3.2858 \(\int \frac{(2+3 x)^{9/2}}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=187 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}-\frac{668 \sqrt{1-2 x} (3 x+2)^{5/2}}{9075 \sqrt{5 x+3}}+\frac{403 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{75625}-\frac{87476 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{378125}-\frac{104663 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{171875 \sqrt{33}}-\frac{6515539 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343750 \sqrt{33}} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(165*(3 + 5*x)^(3/2)) - (668*Sqrt[1 - 2*x]*(2
 + 3*x)^(5/2))/(9075*Sqrt[3 + 5*x]) - (87476*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3
+ 5*x])/378125 + (403*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/75625 - (6515
539*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(343750*Sqrt[33]) - (1046
63*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(171875*Sqrt[33])

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Rubi [A]  time = 0.417154, 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 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}-\frac{668 \sqrt{1-2 x} (3 x+2)^{5/2}}{9075 \sqrt{5 x+3}}+\frac{403 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{75625}-\frac{87476 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{378125}-\frac{104663 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{171875 \sqrt{33}}-\frac{6515539 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343750 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x)^(9/2)/(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)),x]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(165*(3 + 5*x)^(3/2)) - (668*Sqrt[1 - 2*x]*(2
 + 3*x)^(5/2))/(9075*Sqrt[3 + 5*x]) - (87476*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3
+ 5*x])/378125 + (403*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/75625 - (6515
539*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(343750*Sqrt[33]) - (1046
63*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(171875*Sqrt[33])

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Rubi in Sympy [A]  time = 41.2601, size = 172, normalized size = 0.92 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}}}{165 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{668 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}}{9075 \sqrt{5 x + 3}} + \frac{403 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{75625} - \frac{87476 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{378125} - \frac{6515539 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{11343750} - \frac{104663 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{6015625} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**(9/2)/(3+5*x)**(5/2)/(1-2*x)**(1/2),x)

[Out]

-2*sqrt(-2*x + 1)*(3*x + 2)**(7/2)/(165*(5*x + 3)**(3/2)) - 668*sqrt(-2*x + 1)*(
3*x + 2)**(5/2)/(9075*sqrt(5*x + 3)) + 403*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(
5*x + 3)/75625 - 87476*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/378125 - 65155
39*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/11343750 - 104663
*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/6015625

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Mathematica [A]  time = 0.368333, size = 107, normalized size = 0.57 \[ \frac{-\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (3675375 x^3+13721400 x^2+12517925 x+3365042\right )}{(5 x+3)^{3/2}}-3061660 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+6515539 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{11343750} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x)^(9/2)/(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)),x]

[Out]

((-10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3365042 + 12517925*x + 13721400*x^2 + 3675375
*x^3))/(3 + 5*x)^(3/2) + 6515539*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]], -33/2] - 3061660*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]
)/11343750

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Maple [C]  time = 0.03, size = 277, normalized size = 1.5 \[{\frac{1}{68062500\,{x}^{2}+11343750\,x-22687500} \left ( 15308300\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-32577695\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+9184980\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -19546617\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -220522500\,{x}^{5}-860037750\,{x}^{4}-814782000\,{x}^{3}-52653770\,{x}^{2}+216708080\,x+67300840 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^(9/2)/(3+5*x)^(5/2)/(1-2*x)^(1/2),x)

[Out]

1/11343750*(15308300*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)-32577695*
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)+9184980*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))-19546617*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
))-220522500*x^5-860037750*x^4-814782000*x^3-52653770*x^2+216708080*x+67300840)*
(1-2*x)^(1/2)*(2+3*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*sqrt(-2*x + 1)),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*sqrt(-2*x + 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{3 \, x + 2}}{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*sqrt(-2*x + 1)),x, algorithm="fricas")

[Out]

integral((81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*sqrt(3*x + 2)/((25*x^2 + 30*x
+ 9)*sqrt(5*x + 3)*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)**(9/2)/(3+5*x)**(5/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*sqrt(-2*x + 1)),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*sqrt(-2*x + 1)), x)