3.2719 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^{9/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=191 \[ \frac{1255552 \sqrt{1-2 x} \sqrt{5 x+3}}{5145 \sqrt{3 x+2}}+\frac{18068 \sqrt{1-2 x} \sqrt{5 x+3}}{735 (3 x+2)^{3/2}}+\frac{388 \sqrt{1-2 x} \sqrt{5 x+3}}{105 (3 x+2)^{5/2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)^{7/2}}-\frac{37768 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5145}-\frac{1255552 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5145} \]

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^(7/2)) + (388*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(105*(2 + 3*x)^(5/2)) + (18068*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(735*(2 + 3*
x)^(3/2)) + (1255552*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5145*Sqrt[2 + 3*x]) - (125555
2*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5145 - (37768*Sq
rt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5145

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Rubi [A]  time = 0.414102, antiderivative size = 191, 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 \[ \frac{1255552 \sqrt{1-2 x} \sqrt{5 x+3}}{5145 \sqrt{3 x+2}}+\frac{18068 \sqrt{1-2 x} \sqrt{5 x+3}}{735 (3 x+2)^{3/2}}+\frac{388 \sqrt{1-2 x} \sqrt{5 x+3}}{105 (3 x+2)^{5/2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)^{7/2}}-\frac{37768 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5145}-\frac{1255552 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5145} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^(7/2)) + (388*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(105*(2 + 3*x)^(5/2)) + (18068*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(735*(2 + 3*
x)^(3/2)) + (1255552*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5145*Sqrt[2 + 3*x]) - (125555
2*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5145 - (37768*Sq
rt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5145

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Rubi in Sympy [A]  time = 38.3338, size = 172, normalized size = 0.9 \[ \frac{1255552 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{5145 \sqrt{3 x + 2}} + \frac{18068 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{735 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{388 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{105 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{2 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{1255552 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{15435} - \frac{415448 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{180075} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1255552*sqrt(-2*x + 1)*sqrt(5*x + 3)/(5145*sqrt(3*x + 2)) + 18068*sqrt(-2*x + 1)
*sqrt(5*x + 3)/(735*(3*x + 2)**(3/2)) + 388*sqrt(-2*x + 1)*sqrt(5*x + 3)/(105*(3
*x + 2)**(5/2)) + 2*sqrt(-2*x + 1)*sqrt(5*x + 3)/(3*(3*x + 2)**(7/2)) - 1255552*
sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/15435 - 415448*sqrt(
35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/180075

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Mathematica [A]  time = 0.337439, size = 106, normalized size = 0.55 \[ \frac{4 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (16949952 x^3+34469046 x^2+23387310 x+5295887\right )}{2 (3 x+2)^{7/2}}+\sqrt{2} \left (313888 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-158095 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{15435} \]

Antiderivative was successfully verified.

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

[Out]

(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5295887 + 23387310*x + 34469046*x^2 + 169499
52*x^3))/(2*(2 + 3*x)^(7/2)) + Sqrt[2]*(313888*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[
3 + 5*x]], -33/2] - 158095*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))
/15435

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Maple [C]  time = 0.033, size = 505, normalized size = 2.6 \[{\frac{2}{154350\,{x}^{2}+15435\,x-46305} \left ( 8537130\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-16949952\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-33899904\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-22599936\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+2529520\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -5022208\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-81182874\,{x}^{2}-194598129\,x-47662983 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15435*(8537130*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)-16949952*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)[phone]*2^(1/2)*EllipticF(1/1
1*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)-33899904*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)[phone]*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)-22599936*
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)+2529520*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))-5022208*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+1084921236*x^4+652476870*x^3-81182874*x^2-194598129*x-47662983)*
(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(7/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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