3.2961 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=187 \[ \frac{184636 \sqrt{1-2 x} \sqrt{5 x+3}}{290521 \sqrt{3 x+2}}+\frac{974 \sqrt{1-2 x} \sqrt{5 x+3}}{41503 (3 x+2)^{3/2}}+\frac{1072 \sqrt{5 x+3}}{17787 \sqrt{1-2 x} (3 x+2)^{3/2}}+\frac{4 \sqrt{5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac{9124 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{26411 \sqrt{33}}-\frac{184636 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{26411 \sqrt{33}} \]

[Out]

(4*Sqrt[3 + 5*x])/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)) + (1072*Sqrt[3 + 5*x])/(
17787*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) + (974*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(41503*
(2 + 3*x)^(3/2)) + (184636*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(290521*Sqrt[2 + 3*x]) -
(184636*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(26411*Sqrt[33]) - (
9124*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(26411*Sqrt[33])

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Rubi [A]  time = 0.429983, antiderivative size = 187, 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 \[ \frac{184636 \sqrt{1-2 x} \sqrt{5 x+3}}{290521 \sqrt{3 x+2}}+\frac{974 \sqrt{1-2 x} \sqrt{5 x+3}}{41503 (3 x+2)^{3/2}}+\frac{1072 \sqrt{5 x+3}}{17787 \sqrt{1-2 x} (3 x+2)^{3/2}}+\frac{4 \sqrt{5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac{9124 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{26411 \sqrt{33}}-\frac{184636 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{26411 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[3 + 5*x])/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)) + (1072*Sqrt[3 + 5*x])/(
17787*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) + (974*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(41503*
(2 + 3*x)^(3/2)) + (184636*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(290521*Sqrt[2 + 3*x]) -
(184636*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(26411*Sqrt[33]) - (
9124*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(26411*Sqrt[33])

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Rubi in Sympy [A]  time = 37.49, size = 172, normalized size = 0.92 \[ - \frac{184636 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{871563} - \frac{9124 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{871563} - \frac{369272 \sqrt{3 x + 2} \sqrt{5 x + 3}}{871563 \sqrt{- 2 x + 1}} + \frac{5536 \sqrt{5 x + 3}}{3773 \sqrt{- 2 x + 1} \sqrt{3 x + 2}} + \frac{62 \sqrt{5 x + 3}}{539 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{4 \sqrt{5 x + 3}}{231 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-184636*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/871563 - 912
4*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/871563 - 369272*sq
rt(3*x + 2)*sqrt(5*x + 3)/(871563*sqrt(-2*x + 1)) + 5536*sqrt(5*x + 3)/(3773*sqr
t(-2*x + 1)*sqrt(3*x + 2)) + 62*sqrt(5*x + 3)/(539*sqrt(-2*x + 1)*(3*x + 2)**(3/
2)) + 4*sqrt(5*x + 3)/(231*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2))

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Mathematica [A]  time = 0.298264, size = 103, normalized size = 0.55 \[ \frac{2 \left (\frac{\sqrt{5 x+3} \left (3323448 x^3-1066908 x^2-1478206 x+597945\right )}{(1-2 x)^{3/2} (3 x+2)^{3/2}}+\sqrt{2} \left (92318 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-17045 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{871563} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[3 + 5*x]*(597945 - 1478206*x - 1066908*x^2 + 3323448*x^3))/((1 - 2*x)^
(3/2)*(2 + 3*x)^(3/2)) + Sqrt[2]*(92318*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x
]], -33/2] - 17045*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/871563

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Maple [C]  time = 0.036, size = 383, normalized size = 2.1 \[{\frac{2}{871563\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 102270\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-553908\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+17045\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-92318\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-34090\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +184636\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{4}+4635804\,{x}^{3}-10591754\,{x}^{2}-1444893\,x+1793835 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/871563*(1-2*x)^(1/2)*(102270*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)-553908*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)+17045*2^(1/2)*Elli
pticF(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)-92318*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)-34090*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))+184636*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^4+4635804*x^3-10591754*x^2
-1444893*x+1793835)/(2+3*x)^(3/2)/(-1+2*x)^2/(3+5*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqr
t(-2*x + 1)), 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/(1-2*x)**(5/2)/(2+3*x)**(5/2)/(3+5*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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