3.2735 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=185 \[ \frac{5440 \sqrt{1-2 x} \sqrt{3 x+2}}{3 \sqrt{5 x+3}}-\frac{300 \sqrt{1-2 x} \sqrt{3 x+2}}{(5 x+3)^{3/2}}+\frac{404 \sqrt{1-2 x}}{9 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{14 \sqrt{1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}-120 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-1088 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(14*Sqrt[1 - 2*x])/(9*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (404*Sqrt[1 - 2*x])/(9*
Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (300*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3 + 5*x)^(3/
2) + (5440*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3*Sqrt[3 + 5*x]) - 1088*Sqrt[11/3]*Elli
pticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33] - 120*Sqrt[3/11]*EllipticF[ArcSin[
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]

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Rubi [A]  time = 0.433668, antiderivative size = 185, 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 \[ \frac{5440 \sqrt{1-2 x} \sqrt{3 x+2}}{3 \sqrt{5 x+3}}-\frac{300 \sqrt{1-2 x} \sqrt{3 x+2}}{(5 x+3)^{3/2}}+\frac{404 \sqrt{1-2 x}}{9 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{14 \sqrt{1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}-120 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-1088 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(14*Sqrt[1 - 2*x])/(9*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (404*Sqrt[1 - 2*x])/(9*
Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (300*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3 + 5*x)^(3/
2) + (5440*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3*Sqrt[3 + 5*x]) - 1088*Sqrt[11/3]*Elli
pticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33] - 120*Sqrt[3/11]*EllipticF[ArcSin[
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]

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Rubi in Sympy [A]  time = 40.3476, size = 170, normalized size = 0.92 \[ \frac{5440 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{3 \sqrt{5 x + 3}} - \frac{300 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{\left (5 x + 3\right )^{\frac{3}{2}}} + \frac{404 \sqrt{- 2 x + 1}}{9 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{14 \sqrt{- 2 x + 1}}{9 \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{1088 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3} - \frac{120 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5440*sqrt(-2*x + 1)*sqrt(3*x + 2)/(3*sqrt(5*x + 3)) - 300*sqrt(-2*x + 1)*sqrt(3*
x + 2)/(5*x + 3)**(3/2) + 404*sqrt(-2*x + 1)/(9*sqrt(3*x + 2)*(5*x + 3)**(3/2))
+ 14*sqrt(-2*x + 1)/(9*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)) - 1088*sqrt(33)*ellipt
ic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/3 - 120*sqrt(33)*elliptic_f(asin(sq
rt(21)*sqrt(-2*x + 1)/7), 35/33)/11

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Mathematica [A]  time = 0.306055, size = 104, normalized size = 0.56 \[ \frac{2}{3} \left (\frac{\sqrt{1-2 x} \left (122400 x^3+232590 x^2+147122 x+30977\right )}{(3 x+2)^{3/2} (5 x+3)^{3/2}}+2 \sqrt{2} \left (272 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-137 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[1 - 2*x]*(30977 + 147122*x + 232590*x^2 + 122400*x^3))/((2 + 3*x)^(3/2
)*(3 + 5*x)^(3/2)) + 2*Sqrt[2]*(272*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]],
-33/2] - 137*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/3

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Maple [C]  time = 0.035, size = 383, normalized size = 2.1 \[ -{\frac{2}{-3+6\,x}\sqrt{1-2\,x} \left ( 8160\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-4110\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+10336\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-5206\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+3264\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1644\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -244800\,{x}^{4}-342780\,{x}^{3}-61654\,{x}^{2}+85168\,x+30977 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(1-2*x)^(1/2)*(8160*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)-411
0*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)+10336*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)-5206*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)+3264*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))-1644*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))-244800*x^4-342780*x^3-61654*x^2+85168*x+30977)/(2
+3*x)^(3/2)/(3+5*x)^(3/2)/(-1+2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-2*x + 1)^(3/2)/((225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*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)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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