3.2628 \(\int \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=191 \[ \frac{2}{45} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}-\frac{23 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}}{1575}-\frac{1244 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{13125}-\frac{175111 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{236250}-\frac{175111 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1181250}-\frac{2911577 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{590625} \]

[Out]

(-175111*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/236250 - (1244*Sqrt[1 - 2*x]
*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/13125 - (23*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5
*x)^(3/2))/1575 + (2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/45 - (291157
7*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/590625 - (175111
*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1181250

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Rubi [A]  time = 0.422338, 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 \[ \frac{2}{45} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}-\frac{23 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}}{1575}-\frac{1244 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{13125}-\frac{175111 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{236250}-\frac{175111 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1181250}-\frac{2911577 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{590625} \]

Antiderivative was successfully verified.

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

[Out]

(-175111*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/236250 - (1244*Sqrt[1 - 2*x]
*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/13125 - (23*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5
*x)^(3/2))/1575 + (2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/45 - (291157
7*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/590625 - (175111
*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1181250

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Rubi in Sympy [A]  time = 39.1561, size = 172, normalized size = 0.9 \[ \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{27} - \frac{37 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{945} - \frac{3617 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{23625} - \frac{167647 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{236250} - \frac{2911577 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1771875} - \frac{175111 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3543750} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(-2*x + 1)*(3*x + 2)**(7/2)*sqrt(5*x + 3)/27 - 37*sqrt(-2*x + 1)*(3*x + 2)
**(5/2)*sqrt(5*x + 3)/945 - 3617*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/2
3625 - 167647*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/236250 - 2911577*sqrt(3
3)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1771875 - 175111*sqrt(33)*
elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/3543750

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Mathematica [A]  time = 0.366293, size = 102, normalized size = 0.53 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (472500 x^3+861750 x^2+410490 x-136987\right )-5867645 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{3543750 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-136987 + 410490*x + 861750*x^2 +
472500*x^3) + 11646308*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 586
7645*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(3543750*Sqrt[2])

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Maple [C]  time = 0.112, size = 179, normalized size = 0.9 \[{\frac{1}{212625000\,{x}^{3}[phone]\,{x}^{2}-49612500\,x-42525000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 425250000\,{x}^{6}+5867645\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -11646308\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}-106067700\,{x}^{3}-335838930\,{x}^{2}-45120930\,x+24657660 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7087500*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(425250000*x^6+5867645*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))-11646308*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+864823500*x^4-106067700*x^3-335838930*x^2-451
20930*x+24657660)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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