3.2964 \(\int \frac{(2+3 x)^{9/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=187 \[ \frac{7 (3 x+2)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{665 (3 x+2)^{5/2}}{363 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{3284 \sqrt{1-2 x} (3 x+2)^{3/2}}{19965 \sqrt{5 x+3}}-\frac{153319 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{66550}-\frac{160297 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{30250 \sqrt{33}}-\frac{5327983 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{30250 \sqrt{33}} \]

[Out]

(3284*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(19965*Sqrt[3 + 5*x]) - (665*(2 + 3*x)^(5/2
))/(363*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^(7/2))/(33*(1 - 2*x)^(3/2)*S
qrt[3 + 5*x]) - (153319*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/66550 - (5327
983*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(30250*Sqrt[33]) - (16029
7*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(30250*Sqrt[33])

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Rubi [A]  time = 0.415272, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{7 (3 x+2)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{665 (3 x+2)^{5/2}}{363 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{3284 \sqrt{1-2 x} (3 x+2)^{3/2}}{19965 \sqrt{5 x+3}}-\frac{153319 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{66550}-\frac{160297 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{30250 \sqrt{33}}-\frac{5327983 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{30250 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(3284*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(19965*Sqrt[3 + 5*x]) - (665*(2 + 3*x)^(5/2
))/(363*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^(7/2))/(33*(1 - 2*x)^(3/2)*S
qrt[3 + 5*x]) - (153319*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/66550 - (5327
983*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(30250*Sqrt[33]) - (16029
7*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(30250*Sqrt[33])

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Rubi in Sympy [A]  time = 38.7033, size = 172, normalized size = 0.92 \[ \frac{3284 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}}{19965 \sqrt{5 x + 3}} - \frac{153319 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{66550} - \frac{5327983 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{998250} - \frac{160297 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1058750} - \frac{665 \left (3 x + 2\right )^{\frac{5}{2}}}{363 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} + \frac{7 \left (3 x + 2\right )^{\frac{7}{2}}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3284*sqrt(-2*x + 1)*(3*x + 2)**(3/2)/(19965*sqrt(5*x + 3)) - 153319*sqrt(-2*x +
1)*sqrt(3*x + 2)*sqrt(5*x + 3)/66550 - 5327983*sqrt(33)*elliptic_e(asin(sqrt(21)
*sqrt(-2*x + 1)/7), 35/33)/998250 - 160297*sqrt(35)*elliptic_f(asin(sqrt(55)*sqr
t(-2*x + 1)/11), 33/35)/1058750 - 665*(3*x + 2)**(5/2)/(363*sqrt(-2*x + 1)*sqrt(
5*x + 3)) + 7*(3*x + 2)**(7/2)/(33*(-2*x + 1)**(3/2)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.378951, size = 102, normalized size = 0.55 \[ \frac{-\frac{5 \sqrt{6 x+4} \left (1078110 x^3-11321446 x^2-3117099 x+2438391\right )}{(1-2 x)^{3/2} \sqrt{5 x+3}}-5366165 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{998250 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

((-5*Sqrt[4 + 6*x]*(2438391 - 3117099*x - 11321446*x^2 + 1078110*x^3))/((1 - 2*x
)^(3/2)*Sqrt[3 + 5*x]) + 10655966*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -3
3/2] - 5366165*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(998250*Sqrt[
2])

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Maple [C]  time = 0.036, size = 281, normalized size = 1.5 \[{\frac{1}{ \left ( 29947500\,{x}^{2}[phone]\,x+11979000 \right ) \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 10732330\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-21311932\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-5366165\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -32343300\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-10809750\,x-48767820 \right ) } \]

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)^(3/2),x)

[Out]

1/1996500*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(10732330*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)-21311932*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)-5366165*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))[phone]*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))-32343300*x^4+318081180*x^3+319941890
*x^2-10809750*x-48767820)/(15*x^2+19*x+6)/(-1+2*x)^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{3 \, x + 2}}{{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(9/2)/((5*x + 3)^(3/2)*(-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)/((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)**(9/2)/(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. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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