3.2929 \(\int \frac{(2+3 x)^{7/2} \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=187 \[ \frac{\sqrt{5 x+3} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}-\frac{133 \sqrt{5 x+3} (3 x+2)^{5/2}}{33 \sqrt{1-2 x}}-\frac{797}{110} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{18551}{550} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{9694 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{125 \sqrt{33}}-\frac{1289089 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{500 \sqrt{33}} \]

[Out]

(-18551*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/550 - (797*Sqrt[1 - 2*x]*(2 +
3*x)^(3/2)*Sqrt[3 + 5*x])/110 - (133*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/(33*Sqrt[1
- 2*x]) + ((2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/(3*(1 - 2*x)^(3/2)) - (1289089*Ellipti
cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(500*Sqrt[33]) - (9694*EllipticF[Arc
Sin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(125*Sqrt[33])

_______________________________________________________________________________________

Rubi [A]  time = 0.405214, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{\sqrt{5 x+3} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}-\frac{133 \sqrt{5 x+3} (3 x+2)^{5/2}}{33 \sqrt{1-2 x}}-\frac{797}{110} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{18551}{550} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{9694 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{125 \sqrt{33}}-\frac{1289089 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{500 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[((2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/(1 - 2*x)^(5/2),x]

[Out]

(-18551*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/550 - (797*Sqrt[1 - 2*x]*(2 +
3*x)^(3/2)*Sqrt[3 + 5*x])/110 - (133*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/(33*Sqrt[1
- 2*x]) + ((2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/(3*(1 - 2*x)^(3/2)) - (1289089*Ellipti
cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(500*Sqrt[33]) - (9694*EllipticF[Arc
Sin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(125*Sqrt[33])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 40.3536, size = 170, normalized size = 0.91 \[ - \frac{797 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{110} - \frac{18551 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{550} - \frac{1289089 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{16500} - \frac{9694 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{4125} - \frac{133 \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{33 \sqrt{- 2 x + 1}} + \frac{\left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**(7/2)*(3+5*x)**(1/2)/(1-2*x)**(5/2),x)

[Out]

-797*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/110 - 18551*sqrt(-2*x + 1)*sq
rt(3*x + 2)*sqrt(5*x + 3)/550 - 1289089*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-
2*x + 1)/7), 35/33)/16500 - 9694*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1
)/7), 35/33)/4125 - 133*(3*x + 2)**(5/2)*sqrt(5*x + 3)/(33*sqrt(-2*x + 1)) + (3*
x + 2)**(7/2)*sqrt(5*x + 3)/(3*(-2*x + 1)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.325446, size = 125, normalized size = 0.67 \[ -\frac{10 \sqrt{3 x+2} \sqrt{5 x+3} \left (8910 x^3+45342 x^2-275587 x+101763\right )-649285 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+1289089 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{16500 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/(1 - 2*x)^(5/2),x]

[Out]

-(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(101763 - 275587*x + 45342*x^2 + 8910*x^3) + 12
89089*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2
] - 649285*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]],
-33/2])/(16500*(1 - 2*x)^(3/2))

_______________________________________________________________________________________

Maple [C]  time = 0.029, size = 286, normalized size = 1.5 \[{\frac{1}{16500\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 1298570\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2578178\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-649285\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1289089\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1336500\,{x}^{5}-8494200\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-2799750\,x-6105780 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^(7/2)*(3+5*x)^(1/2)/(1-2*x)^(5/2),x)

[Out]

1/16500*(1298570*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)-2578178*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)-649285*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))+1289089*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*El
lipticE(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))-1336
500*x^5-8494200*x^4+32188470*x^3+34376560*x^2-2799750*x-6105780)*(1-2*x)^(1/2)*(
3+5*x)^(1/2)*(2+3*x)^(1/2)/(-1+2*x)^2/(15*x^2+19*x+6)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(3*x + 2)^(7/2)/(-2*x + 1)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(5*x + 3)*(3*x + 2)^(7/2)/(-2*x + 1)^(5/2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(3*x + 2)^(7/2)/(-2*x + 1)^(5/2),x, algorithm="fricas")

[Out]

integral((27*x^3 + 54*x^2 + 36*x + 8)*sqrt(5*x + 3)*sqrt(3*x + 2)/((4*x^2 - 4*x
+ 1)*sqrt(-2*x + 1)), x)

_______________________________________________________________________________________

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)**(7/2)*(3+5*x)**(1/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(3*x + 2)^(7/2)/(-2*x + 1)^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(5*x + 3)*(3*x + 2)^(7/2)/(-2*x + 1)^(5/2), x)