3.2975 \(\int \frac{(2+3 x)^{9/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=187 \[ \frac{7 (3 x+2)^{7/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{140 (3 x+2)^{5/2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{2063 \sqrt{1-2 x} (3 x+2)^{3/2}}{19965 (5 x+3)^{3/2}}+\frac{70226 \sqrt{1-2 x} \sqrt{3 x+2}}{1098075 \sqrt{5 x+3}}-\frac{76163 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{166375 \sqrt{33}}-\frac{4971289 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{332750 \sqrt{33}} \]

[Out]

(2063*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(19965*(3 + 5*x)^(3/2)) - (140*(2 + 3*x)^(5
/2))/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^(7/2))/(33*(1 - 2*x)^(3/
2)*(3 + 5*x)^(3/2)) + (70226*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1098075*Sqrt[3 + 5*x]
) - (4971289*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(332750*Sqrt[33]
) - (76163*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(166375*Sqrt[33])

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Rubi [A]  time = 0.428905, 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 \[ \frac{7 (3 x+2)^{7/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{140 (3 x+2)^{5/2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{2063 \sqrt{1-2 x} (3 x+2)^{3/2}}{19965 (5 x+3)^{3/2}}+\frac{70226 \sqrt{1-2 x} \sqrt{3 x+2}}{1098075 \sqrt{5 x+3}}-\frac{76163 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{166375 \sqrt{33}}-\frac{4971289 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{332750 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x)^(9/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

(2063*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(19965*(3 + 5*x)^(3/2)) - (140*(2 + 3*x)^(5
/2))/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^(7/2))/(33*(1 - 2*x)^(3/
2)*(3 + 5*x)^(3/2)) + (70226*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1098075*Sqrt[3 + 5*x]
) - (4971289*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(332750*Sqrt[33]
) - (76163*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(166375*Sqrt[33])

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Rubi in Sympy [A]  time = 38.3454, size = 172, normalized size = 0.92 \[ \frac{2063 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}}{19965 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{70226 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{1098075 \sqrt{5 x + 3}} - \frac{4971289 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{10980750} - \frac{76163 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{5823125} - \frac{140 \left (3 x + 2\right )^{\frac{5}{2}}}{121 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{7 \left (3 x + 2\right )^{\frac{7}{2}}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**(9/2)/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

2063*sqrt(-2*x + 1)*(3*x + 2)**(3/2)/(19965*(5*x + 3)**(3/2)) + 70226*sqrt(-2*x
+ 1)*sqrt(3*x + 2)/(1098075*sqrt(5*x + 3)) - 4971289*sqrt(33)*elliptic_e(asin(sq
rt(21)*sqrt(-2*x + 1)/7), 35/33)/10980750 - 76163*sqrt(35)*elliptic_f(asin(sqrt(
55)*sqrt(-2*x + 1)/11), 33/35)/5823125 - 140*(3*x + 2)**(5/2)/(121*sqrt(-2*x + 1
)*(5*x + 3)**(3/2)) + 7*(3*x + 2)**(7/2)/(33*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.367931, size = 107, normalized size = 0.57 \[ \frac{\frac{10 \sqrt{3 x+2} \left (31924075 x^3+30619782 x^2+2244393 x-2780992\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}-2457910 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+4971289 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{10980750} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x)^(9/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

((10*Sqrt[2 + 3*x]*(-2780992 + 2244393*x + 30619782*x^2 + 31924075*x^3))/((1 - 2
*x)^(3/2)*(3 + 5*x)^(3/2)) + 4971289*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]], -33/2] - 2457910*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -3
3/2])/10980750

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Maple [C]  time = 0.034, size = 383, normalized size = 2.1 \[{\frac{1}{10980750\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 24579100\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-49712890\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+2457910\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-4971289\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-7373730\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-38541900\,x-55619840 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{2+3\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^(9/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x)

[Out]

1/10980750*(1-2*x)^(1/2)*(24579100*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)-49712890*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)+2457910*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*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-4971289*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)-7373730*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))+
14913867*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))[phone]*x^4+155707496
0*x^3+679727430*x^2-38541900*x-55619840)/(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. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{3 \, x + 2}}{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)),x, algorithm="fricas")

[Out]

integral((81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*sqrt(3*x + 2)/((100*x^4 + 20*x
^3 - 59*x^2 - 6*x + 9)*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)**(9/2)/(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. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)), x)