3.2859 \(\int \frac{(2+3 x)^{7/2}}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=156 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}-\frac{536 \sqrt{1-2 x} (3 x+2)^{3/2}}{9075 \sqrt{5 x+3}}-\frac{487 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{15125}-\frac{2281 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6875 \sqrt{33}}-\frac{46159 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6875 \sqrt{33}} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(165*(3 + 5*x)^(3/2)) - (536*Sqrt[1 - 2*x]*(2
+ 3*x)^(3/2))/(9075*Sqrt[3 + 5*x]) - (487*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 +
5*x])/15125 - (46159*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6875*Sq
rt[33]) - (2281*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6875*Sqrt[33
])

_______________________________________________________________________________________

Rubi [A]  time = 0.335928, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}-\frac{536 \sqrt{1-2 x} (3 x+2)^{3/2}}{9075 \sqrt{5 x+3}}-\frac{487 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{15125}-\frac{2281 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6875 \sqrt{33}}-\frac{46159 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6875 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(165*(3 + 5*x)^(3/2)) - (536*Sqrt[1 - 2*x]*(2
+ 3*x)^(3/2))/(9075*Sqrt[3 + 5*x]) - (487*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 +
5*x])/15125 - (46159*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6875*Sq
rt[33]) - (2281*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6875*Sqrt[33
])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 33.2552, size = 144, normalized size = 0.92 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}}{165 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{536 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}}{9075 \sqrt{5 x + 3}} - \frac{487 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{15125} - \frac{46159 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{226875} - \frac{2281 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{240625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-2*x + 1)*(3*x + 2)**(5/2)/(165*(5*x + 3)**(3/2)) - 536*sqrt(-2*x + 1)*(
3*x + 2)**(3/2)/(9075*sqrt(5*x + 3)) - 487*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x
+ 3)/15125 - 46159*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/
226875 - 2281*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/24062
5

_______________________________________________________________________________________

Mathematica [A]  time = 0.316742, size = 102, normalized size = 0.65 \[ \frac{-\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (81675 x^2+101350 x+31429\right )}{(5 x+3)^{3/2}}-17045 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+92318 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{453750} \]

Antiderivative was successfully verified.

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

[Out]

((-10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(31429 + 101350*x + 81675*x^2))/(3 + 5*x)^(3/2
) + 92318*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 17045*Sqr
t[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/453750

_______________________________________________________________________________________

Maple [C]  time = 0.029, size = 272, normalized size = 1.7 \[{\frac{1}{2722500\,{x}^{2}+453750\,x-907500} \left ( 85225\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-461590\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+51135\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -276954\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -4900500\,{x}^{4}-6897750\,{x}^{3}-1265740\,{x}^{2}+1712710\,x+628580 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/453750*(85225*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)-461590*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)+51135*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))-276954*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellipt
icE(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))-4900500*
x^4-6897750*x^3-1265740*x^2+1712710*x+628580)*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(6*x^2
+x-2)/(3+5*x)^(3/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{3 \, x + 2}}{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((27*x^3 + 54*x^2 + 36*x + 8)*sqrt(3*x + 2)/((25*x^2 + 30*x + 9)*sqrt(5*
x + 3)*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)**(5/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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