3.2743 \(\int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^{7/2}} \, dx\)

Optimal. Leaf size=156 \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{15 (3 x+2)^{5/2}}+\frac{2 \sqrt{5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{3/2}}+\frac{8 \sqrt{5 x+3} \sqrt{1-2 x}}{\sqrt{3 x+2}}-\frac{4}{5} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{12}{5} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(15*(2 + 3*x)^(5/2)) + (2*(1 - 2*x)^(3/2)*Sqr
t[3 + 5*x])/(3*(2 + 3*x)^(3/2)) + (8*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/Sqrt[2 + 3*x]
- (12*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (4*Sqrt[11
/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5

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Rubi [A]  time = 0.338234, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{15 (3 x+2)^{5/2}}+\frac{2 \sqrt{5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{3/2}}+\frac{8 \sqrt{5 x+3} \sqrt{1-2 x}}{\sqrt{3 x+2}}-\frac{4}{5} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{12}{5} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(15*(2 + 3*x)^(5/2)) + (2*(1 - 2*x)^(3/2)*Sqr
t[3 + 5*x])/(3*(2 + 3*x)^(3/2)) + (8*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/Sqrt[2 + 3*x]
- (12*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (4*Sqrt[11
/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5

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Rubi in Sympy [A]  time = 30.7352, size = 141, normalized size = 0.9 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{15 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{3 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{8 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{\sqrt{3 x + 2}} - \frac{12 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{5} - \frac{44 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{175} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(15*(3*x + 2)**(5/2)) + 2*(-2*x + 1)**(3/2)*s
qrt(5*x + 3)/(3*(3*x + 2)**(3/2)) + 8*sqrt(-2*x + 1)*sqrt(5*x + 3)/sqrt(3*x + 2)
- 12*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/5 - 44*sqrt(35
)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/175

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Mathematica [A]  time = 0.204114, size = 99, normalized size = 0.63 \[ \frac{2}{15} \left (\frac{\sqrt{1-2 x} \sqrt{5 x+3} \left (506 x^2+719 x+249\right )}{(3 x+2)^{5/2}}+3 \sqrt{2} \left (5 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+6 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(249 + 719*x + 506*x^2))/(2 + 3*x)^(5/2) + 3*Sq
rt[2]*(6*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 5*EllipticF[ArcSin
[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/15

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Maple [C]  time = 0.028, size = 386, normalized size = 2.5 \[ -{\frac{2}{150\,{x}^{2}+15\,x-45} \left ( 135\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+162\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+180\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+216\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+60\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +72\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -5060\,{x}^{4}-7696\,{x}^{3}-1691\,{x}^{2}+1908\,x+747 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(135*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)+162*2^(1/2)*Ellip
ticE(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)+180*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)+216*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)+60*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))+72*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))-5060*x^4-7696*x^3-1691*x^2+1908*x+747)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*
x^2+x-3)/(2+3*x)^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((27*x^3 + 54*x^2 + 36*x
+ 8)*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)**(5/2)*(3+5*x)**(1/2)/(2+3*x)**(7/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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