3.2911 \(\int \frac{(2+3 x)^{5/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=129 \[ \frac{7 (3 x+2)^{3/2}}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} \sqrt{3 x+2}}{605 \sqrt{5 x+3}}+\frac{31}{275} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{1159}{275} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-37*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(605*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^(3/2))/(11*
Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (1159*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[
1 - 2*x]], 35/33])/275 + (31*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/275

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Rubi [A]  time = 0.258724, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{7 (3 x+2)^{3/2}}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} \sqrt{3 x+2}}{605 \sqrt{5 x+3}}+\frac{31}{275} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{1159}{275} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-37*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(605*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^(3/2))/(11*
Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (1159*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[
1 - 2*x]], 35/33])/275 + (31*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/275

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Rubi in Sympy [A]  time = 25.5945, size = 114, normalized size = 0.88 \[ - \frac{37 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{605 \sqrt{5 x + 3}} + \frac{1159 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3025} + \frac{93 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{9625} + \frac{7 \left (3 x + 2\right )^{\frac{3}{2}}}{11 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-37*sqrt(-2*x + 1)*sqrt(3*x + 2)/(605*sqrt(5*x + 3)) + 1159*sqrt(33)*elliptic_e(
asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/3025 + 93*sqrt(35)*elliptic_f(asin(sqrt(
55)*sqrt(-2*x + 1)/11), 33/35)/9625 + 7*(3*x + 2)**(3/2)/(11*sqrt(-2*x + 1)*sqrt
(5*x + 3))

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Mathematica [A]  time = 0.21645, size = 122, normalized size = 0.95 \[ \frac{10 \sqrt{3 x+2} \sqrt{5 x+3} (1229 x+733)+1295 \sqrt{2-4 x} (5 x+3) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2318 \sqrt{2-4 x} (5 x+3) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{6050 \sqrt{1-2 x} (5 x+3)} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(733 + 1229*x) - 2318*Sqrt[2 - 4*x]*(3 + 5*x)*El
lipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 1295*Sqrt[2 - 4*x]*(3 + 5*x)*
EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(6050*Sqrt[1 - 2*x]*(3 + 5*x
))

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Maple [C]  time = 0.029, size = 159, normalized size = 1.2 \[ -{\frac{1}{181500\,{x}^{3}+139150\,{x}^{2}-42350\,x-36300}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 1295\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2318\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +36870\,{x}^{2}+46570\,x+14660 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6050*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(1295*2^(1/2)*(3+5*x)^(1/2)*(2
+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11
^(1/2)*3^(1/2)*2^(1/2))-2318*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*E
llipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))+368
70*x^2+46570*x+14660)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{3 \, x + 2}}{{\left (10 \, x^{2} + x - 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(9*x^2 + 12*x + 4)*sqrt(3*x + 2)/((10*x^2 + x - 3)*sqrt(5*x + 3)*sqrt(
-2*x + 1)), 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((2+3*x)**(5/2)/(1-2*x)**(3/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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