3.2750 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{\sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=191 \[ \frac{2}{27} \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac{362 \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{2835}+\frac{14318 \sqrt{3 x+2} (5 x+3)^{3/2} \sqrt{1-2 x}}{70875}-\frac{429479 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{637875}-\frac{429479 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3189375}-\frac{4457606 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3189375} \]

[Out]

(-429479*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/637875 + (14318*Sqrt[1 - 2*x
]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/70875 + (362*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 +
5*x)^(3/2))/2835 + (2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/27 - (4457
606*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3189375 - (429
479*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3189375

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Rubi [A]  time = 0.395649, 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 \[ \frac{2}{27} \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac{362 \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{2835}+\frac{14318 \sqrt{3 x+2} (5 x+3)^{3/2} \sqrt{1-2 x}}{70875}-\frac{429479 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{637875}-\frac{429479 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3189375}-\frac{4457606 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3189375} \]

Antiderivative was successfully verified.

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

[Out]

(-429479*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/637875 + (14318*Sqrt[1 - 2*x
]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/70875 + (362*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 +
5*x)^(3/2))/2835 + (2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/27 - (4457
606*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3189375 - (429
479*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3189375

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Rubi in Sympy [A]  time = 39.4964, size = 172, normalized size = 0.9 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{27} - \frac{181 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{567} + \frac{932 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{4725} + \frac{279262 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{637875} - \frac{4457606 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{9568125} - \frac{429479 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{9568125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(5/2)*sqrt(3*x + 2)*(5*x + 3)**(3/2)/27 - 181*(-2*x + 1)**(5/2)*sq
rt(3*x + 2)*sqrt(5*x + 3)/567 + 932*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*sqrt(5*x + 3
)/4725 + 279262*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/637875 - 4457606*sqrt
(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/9568125 - 429479*sqrt(33
)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/9568125

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Mathematica [A]  time = 0.246345, size = 102, normalized size = 0.53 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (945000 x^3-1192500 x^2+232110 x+343207\right )+5257595 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+8915212 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{9568125 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(343207 + 232110*x - 1192500*x^2 +
945000*x^3) + 8915212*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 5257
595*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(9568125*Sqrt[2])

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Maple [C]  time = 0.017, size = 179, normalized size = 0.9 \[ -{\frac{1}{574087500\,{x}^{3}[phone]\,{x}^{2}-133953750\,x-114817500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -850500000\,{x}^{6}+5257595\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +8915212\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}-549367200\,{x}^{3}-402719730\,{x}^{2}[phone]\,x+61777260 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/19136250*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(-850500000*x^6+5257595*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))+8915212*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))[phone]*x^5+812376000*x^4-549367200*x^3-402719730*x^2+11
3853270*x+61777260)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((20*x^3 - 8*x^2 - 7*x + 3)*sqrt(5*x + 3)*sqrt(-2*x + 1)/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)**(3/2)/(2+3*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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