3.2966 \(\int \frac{(2+3 x)^{5/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=156 \[ \frac{7 (3 x+2)^{3/2}}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}+\frac{974 \sqrt{1-2 x} \sqrt{3 x+2}}{3993 \sqrt{5 x+3}}-\frac{203 \sqrt{3 x+2}}{363 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{41 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{605 \sqrt{33}}-\frac{974 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{605 \sqrt{33}} \]

[Out]

(-203*Sqrt[2 + 3*x])/(363*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (974*Sqrt[1 - 2*x]*Sqrt
[2 + 3*x])/(3993*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^(3/2))/(33*(1 - 2*x)^(3/2)*Sqrt[3
+ 5*x]) - (974*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(605*Sqrt[33]
) - (41*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(605*Sqrt[33])

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Rubi [A]  time = 0.341992, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{7 (3 x+2)^{3/2}}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}+\frac{974 \sqrt{1-2 x} \sqrt{3 x+2}}{3993 \sqrt{5 x+3}}-\frac{203 \sqrt{3 x+2}}{363 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{41 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{605 \sqrt{33}}-\frac{974 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{605 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-203*Sqrt[2 + 3*x])/(363*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (974*Sqrt[1 - 2*x]*Sqrt
[2 + 3*x])/(3993*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^(3/2))/(33*(1 - 2*x)^(3/2)*Sqrt[3
+ 5*x]) - (974*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(605*Sqrt[33]
) - (41*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(605*Sqrt[33])

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Rubi in Sympy [A]  time = 30.6082, size = 143, normalized size = 0.92 \[ \frac{974 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{3993 \sqrt{5 x + 3}} - \frac{974 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{19965} - \frac{41 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{21175} - \frac{203 \sqrt{3 x + 2}}{363 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} + \frac{7 \left (3 x + 2\right )^{\frac{3}{2}}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

974*sqrt(-2*x + 1)*sqrt(3*x + 2)/(3993*sqrt(5*x + 3)) - 974*sqrt(33)*elliptic_e(
asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/19965 - 41*sqrt(35)*elliptic_f(asin(sqrt
(55)*sqrt(-2*x + 1)/11), 33/35)/21175 - 203*sqrt(3*x + 2)/(363*sqrt(-2*x + 1)*sq
rt(5*x + 3)) + 7*(3*x + 2)**(3/2)/(33*(-2*x + 1)**(3/2)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.219056, size = 102, normalized size = 0.65 \[ \frac{\frac{10 \sqrt{3 x+2} \left (3896 x^2+3111 x+435\right )}{(1-2 x)^{3/2} \sqrt{5 x+3}}-595 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+1948 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{39930} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[2 + 3*x]*(435 + 3111*x + 3896*x^2))/((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) +
1948*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 595*Sqrt[2]*El
lipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/39930

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Maple [C]  time = 0.034, size = 276, normalized size = 1.8 \[{\frac{1}{ \left ( 598950\,{x}^{2}+758670\,x+239580 \right ) \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 1190\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3896\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-595\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1948\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +116880\,{x}^{3}+171250\,{x}^{2}+75270\,x+8700 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/39930*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(1190*2^(1/2)*EllipticF(1/11*1
1^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x*(3+5*x)^(1/2)*(2
+3*x)^(1/2)*(1-2*x)^(1/2)-3896*2^(1/2)*EllipticE(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))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)
-595*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))+1948*2^(1/2)*(3+5*x)^(1/2)*
(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(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))+116880*x^3+171250*x^2+75270*x+8700)/(15*x^2+19*x+6)/(-
1+2*x)^2

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

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)^(5/2)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{3 \, x + 2}}{{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

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)^(5/2)),x, algorithm="fricas")

[Out]

integral((9*x^2 + 12*x + 4)*sqrt(3*x + 2)/((20*x^3 - 8*x^2 - 7*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)**(5/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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

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)^(5/2)),x, algorithm="giac")

[Out]

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