3.2881 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx\)

Optimal. Leaf size=189 \[ \frac{5636 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 \sqrt{3 x+2}}-\frac{26 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 (3 x+2)^{3/2}}-\frac{36 \sqrt{1-2 x} \sqrt{5 x+3}}{245 (3 x+2)^{5/2}}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{4364 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005 \sqrt{33}}-\frac{5636 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005} \]

[Out]

(2*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) - (36*Sqrt[1 - 2*x]*Sqrt[3 +
5*x])/(245*(2 + 3*x)^(5/2)) - (26*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1715*(2 + 3*x)^
(3/2)) + (5636*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12005*Sqrt[2 + 3*x]) - (5636*Sqrt[1
1/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/12005 - (4364*EllipticF[
ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(12005*Sqrt[33])

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Rubi [A]  time = 0.426619, antiderivative size = 189, 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 \[ \frac{5636 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 \sqrt{3 x+2}}-\frac{26 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 (3 x+2)^{3/2}}-\frac{36 \sqrt{1-2 x} \sqrt{5 x+3}}{245 (3 x+2)^{5/2}}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{4364 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005 \sqrt{33}}-\frac{5636 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) - (36*Sqrt[1 - 2*x]*Sqrt[3 +
5*x])/(245*(2 + 3*x)^(5/2)) - (26*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1715*(2 + 3*x)^
(3/2)) + (5636*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12005*Sqrt[2 + 3*x]) - (5636*Sqrt[1
1/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/12005 - (4364*EllipticF[
ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(12005*Sqrt[33])

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Rubi in Sympy [A]  time = 38.9249, size = 172, normalized size = 0.91 \[ \frac{5636 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{12005 \sqrt{3 x + 2}} - \frac{26 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1715 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{36 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{245 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{5636 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{36015} - \frac{4364 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{396165} + \frac{2 \sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5636*sqrt(-2*x + 1)*sqrt(5*x + 3)/(12005*sqrt(3*x + 2)) - 26*sqrt(-2*x + 1)*sqrt
(5*x + 3)/(1715*(3*x + 2)**(3/2)) - 36*sqrt(-2*x + 1)*sqrt(5*x + 3)/(245*(3*x +
2)**(5/2)) - 5636*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/36
015 - 4364*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/396165 +
2*sqrt(5*x + 3)/(7*sqrt(-2*x + 1)*(3*x + 2)**(5/2))

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Mathematica [A]  time = 0.257332, size = 104, normalized size = 0.55 \[ \frac{2 \left (\sqrt{2} \left (455 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+2818 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{3 \sqrt{5 x+3} \left (50724 x^3+41724 x^2-13127 x-11923\right )}{\sqrt{1-2 x} (3 x+2)^{5/2}}\right )}{36015} \]

Antiderivative was successfully verified.

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

[Out]

(2*((-3*Sqrt[3 + 5*x]*(-11923 - 13127*x + 41724*x^2 + 50724*x^3))/(Sqrt[1 - 2*x]
*(2 + 3*x)^(5/2)) + Sqrt[2]*(2818*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -3
3/2] + 455*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/36015

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Maple [C]  time = 0.036, size = 386, normalized size = 2. \[ -{\frac{2}{360150\,{x}^{2}+36015\,x-108045}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 4095\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+25362\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+5460\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+33816\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1820\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +11272\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -760860\,{x}^{4}-1082376\,{x}^{3}-178611\,{x}^{2}+296988\,x+107307 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/36015*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(4095*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)+25362*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)+5460*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)+33816*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)+1820*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*E
llipticF(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))+112
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))-760860*x^4-1082376*x^3-178611
*x^2+296988*x+107307)/(2+3*x)^(5/2)/(10*x^2+x-3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{\sqrt{5 \, x + 3}}{{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-sqrt(5*x + 3)/((54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*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((3+5*x)**(1/2)/(1-2*x)**(3/2)/(2+3*x)**(7/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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