3.2902 \(\int \frac{(2+3 x)^{5/2}}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=129 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{3/2}}{11 \sqrt{1-2 x}}+\frac{69}{55} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{24}{25} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{1597}{50} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(69*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/55 + (7*(2 + 3*x)^(3/2)*Sqrt[3 +
5*x])/(11*Sqrt[1 - 2*x]) + (1597*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/50 + (24*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/25

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Rubi [A]  time = 0.257777, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{3/2}}{11 \sqrt{1-2 x}}+\frac{69}{55} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{24}{25} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{1597}{50} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(69*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/55 + (7*(2 + 3*x)^(3/2)*Sqrt[3 +
5*x])/(11*Sqrt[1 - 2*x]) + (1597*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/50 + (24*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/25

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Rubi in Sympy [A]  time = 24.7397, size = 114, normalized size = 0.88 \[ \frac{69 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{55} + \frac{1597 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{550} + \frac{24 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{275} + \frac{7 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{11 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

69*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/55 + 1597*sqrt(33)*elliptic_e(asin
(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/550 + 24*sqrt(33)*elliptic_f(asin(sqrt(21)*s
qrt(-2*x + 1)/7), 35/33)/275 + 7*(3*x + 2)**(3/2)*sqrt(5*x + 3)/(11*sqrt(-2*x +
1))

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Mathematica [A]  time = 0.139226, size = 105, normalized size = 0.81 \[ \frac{10 \sqrt{3 x+2} \sqrt{5 x+3} (139-33 x)+805 \sqrt{2-4 x} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-1597 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{550 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(10*(139 - 33*x)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] - 1597*Sqrt[2 - 4*x]*EllipticE[ArcS
in[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 805*Sqrt[2 - 4*x]*EllipticF[ArcSin[Sqrt[2
/11]*Sqrt[3 + 5*x]], -33/2])/(550*Sqrt[1 - 2*x])

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Maple [C]  time = 0.026, size = 164, normalized size = 1.3 \[ -{\frac{1}{16500\,{x}^{3}+12650\,{x}^{2}-3850\,x-3300}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 805\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1597\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -4950\,{x}^{3}+14580\,{x}^{2}+24430\,x+8340 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/550*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(805*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))-1597*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ell
ipticE(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))-4950*
x^3+14580*x^2+24430*x+8340)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{3 \, x + 2}}{\sqrt{5 \, x + 3}{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(9*x^2 + 12*x + 4)*sqrt(3*x + 2)/(sqrt(5*x + 3)*(2*x - 1)*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)**(5/2)/(1-2*x)**(3/2)/(3+5*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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