3.2889 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{338 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 \sqrt{3 x+2}}-\frac{458 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 (3 x+2)^{3/2}}-\frac{163 \sqrt{1-2 x} \sqrt{5 x+3}}{245 (3 x+2)^{5/2}}+\frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{992 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005}-\frac{338 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005} \]

[Out]

(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) - (163*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(245*(2 + 3*x)^(5/2)) - (458*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1715*(2 + 3*
x)^(3/2)) + (338*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12005*Sqrt[2 + 3*x]) - (338*Sqrt[
11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/12005 - (992*Sqrt[11/3]
*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/12005

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Rubi [A]  time = 0.433784, antiderivative size = 191, 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 \[ \frac{338 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 \sqrt{3 x+2}}-\frac{458 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 (3 x+2)^{3/2}}-\frac{163 \sqrt{1-2 x} \sqrt{5 x+3}}{245 (3 x+2)^{5/2}}+\frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{992 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005}-\frac{338 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005} \]

Antiderivative was successfully verified.

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

[Out]

(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) - (163*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(245*(2 + 3*x)^(5/2)) - (458*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1715*(2 + 3*
x)^(3/2)) + (338*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12005*Sqrt[2 + 3*x]) - (338*Sqrt[
11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/12005 - (992*Sqrt[11/3]
*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/12005

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Rubi in Sympy [A]  time = 38.6245, size = 172, normalized size = 0.9 \[ \frac{338 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{12005 \sqrt{3 x + 2}} - \frac{458 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1715 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{163 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{245 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{338 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{36015} - \frac{992 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{36015} + \frac{11 \sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

338*sqrt(-2*x + 1)*sqrt(5*x + 3)/(12005*sqrt(3*x + 2)) - 458*sqrt(-2*x + 1)*sqrt
(5*x + 3)/(1715*(3*x + 2)**(3/2)) - 163*sqrt(-2*x + 1)*sqrt(5*x + 3)/(245*(3*x +
2)**(5/2)) - 338*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/36
015 - 992*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/36015 + 11
*sqrt(5*x + 3)/(7*sqrt(-2*x + 1)*(3*x + 2)**(5/2))

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Mathematica [A]  time = 0.243253, size = 104, normalized size = 0.54 \[ \frac{2 \left (\sqrt{2} \left (8015 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+169 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{3 \sqrt{5 x+3} \left (3042 x^3-7083 x^2-10266 x-2909\right )}{\sqrt{1-2 x} (3 x+2)^{5/2}}\right )}{36015} \]

Antiderivative was successfully verified.

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

[Out]

(2*((-3*Sqrt[3 + 5*x]*(-2909 - 10266*x - 7083*x^2 + 3042*x^3))/(Sqrt[1 - 2*x]*(2
+ 3*x)^(5/2)) + Sqrt[2]*(169*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]
+ 8015*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/36015

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Maple [C]  time = 0.036, size = 386, normalized size = 2. \[ -{\frac{2}{360150\,{x}^{2}+36015\,x-108045}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 72135\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1521\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+96180\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+2028\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+32060\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +676\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -45630\,{x}^{4}+78867\,{x}^{3}+217737\,{x}^{2}+136029\,x+26181 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/36015*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(72135*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)+1521*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)+96180*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)+2028*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)+32060*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))+67
6*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))-45630*x^4+78867*x^3+217737*x^2
+136029*x+26181)/(2+3*x)^(5/2)/(10*x^2+x-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(5*x + 3)^(3/2)/((54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*sqrt(3*x + 2)*s
qrt(-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((3+5*x)**(3/2)/(1-2*x)**(3/2)/(2+3*x)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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