3.2910 \(\int \frac{(2+3 x)^{7/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=160 \[ \frac{7 (3 x+2)^{5/2}}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} (3 x+2)^{3/2}}{605 \sqrt{5 x+3}}+\frac{2388 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{3025}+\frac{823 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1375}+\frac{55019 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2750} \]

[Out]

(-37*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(605*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^(5/2))/(1
1*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (2388*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]
)/3025 + (55019*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/27
50 + (823*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1375

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Rubi [A]  time = 0.330499, antiderivative size = 160, 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 \[ \frac{7 (3 x+2)^{5/2}}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} (3 x+2)^{3/2}}{605 \sqrt{5 x+3}}+\frac{2388 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{3025}+\frac{823 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1375}+\frac{55019 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2750} \]

Antiderivative was successfully verified.

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

[Out]

(-37*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(605*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^(5/2))/(1
1*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (2388*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]
)/3025 + (55019*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/27
50 + (823*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1375

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Rubi in Sympy [A]  time = 32.6835, size = 143, normalized size = 0.89 \[ - \frac{37 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}}{605 \sqrt{5 x + 3}} + \frac{2388 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{3025} + \frac{55019 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{30250} + \frac{2469 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{48125} + \frac{7 \left (3 x + 2\right )^{\frac{5}{2}}}{11 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-37*sqrt(-2*x + 1)*(3*x + 2)**(3/2)/(605*sqrt(5*x + 3)) + 2388*sqrt(-2*x + 1)*sq
rt(3*x + 2)*sqrt(5*x + 3)/3025 + 55019*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2
*x + 1)/7), 35/33)/30250 + 2469*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)
/11), 33/35)/48125 + 7*(3*x + 2)**(5/2)/(11*sqrt(-2*x + 1)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.236323, size = 127, normalized size = 0.79 \[ \frac{10 \sqrt{3 x+2} \sqrt{5 x+3} \left (-5445 x^2+20897 x+14494\right )+27860 \sqrt{2-4 x} (5 x+3) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-55019 \sqrt{2-4 x} (5 x+3) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{30250 \sqrt{1-2 x} (5 x+3)} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(14494 + 20897*x - 5445*x^2) - 55019*Sqrt[2 - 4*
x]*(3 + 5*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 27860*Sqrt[2 -
4*x]*(3 + 5*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(30250*Sqrt[
1 - 2*x]*(3 + 5*x))

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Maple [C]  time = 0.03, size = 164, normalized size = 1. \[ -{\frac{1}{907500\,{x}^{3}+695750\,{x}^{2}-211750\,x-181500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 27860\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -55019\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -163350\,{x}^{3}+518010\,{x}^{2}+852760\,x+289880 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30250*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(27860*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))-55019*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))-
163350*x^3+518010*x^2+852760*x+289880)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{3 \, x + 2}}{{\left (10 \, x^{2} + x - 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(27*x^3 + 54*x^2 + 36*x + 8)*sqrt(3*x + 2)/((10*x^2 + x - 3)*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)**(7/2)/(1-2*x)**(3/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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