3.2943 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{496 \sqrt{1-2 x} \sqrt{5 x+3}}{2401 \sqrt{3 x+2}}-\frac{89 \sqrt{1-2 x} \sqrt{5 x+3}}{343 (3 x+2)^{3/2}}+\frac{58 \sqrt{5 x+3}}{147 \sqrt{1-2 x} (3 x+2)^{3/2}}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac{582 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2401}+\frac{496 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2401} \]

[Out]

(11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)) + (58*Sqrt[3 + 5*x])/(14
7*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) - (89*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(343*(2 + 3*
x)^(3/2)) - (496*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2401*Sqrt[2 + 3*x]) + (496*Sqrt[1
1/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2401 - (582*Sqrt[3/11]*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2401

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Rubi [A]  time = 0.429901, 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 \[ -\frac{496 \sqrt{1-2 x} \sqrt{5 x+3}}{2401 \sqrt{3 x+2}}-\frac{89 \sqrt{1-2 x} \sqrt{5 x+3}}{343 (3 x+2)^{3/2}}+\frac{58 \sqrt{5 x+3}}{147 \sqrt{1-2 x} (3 x+2)^{3/2}}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac{582 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2401}+\frac{496 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2401} \]

Antiderivative was successfully verified.

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

[Out]

(11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)) + (58*Sqrt[3 + 5*x])/(14
7*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) - (89*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(343*(2 + 3*
x)^(3/2)) - (496*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2401*Sqrt[2 + 3*x]) + (496*Sqrt[1
1/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2401 - (582*Sqrt[3/11]*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2401

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Rubi in Sympy [A]  time = 37.7118, size = 172, normalized size = 0.9 \[ - \frac{496 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2401 \sqrt{3 x + 2}} + \frac{496 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{7203} - \frac{582 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{26411} + \frac{178 \sqrt{5 x + 3}}{1029 \sqrt{- 2 x + 1} \sqrt{3 x + 2}} - \frac{31 \sqrt{5 x + 3}}{147 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{11 \sqrt{5 x + 3}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-496*sqrt(-2*x + 1)*sqrt(5*x + 3)/(2401*sqrt(3*x + 2)) + 496*sqrt(33)*elliptic_e
(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/7203 - 582*sqrt(33)*elliptic_f(asin(sqr
t(21)*sqrt(-2*x + 1)/7), 35/33)/26411 + 178*sqrt(5*x + 3)/(1029*sqrt(-2*x + 1)*s
qrt(3*x + 2)) - 31*sqrt(5*x + 3)/(147*sqrt(-2*x + 1)*(3*x + 2)**(3/2)) + 11*sqrt
(5*x + 3)/(21*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2))

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Mathematica [A]  time = 0.258588, size = 104, normalized size = 0.54 \[ \frac{\sqrt{2} \left (3115 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-496 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{2 \sqrt{5 x+3} \left (8928 x^3+762 x^2-4616 x-885\right )}{(1-2 x)^{3/2} (3 x+2)^{3/2}}}{7203} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[3 + 5*x]*(-885 - 4616*x + 762*x^2 + 8928*x^3))/((1 - 2*x)^(3/2)*(2 + 3
*x)^(3/2)) + Sqrt[2]*(-496*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] +
3115*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/7203

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Maple [C]  time = 0.035, size = 383, normalized size = 2. \[ -{\frac{1}{7203\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 18690\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2976\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+3115\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-496\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-6230\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +992\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +89280\,{x}^{4}+61188\,{x}^{3}-41588\,{x}^{2}-36546\,x-5310 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7203*(1-2*x)^(1/2)*(18690*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)
-2976*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)+3115*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)-496*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)-6230*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))+992*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))+89280*x^4+61188*x^3-41588*x^2-36546*x-5310)/(2+3
*x)^(3/2)/(-1+2*x)^2/(3+5*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((5*x + 3)^(3/2)/((36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*sqrt(3*x + 2)*sqr
t(-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)**(5/2)/(2+3*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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