3.2762 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx\)

Optimal. Leaf size=222 \[ \frac{5260}{567} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 \sqrt{3 x+2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}-\frac{31298}{567} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{135334 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{5103}+\frac{135334 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25515}-\frac{452399 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25515} \]

[Out]

(135334*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/5103 - (31298*Sqrt[1 - 2*x]*S
qrt[2 + 3*x]*(3 + 5*x)^(3/2))/567 - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(9*(2 +
3*x)^(3/2)) + (370*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(27*Sqrt[2 + 3*x]) + (5260*S
qrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/567 - (452399*Sqrt[11/3]*EllipticE[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/25515 + (135334*Sqrt[11/3]*EllipticF[Arc
Sin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/25515

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Rubi [A]  time = 0.484889, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{5260}{567} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 \sqrt{3 x+2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}-\frac{31298}{567} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{135334 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{5103}+\frac{135334 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25515}-\frac{452399 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25515} \]

Antiderivative was successfully verified.

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

[Out]

(135334*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/5103 - (31298*Sqrt[1 - 2*x]*S
qrt[2 + 3*x]*(3 + 5*x)^(3/2))/567 - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(9*(2 +
3*x)^(3/2)) + (370*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(27*Sqrt[2 + 3*x]) + (5260*S
qrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/567 - (452399*Sqrt[11/3]*EllipticE[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/25515 + (135334*Sqrt[11/3]*EllipticF[Arc
Sin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/25515

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Rubi in Sympy [A]  time = 49.7736, size = 201, normalized size = 0.91 \[ - \frac{370 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{189 \sqrt{3 x + 2}} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{9 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{940 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{567} - \frac{2368 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{567} + \frac{135334 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{5103} - \frac{452399 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{76545} + \frac{135334 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{76545} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-370*(-2*x + 1)**(5/2)*(5*x + 3)**(3/2)/(189*sqrt(3*x + 2)) - 2*(-2*x + 1)**(5/2
)*(5*x + 3)**(5/2)/(9*(3*x + 2)**(3/2)) - 940*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*(5
*x + 3)**(3/2)/567 - 2368*sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(3/2)/567 + 13
5334*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/5103 - 452399*sqrt(33)*elliptic_
e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/76545 + 135334*sqrt(33)*elliptic_f(asi
n(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/76545

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Mathematica [A]  time = 0.418589, size = 112, normalized size = 0.5 \[ \frac{\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} \left (24300 x^4-25110 x^3+5949 x^2+108285 x+56963\right )}{(3 x+2)^{3/2}}-2685410 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+452399 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{76545} \]

Antiderivative was successfully verified.

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(56963 + 108285*x + 5949*x^2 - 25110*x^3 + 2430
0*x^4))/(2 + 3*x)^(3/2) + 452399*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]], -33/2] - 2685410*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]
)/76545

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Maple [C]  time = 0.029, size = 282, normalized size = 1.3 \[{\frac{1}{765450\,{x}^{2}+76545\,x-229635} \left ( 8056230\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1357197\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+7290000\,{x}^{6}+5370820\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -904798\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -6804000\,{x}^{5}-1155600\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-8036760\,x-5126670 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/76545*(8056230*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)-1357197*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)+7290000*x^6+5370820*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))-904798*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))-6804000*x^5-1155600*x^4+34923870*x^3+19802040*x^2-8036760*x-5126670)*(3+
5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((9*
x^2 + 12*x + 4)*sqrt(3*x + 2)), 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((1-2*x)**(5/2)*(3+5*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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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