3.2663 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=129 \[ \frac{136 \sqrt{1-2 x} \sqrt{5 x+3}}{21 \sqrt{3 x+2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)^{3/2}}-\frac{4}{21} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{136}{21} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^(3/2)) + (136*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(21*Sqrt[2 + 3*x]) - (136*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/21 - (4*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/21

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Rubi [A]  time = 0.261951, antiderivative size = 129, 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 \[ \frac{136 \sqrt{1-2 x} \sqrt{5 x+3}}{21 \sqrt{3 x+2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)^{3/2}}-\frac{4}{21} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{136}{21} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^(3/2)) + (136*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(21*Sqrt[2 + 3*x]) - (136*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/21 - (4*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/21

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Rubi in Sympy [A]  time = 25.2261, size = 114, normalized size = 0.88 \[ \frac{136 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{21 \sqrt{3 x + 2}} + \frac{2 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{136 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{63} - \frac{4 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{63} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

136*sqrt(-2*x + 1)*sqrt(5*x + 3)/(21*sqrt(3*x + 2)) + 2*sqrt(-2*x + 1)*sqrt(5*x
+ 3)/(3*(3*x + 2)**(3/2)) - 136*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)
/7), 35/33)/63 - 4*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/6
3

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Mathematica [A]  time = 0.323401, size = 97, normalized size = 0.75 \[ \frac{2}{63} \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} (204 x+143)}{(3 x+2)^{3/2}}-35 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+68 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[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]*(143 + 204*x))/(2 + 3*x)^(3/2) + 68*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]))/63

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Maple [C]  time = 0.028, size = 267, normalized size = 2.1 \[{\frac{2}{630\,{x}^{2}+63\,x-189} \left ( 105\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-204\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+70\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -136\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +6120\,{x}^{3}+4902\,{x}^{2}-1407\,x-1287 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/63*(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)-204*2^(1/2)*Elliptic
E(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))-136*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))+6120*x^3+4902*x^2-1407*
x-1287)*(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. \[ \int \frac{\sqrt{-2 \, x + 1}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(-2*x + 1)/((9*x^2 + 12*x + 4)*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)**(1/2)/(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. \[ \int \frac{\sqrt{-2 \, x + 1}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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