3.2718 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=160 \[ \frac{17804 \sqrt{1-2 x} \sqrt{5 x+3}}{315 \sqrt{3 x+2}}+\frac{256 \sqrt{1-2 x} \sqrt{5 x+3}}{45 (3 x+2)^{3/2}}+\frac{14 \sqrt{1-2 x} \sqrt{5 x+3}}{15 (3 x+2)^{5/2}}-\frac{536}{315} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{17804}{315} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15*(2 + 3*x)^(5/2)) + (256*Sqrt[1 - 2*x]*Sqrt[
3 + 5*x])/(45*(2 + 3*x)^(3/2)) + (17804*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(315*Sqrt[2
+ 3*x]) - (17804*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/
315 - (536*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/315

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Rubi [A]  time = 0.342947, antiderivative size = 160, 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 \[ \frac{17804 \sqrt{1-2 x} \sqrt{5 x+3}}{315 \sqrt{3 x+2}}+\frac{256 \sqrt{1-2 x} \sqrt{5 x+3}}{45 (3 x+2)^{3/2}}+\frac{14 \sqrt{1-2 x} \sqrt{5 x+3}}{15 (3 x+2)^{5/2}}-\frac{536}{315} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{17804}{315} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15*(2 + 3*x)^(5/2)) + (256*Sqrt[1 - 2*x]*Sqrt[
3 + 5*x])/(45*(2 + 3*x)^(3/2)) + (17804*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(315*Sqrt[2
+ 3*x]) - (17804*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/
315 - (536*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/315

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Rubi in Sympy [A]  time = 30.5885, size = 143, normalized size = 0.89 \[ \frac{17804 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{315 \sqrt{3 x + 2}} + \frac{256 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{45 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{14 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{15 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{17804 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{945} - \frac{5896 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{11025} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

17804*sqrt(-2*x + 1)*sqrt(5*x + 3)/(315*sqrt(3*x + 2)) + 256*sqrt(-2*x + 1)*sqrt
(5*x + 3)/(45*(3*x + 2)**(3/2)) + 14*sqrt(-2*x + 1)*sqrt(5*x + 3)/(15*(3*x + 2)*
*(5/2)) - 17804*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/945
- 5896*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/11025

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Mathematica [A]  time = 0.305989, size = 101, normalized size = 0.63 \[ \frac{4}{945} \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (80118 x^2+109512 x+37547\right )}{2 (3 x+2)^{5/2}}+\sqrt{2} \left (4451 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2240 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(37547 + 109512*x + 80118*x^2))/(2*(2 + 3*x)^
(5/2)) + Sqrt[2]*(4451*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2240
*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/945

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Maple [C]  time = 0.03, size = 386, normalized size = 2.4 \[{\frac{2}{9450\,{x}^{2}+945\,x-2835} \left ( 40320\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-80118\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+53760\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-106824\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+17920\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -35608\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +2403540\,{x}^{4}+3525714\,{x}^{3}+733884\,{x}^{2}-872967\,x-337923 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/945*(40320*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)-80118*2^(1/2)*E
llipticE(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)+53760*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)-106824*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)+1792
0*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))-35608*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))+2403540*x^4+3525714*x^3+733884*x^2-872967*x-337923)*(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. \[ \int \frac{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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