2)*sqrt(2*x - 5)*sqrt(4*x + 1)/9720 - 17746949*sqrt(11)*sqrt(12*x
/11 + 3/11)*sqrt(2*x - 5)*elliptic_e(asin(2*sqrt(11)*sqrt(-3*x + 2)/11), -1/2)/(
29160*sqrt(-6*x/11 + 15/11)*sqrt(4*x + 1)) + 61517489*sqrt(33)*sqrt(-12*x/11 + 8
/11)*sqrt(-4*x/11 + 10/11)*elliptic_f(asin(sqrt(33)*sqrt(4*x + 1)/11), 1/3)/(466
56*sqrt(-3*x + 2)*sqrt(2*x - 5))

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Mathematica [A]  time = 0.342668, size = 130, normalized size = 0.53 \[ \frac{6 \sqrt{2-3 x} \sqrt{4 x+1} \left (216000 x^4+147600 x^3-1649952 x^2-2933650 x+6902575\right )[phone] \sqrt{66} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )-35493898 \sqrt{66} \sqrt{5-2 x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{116640 \sqrt{2 x-5}} \]

Antiderivative was successfully verified.

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

[Out]

(6*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(6902575 - 2933650*x - 1649952*x^2 + 147600*x^3 +
216000*x^4) - 35493898*Sqrt[66]*Sqrt[5 - 2*x]*EllipticE[ArcSin[Sqrt[3/11]*Sqrt[
1 + 4*x]], 1/3] + 27962495*Sqrt[66]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sq
rt[1 + 4*x]], 1/3])/(116640*Sqrt[-5 + 2*x])

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Maple [A]  time = 0.017, size = 161, normalized size = 0.7 \[{\frac{1}{2799360\,{x}^{3}-8164800\,{x}^{2}+2449440\,x+1166400}\sqrt{2-3\,x}\sqrt{-5+2\,x}\sqrt{1+4\,x} \left ( 15552000\,{x}^{6}[phone]\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticF} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -70987796\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticE} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) +4147200\,{x}^{5}-125816544\,{x}^{4}-163495440\,{x}^{3}[phone]\,{x}^{2}-171873450\,x-82830900 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/116640*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)*(15552000*x^6+83887485*11^(1
/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(1+4*x)^(1/2)*EllipticF(2/11*(2-3*x)^(1/2)*11^(1
/2),1/2*I*2^(1/2))-70987796*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(1+4*x)^(1/2)*E
llipticE(2/11*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))+4147200*x^5-125816544*x^4-16
3495440*x^3+604794324*x^2-171873450*x-82830900)/(24*x^3-70*x^2+21*x+10)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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