3.2941 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{5/2} \sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=127 \[ -\frac{74 \sqrt{3 x+2} \sqrt{5 x+3}}{147 \sqrt{1-2 x}}+\frac{11 \sqrt{3 x+2} \sqrt{5 x+3}}{21 (1-2 x)^{3/2}}-\frac{13 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{49 \sqrt{33}}-\frac{37}{49} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(11*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)) - (74*Sqrt[2 + 3*x]*Sqrt[3
+ 5*x])/(147*Sqrt[1 - 2*x]) - (37*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/49 - (13*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(49
*Sqrt[33])

_______________________________________________________________________________________

Rubi [A]  time = 0.256445, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{74 \sqrt{3 x+2} \sqrt{5 x+3}}{147 \sqrt{1-2 x}}+\frac{11 \sqrt{3 x+2} \sqrt{5 x+3}}{21 (1-2 x)^{3/2}}-\frac{13 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{49 \sqrt{33}}-\frac{37}{49} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(11*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)) - (74*Sqrt[2 + 3*x]*Sqrt[3
+ 5*x])/(147*Sqrt[1 - 2*x]) - (37*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/49 - (13*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(49
*Sqrt[33])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.791, size = 114, normalized size = 0.9 \[ - \frac{37 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{147} - \frac{13 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1617} - \frac{74 \sqrt{3 x + 2} \sqrt{5 x + 3}}{147 \sqrt{- 2 x + 1}} + \frac{11 \sqrt{3 x + 2} \sqrt{5 x + 3}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

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)**(1/2),x)

[Out]

-37*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/147 - 13*sqrt(33
)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1617 - 74*sqrt(3*x + 2)*sqr
t(5*x + 3)/(147*sqrt(-2*x + 1)) + 11*sqrt(3*x + 2)*sqrt(5*x + 3)/(21*(-2*x + 1)*
*(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.25014, size = 115, normalized size = 0.91 \[ \frac{2 \sqrt{3 x+2} \sqrt{5 x+3} (148 x+3)+35 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-74 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{294 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(3 + 148*x) - 74*Sqrt[2 - 4*x]*(-1 + 2*x)*Ellipti
cE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 35*Sqrt[2 - 4*x]*(-1 + 2*x)*Ellipt
icF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(294*(1 - 2*x)^(3/2))

_______________________________________________________________________________________

Maple [C]  time = 0.03, size = 276, normalized size = 2.2 \[{\frac{1}{294\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 70\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-148\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-35\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +74\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +4440\,{x}^{3}+5714\,{x}^{2}+1890\,x+36 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}} \]

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)^(1/2),x)

[Out]

1/294*(70*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)-148*2^(1/2)*Elliptic
E(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)-35*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))+74*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))+4440*x^3+5714*x^2+1890*x
+36)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)^2/(15*x^2+19*x+6)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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