3.2978 \(\int \frac{(2+3 x)^{3/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=187 \[ -\frac{2960 \sqrt{1-2 x} \sqrt{3 x+2}}{43923 \sqrt{5 x+3}}-\frac{575 \sqrt{1-2 x} \sqrt{3 x+2}}{3993 (5 x+3)^{3/2}}+\frac{26 \sqrt{3 x+2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{7 \sqrt{3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{230 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1331 \sqrt{33}}+\frac{592 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1331 \sqrt{33}} \]

[Out]

(7*Sqrt[2 + 3*x])/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (26*Sqrt[2 + 3*x])/(121
*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (575*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3993*(3 + 5
*x)^(3/2)) - (2960*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(43923*Sqrt[3 + 5*x]) + (592*Ell
ipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1331*Sqrt[33]) - (230*EllipticF
[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1331*Sqrt[33])

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Rubi [A]  time = 0.424312, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{2960 \sqrt{1-2 x} \sqrt{3 x+2}}{43923 \sqrt{5 x+3}}-\frac{575 \sqrt{1-2 x} \sqrt{3 x+2}}{3993 (5 x+3)^{3/2}}+\frac{26 \sqrt{3 x+2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{7 \sqrt{3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{230 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1331 \sqrt{33}}+\frac{592 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1331 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[2 + 3*x])/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (26*Sqrt[2 + 3*x])/(121
*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (575*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3993*(3 + 5
*x)^(3/2)) - (2960*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(43923*Sqrt[3 + 5*x]) + (592*Ell
ipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1331*Sqrt[33]) - (230*EllipticF
[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1331*Sqrt[33])

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Rubi in Sympy [A]  time = 38.033, size = 172, normalized size = 0.92 \[ \frac{592 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{43923} - \frac{230 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{43923} + \frac{1184 \sqrt{3 x + 2} \sqrt{5 x + 3}}{43923 \sqrt{- 2 x + 1}} - \frac{362 \sqrt{3 x + 2}}{3993 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} - \frac{37 \sqrt{3 x + 2}}{363 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{7 \sqrt{3 x + 2}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

592*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/43923 - 230*sqrt
(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/43923 + 1184*sqrt(3*x +
2)*sqrt(5*x + 3)/(43923*sqrt(-2*x + 1)) - 362*sqrt(3*x + 2)/(3993*sqrt(-2*x + 1)
*sqrt(5*x + 3)) - 37*sqrt(3*x + 2)/(363*sqrt(-2*x + 1)*(5*x + 3)**(3/2)) + 7*sqr
t(3*x + 2)/(33*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.257283, size = 104, normalized size = 0.56 \[ \frac{\sqrt{2} \left (4387 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-592 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{2 \sqrt{3 x+2} \left (29600 x^3+810 x^2-13572 x-1775\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}}{43923} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[2 + 3*x]*(-1775 - 13572*x + 810*x^2 + 29600*x^3))/((1 - 2*x)^(3/2)*(3
+ 5*x)^(3/2)) + Sqrt[2]*(-592*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]
+ 4387*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/43923

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Maple [C]  time = 0.033, size = 383, normalized size = 2.1 \[ -{\frac{1}{43923\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 43870\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-5920\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+4387\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-592\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-13161\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1776\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +177600\,{x}^{4}+123260\,{x}^{3}-78192\,{x}^{2}-64938\,x-7100 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{2+3\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/43923*(1-2*x)^(1/2)*(43870*2^(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))*x^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2
)-5920*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^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+4387*2^(1/2)*Elliptic
F(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)-592*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)-13161*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(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))+1776*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))+177600*x^4+123260*x^3-78192*x^2-64938*x-7100)
/(3+5*x)^(3/2)/(-1+2*x)^2/(2+3*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (3 \, x + 2\right )}^{\frac{3}{2}}}{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((3*x + 2)^(3/2)/((100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*sqrt(5*x + 3)*sq
rt(-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)**(3/2)/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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