3.2665 \(\int \frac{\sqrt{1-2 x} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=189 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{7/2}}{5 \sqrt{5 x+3}}+\frac{48}{175} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{183 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{4375}-\frac{2486 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{21875}-\frac{38723 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375 \sqrt{33}}-\frac{203179 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{218750} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(5*Sqrt[3 + 5*x]) - (2486*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*Sqrt[3 + 5*x])/21875 + (183*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]
)/4375 + (48*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/175 - (203179*Sqrt[11/
3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/218750 - (38723*EllipticF[
ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(109375*Sqrt[33])

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Rubi [A]  time = 0.408487, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{7/2}}{5 \sqrt{5 x+3}}+\frac{48}{175} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{183 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{4375}-\frac{2486 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{21875}-\frac{38723 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375 \sqrt{33}}-\frac{203179 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{218750} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(5*Sqrt[3 + 5*x]) - (2486*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*Sqrt[3 + 5*x])/21875 + (183*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]
)/4375 + (48*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/175 - (203179*Sqrt[11/
3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/218750 - (38723*EllipticF[
ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(109375*Sqrt[33])

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Rubi in Sympy [A]  time = 41.4733, size = 172, normalized size = 0.91 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}}}{5 \sqrt{5 x + 3}} + \frac{48 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{175} + \frac{183 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{4375} - \frac{2486 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{21875} - \frac{203179 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{656250} - \frac{38723 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{3828125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-2*x + 1)*(3*x + 2)**(7/2)/(5*sqrt(5*x + 3)) + 48*sqrt(-2*x + 1)*(3*x +
2)**(5/2)*sqrt(5*x + 3)/175 + 183*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/
4375 - 2486*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/21875 - 203179*sqrt(33)*e
lliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/656250 - 38723*sqrt(35)*ellipt
ic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/3828125

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Mathematica [A]  time = 0.429746, size = 107, normalized size = 0.57 \[ \frac{\frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \left (33750 x^3+63225 x^2+25955 x+32\right )}{\sqrt{5 x+3}}-87010 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+203179 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{656250} \]

Antiderivative was successfully verified.

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(32 + 25955*x + 63225*x^2 + 33750*x^3))/Sqrt[3
+ 5*x] + 203179*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 870
10*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/656250

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Maple [C]  time = 0.048, size = 174, normalized size = 0.9 \[{\frac{1}{19687500\,{x}^{3}[phone]\,{x}^{2}-4593750\,x-3937500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 87010\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -203179\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +6075000\,{x}^{5}[phone]\,{x}^{4}+4543650\,{x}^{3}-3009090\,{x}^{2}-1556340\,x-1920 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/656250*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(87010*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))-203179*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))
+6075000*x^5+12393000*x^4+4543650*x^3-3009090*x^2-1556340*x-1920)/(30*x^3+23*x^2
-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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