3.2952 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{7/2}} \, dx\)

Optimal. Leaf size=220 \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac{27618 \sqrt{1-2 x} \sqrt{5 x+3}}{84035 \sqrt{3 x+2}}-\frac{4437 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 (3 x+2)^{3/2}}-\frac{1432 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 (3 x+2)^{5/2}}+\frac{99 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{7738 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035}+\frac{9206 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035} \]

[Out]

(99*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) - (1432*Sqrt[1 - 2*x]*Sqrt
[3 + 5*x])/(1715*(2 + 3*x)^(5/2)) - (4437*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12005*(2
+ 3*x)^(3/2)) - (27618*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84035*Sqrt[2 + 3*x]) + (11
*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)) + (9206*Sqrt[33]*Elliptic
E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/84035 - (7738*Sqrt[11/3]*EllipticF[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/84035

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Rubi [A]  time = 0.50505, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac{27618 \sqrt{1-2 x} \sqrt{5 x+3}}{84035 \sqrt{3 x+2}}-\frac{4437 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 (3 x+2)^{3/2}}-\frac{1432 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 (3 x+2)^{5/2}}+\frac{99 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{7738 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035}+\frac{9206 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035} \]

Antiderivative was successfully verified.

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

[Out]

(99*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) - (1432*Sqrt[1 - 2*x]*Sqrt
[3 + 5*x])/(1715*(2 + 3*x)^(5/2)) - (4437*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12005*(2
+ 3*x)^(3/2)) - (27618*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84035*Sqrt[2 + 3*x]) + (11
*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)) + (9206*Sqrt[33]*Elliptic
E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/84035 - (7738*Sqrt[11/3]*EllipticF[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/84035

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Rubi in Sympy [A]  time = 44.9761, size = 201, normalized size = 0.91 \[ \frac{9206 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{84035} - \frac{85118 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{2941225} + \frac{18412 \sqrt{3 x + 2} \sqrt{5 x + 3}}{84035 \sqrt{- 2 x + 1}} - \frac{6248 \sqrt{5 x + 3}}{12005 \sqrt{- 2 x + 1} \sqrt{3 x + 2}} - \frac{1573 \sqrt{5 x + 3}}{5145 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{53 \sqrt{5 x + 3}}{735 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{11 \left (5 x + 3\right )^{\frac{3}{2}}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9206*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/84035 - 85118*s
qrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/2941225 + 18412*sqrt
(3*x + 2)*sqrt(5*x + 3)/(84035*sqrt(-2*x + 1)) - 6248*sqrt(5*x + 3)/(12005*sqrt(
-2*x + 1)*sqrt(3*x + 2)) - 1573*sqrt(5*x + 3)/(5145*sqrt(-2*x + 1)*(3*x + 2)**(3
/2)) + 53*sqrt(5*x + 3)/(735*sqrt(-2*x + 1)*(3*x + 2)**(5/2)) + 11*(5*x + 3)**(3
/2)/(21*(-2*x + 1)**(3/2)*(3*x + 2)**(5/2))

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Mathematica [A]  time = 0.300704, size = 110, normalized size = 0.5 \[ \frac{3 \sqrt{2} \left (51765 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-9206 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{2 \sqrt{5 x+3} \left (1491372 x^4+1056186 x^3-718167 x^2-640441 x-88623\right )}{(1-2 x)^{3/2} (3 x+2)^{5/2}}}{252105} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[3 + 5*x]*(-88623 - 640441*x - 718167*x^2 + 1056186*x^3 + 1491372*x^4))
/((1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)) + 3*Sqrt[2]*(-9206*EllipticE[ArcSin[Sqrt[2/11
]*Sqrt[3 + 5*x]], -33/2] + 51765*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33
/2]))/252105

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Maple [C]  time = 0.036, size = 502, normalized size = 2.3 \[ -{\frac{1}{252105\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 2795310\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-497124\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}+2329425\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-414270\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-621180\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+110472\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-621180\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +110472\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}-844554\,{x}^{3}-10713412\,{x}^{2}-4728876\,x-531738 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/252105*(1-2*x)^(1/2)*(2795310*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(
1/2)-497124*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)+2329425*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^
2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-414270*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^2*(3+5*x)^(1/2)*(2+3*
x)^(1/2)*(1-2*x)^(1/2)-621180*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)+
110472*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)-621180*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))+110472*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+19510092*x^4-844554*x^3-10713412*x^2-4728876*x-531738)/(2+3*
x)^(5/2)/(-1+2*x)^2/(3+5*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}{{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((25*x^2 + 30*x + 9)*sqrt(5*x + 3)/((108*x^5 + 108*x^4 - 45*x^3 - 58*x^2
+ 4*x + 8)*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((3+5*x)**(5/2)/(1-2*x)**(5/2)/(2+3*x)**(7/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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