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

Optimal. Leaf size=98 \[ \frac{13 \sqrt{\frac{3}{22}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{\sqrt{2 x-5}}-\frac{5 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{6 \sqrt{5-2 x}} \]

[Out]

(-5*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])
/(6*Sqrt[5 - 2*x]) + (13*Sqrt[3/22]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sq
rt[1 + 4*x]], 1/3])/Sqrt[-5 + 2*x]

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Rubi [A]  time = 0.252159, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ \frac{13 \sqrt{\frac{3}{22}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{\sqrt{2 x-5}}-\frac{5 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{6 \sqrt{5-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])
/(6*Sqrt[5 - 2*x]) + (13*Sqrt[3/22]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sq
rt[1 + 4*x]], 1/3])/Sqrt[-5 + 2*x]

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Rubi in Sympy [A]  time = 22.2094, size = 141, normalized size = 1.44 \[ - \frac{5 \sqrt{11} \sqrt{\frac{12 x}{11} + \frac{3}{11}} \sqrt{2 x - 5} E\left (\operatorname{asin}{\left (\frac{2 \sqrt{11} \sqrt{- 3 x + 2}}{11} \right )}\middle | - \frac{1}{2}\right )}{6 \sqrt{- \frac{6 x}{11} + \frac{15}{11}} \sqrt{4 x + 1}} + \frac{13 \sqrt{33} \sqrt{- \frac{12 x}{11} + \frac{8}{11}} \sqrt{- \frac{4 x}{11} + \frac{10}{11}} F\left (\operatorname{asin}{\left (\frac{\sqrt{33} \sqrt{4 x + 1}}{11} \right )}\middle | \frac{1}{3}\right )}{4 \sqrt{- 3 x + 2} \sqrt{2 x - 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*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)/(6*sqrt(-6*x/11 + 15/11)*sqrt(4*x + 1)) + 13*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)/(4*sqrt(-3*x + 2)*sqrt(2*x - 5))

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Mathematica [A]  time = 0.725348, size = 187, normalized size = 1.91 \[ \frac{220 \left (6 x^2-19 x+10\right ) \sqrt{4 x+1}-124 \sqrt{22} \sqrt{\frac{2 x-5}{4 x+1}} \sqrt{\frac{3 x-2}{4 x+1}} (4 x+1)^2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{\frac{11}{3}}}{\sqrt{4 x+1}}\right )\right |3\right )+55 \sqrt{22} \sqrt{\frac{2 x-5}{4 x+1}} \sqrt{\frac{3 x-2}{4 x+1}} (4 x+1)^2 E\left (\left .\sin ^{-1}\left (\frac{\sqrt{\frac{11}{3}}}{\sqrt{4 x+1}}\right )\right |3\right )}{132 \sqrt{2-3 x} \sqrt{2 x-5} (4 x+1)} \]

Antiderivative was successfully verified.

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

[Out]

(220*Sqrt[1 + 4*x]*(10 - 19*x + 6*x^2) + 55*Sqrt[22]*Sqrt[(-5 + 2*x)/(1 + 4*x)]*
Sqrt[(-2 + 3*x)/(1 + 4*x)]*(1 + 4*x)^2*EllipticE[ArcSin[Sqrt[11/3]/Sqrt[1 + 4*x]
], 3] - 124*Sqrt[22]*Sqrt[(-5 + 2*x)/(1 + 4*x)]*Sqrt[(-2 + 3*x)/(1 + 4*x)]*(1 +
4*x)^2*EllipticF[ArcSin[Sqrt[11/3]/Sqrt[1 + 4*x]], 3])/(132*Sqrt[2 - 3*x]*Sqrt[-
5 + 2*x]*(1 + 4*x))

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Maple [A]  time = 0.021, size = 63, normalized size = 0.6 \[ -{\frac{\sqrt{11}}{66} \left ( 117\,{\it EllipticF} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -55\,{\it EllipticE} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) \right ) \sqrt{5-2\,x}{\frac{1}{\sqrt{-5+2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/66*(117*EllipticF(2/11*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))-55*EllipticE(2/1
1*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2)))*(5-2*x)^(1/2)*11^(1/2)/(-5+2*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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