3.2812 \(\int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^{9/2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{184636 \sqrt{1-2 x} \sqrt{5 x+3}}{252105 \sqrt{3 x+2}}+\frac{974 \sqrt{1-2 x} \sqrt{5 x+3}}{36015 (3 x+2)^{3/2}}-\frac{536 \sqrt{1-2 x} \sqrt{5 x+3}}{5145 (3 x+2)^{5/2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{147 (3 x+2)^{7/2}}-\frac{9124 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{252105}-\frac{184636 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{252105} \]

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(147*(2 + 3*x)^(7/2)) - (536*Sqrt[1 - 2*x]*Sqrt[
3 + 5*x])/(5145*(2 + 3*x)^(5/2)) + (974*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(36015*(2 +
3*x)^(3/2)) + (184636*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(252105*Sqrt[2 + 3*x]) - (18
4636*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/252105 - (912
4*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/252105

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Rubi [A]  time = 0.422659, 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 \[ \frac{184636 \sqrt{1-2 x} \sqrt{5 x+3}}{252105 \sqrt{3 x+2}}+\frac{974 \sqrt{1-2 x} \sqrt{5 x+3}}{36015 (3 x+2)^{3/2}}-\frac{536 \sqrt{1-2 x} \sqrt{5 x+3}}{5145 (3 x+2)^{5/2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{147 (3 x+2)^{7/2}}-\frac{9124 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{252105}-\frac{184636 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{252105} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(147*(2 + 3*x)^(7/2)) - (536*Sqrt[1 - 2*x]*Sqrt[
3 + 5*x])/(5145*(2 + 3*x)^(5/2)) + (974*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(36015*(2 +
3*x)^(3/2)) + (184636*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(252105*Sqrt[2 + 3*x]) - (18
4636*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/252105 - (912
4*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/252105

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Rubi in Sympy [A]  time = 40.4077, size = 172, normalized size = 0.9 \[ \frac{184636 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{252105 \sqrt{3 x + 2}} + \frac{974 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{36015 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{536 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{5145 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{2 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{147 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{184636 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{756315} - \frac{9124 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{756315} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

184636*sqrt(-2*x + 1)*sqrt(5*x + 3)/(252105*sqrt(3*x + 2)) + 974*sqrt(-2*x + 1)*
sqrt(5*x + 3)/(36015*(3*x + 2)**(3/2)) - 536*sqrt(-2*x + 1)*sqrt(5*x + 3)/(5145*
(3*x + 2)**(5/2)) + 2*sqrt(-2*x + 1)*sqrt(5*x + 3)/(147*(3*x + 2)**(7/2)) - 1846
36*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/756315 - 9124*sqr
t(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/756315

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Mathematica [A]  time = 0.264225, size = 104, normalized size = 0.54 \[ \frac{2 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (2492586 x^3+5015853 x^2+3324960 x+727631\right )}{(3 x+2)^{7/2}}+\sqrt{2} \left (92318 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-17045 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{756315} \]

Antiderivative was successfully verified.

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

[Out]

(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(727631 + 3324960*x + 5015853*x^2 + 2492586*x
^3))/(2 + 3*x)^(7/2) + Sqrt[2]*(92318*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]]
, -33/2] - 17045*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/756315

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Maple [C]  time = 0.031, size = 505, normalized size = 2.6 \[{\frac{2}{7563150\,{x}^{2}+756315\,x-2268945} \left ( 460215\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-2492586\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}+920430\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-4985172\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+613620\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3323448\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+136360\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -738544\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-13338867\,{x}^{2}-27741747\,x-6548679 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/756315*(460215*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)-2492586*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)+920430*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^2*(3+5*x)^(1/2)*
(2+3*x)^(1/2)*(1-2*x)^(1/2)-4985172*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)+613620*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)-3323448*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)+136360*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))-738544*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellip
ticE(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))+7477758
0*x^5+157953348*x^4+92363085*x^3-13338867*x^2-27741747*x-6548679)*(1-2*x)^(1/2)*
(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((5*x + 3)^(3/2)/((81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*sqrt(3*x + 2)
*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((3+5*x)**(3/2)/(2+3*x)**(9/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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