3.2848 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=127 \[ \frac{148 \sqrt{1-2 x} \sqrt{5 x+3}}{49 \sqrt{3 x+2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{7 (3 x+2)^{3/2}}-\frac{52 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{49 \sqrt{33}}-\frac{148}{49} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)^(3/2)) + (148*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(49*Sqrt[2 + 3*x]) - (148*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/49 - (52*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(49*
Sqrt[33])

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Rubi [A]  time = 0.270109, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{148 \sqrt{1-2 x} \sqrt{5 x+3}}{49 \sqrt{3 x+2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{7 (3 x+2)^{3/2}}-\frac{52 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{49 \sqrt{33}}-\frac{148}{49} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)^(3/2)) + (148*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(49*Sqrt[2 + 3*x]) - (148*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/49 - (52*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(49*
Sqrt[33])

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Rubi in Sympy [A]  time = 25.1647, size = 114, normalized size = 0.9 \[ \frac{148 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{49 \sqrt{3 x + 2}} + \frac{2 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{7 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{148 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{147} - \frac{52 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1617} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

148*sqrt(-2*x + 1)*sqrt(5*x + 3)/(49*sqrt(3*x + 2)) + 2*sqrt(-2*x + 1)*sqrt(5*x
+ 3)/(7*(3*x + 2)**(3/2)) - 148*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)
/7), 35/33)/147 - 52*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)
/1617

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Mathematica [A]  time = 0.232014, size = 97, normalized size = 0.76 \[ \frac{2}{147} \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} (222 x+155)}{(3 x+2)^{3/2}}-35 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+74 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(155 + 222*x))/(2 + 3*x)^(3/2) + 74*Sqrt[2]*E
llipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 35*Sqrt[2]*EllipticF[ArcSin[
Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/147

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Maple [C]  time = 0.031, size = 267, normalized size = 2.1 \[{\frac{2}{1470\,{x}^{2}+147\,x-441} \left ( 105\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-222\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+70\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -148\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +6660\,{x}^{3}+5316\,{x}^{2}-1533\,x-1395 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/147*(105*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)-222*2^(1/2)*Ellipti
cE(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)+70*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))-148*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))+6660*x^3+5316*x^2-1533
*x-1395)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((9*x^2 + 12*x + 4)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-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/(2+3*x)**(5/2)/(1-2*x)**(1/2)/(3+5*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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