3.2804 \(\int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{7/2}} \, dx\)

Optimal. Leaf size=158 \[ \frac{1752 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 \sqrt{3 x+2}}+\frac{18 \sqrt{1-2 x} \sqrt{5 x+3}}{245 (3 x+2)^{3/2}}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{35 (3 x+2)^{5/2}}-\frac{68 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715}-\frac{584 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715} \]

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35*(2 + 3*x)^(5/2)) + (18*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(245*(2 + 3*x)^(3/2)) + (1752*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1715*Sqrt[2
+ 3*x]) - (584*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1715
- (68*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1715

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Rubi [A]  time = 0.341106, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{1752 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 \sqrt{3 x+2}}+\frac{18 \sqrt{1-2 x} \sqrt{5 x+3}}{245 (3 x+2)^{3/2}}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{35 (3 x+2)^{5/2}}-\frac{68 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715}-\frac{584 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35*(2 + 3*x)^(5/2)) + (18*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(245*(2 + 3*x)^(3/2)) + (1752*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1715*Sqrt[2
+ 3*x]) - (584*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1715
- (68*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1715

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Rubi in Sympy [A]  time = 32.2061, size = 143, normalized size = 0.91 \[ \frac{1752 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1715 \sqrt{3 x + 2}} + \frac{18 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{245 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{2 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{35 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{584 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1715} - \frac{68 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{5145} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1752*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1715*sqrt(3*x + 2)) + 18*sqrt(-2*x + 1)*sqrt(
5*x + 3)/(245*(3*x + 2)**(3/2)) - 2*sqrt(-2*x + 1)*sqrt(5*x + 3)/(35*(3*x + 2)**
(5/2)) - 584*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1715 -
68*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/5145

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Mathematica [A]  time = 0.212458, size = 98, normalized size = 0.62 \[ \frac{2 \left (\frac{\sqrt{1-2 x} \sqrt{5 x+3} \left (7884 x^2+10701 x+3581\right )}{(3 x+2)^{5/2}}+\sqrt{2} \left (292 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-105 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{1715} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3581 + 10701*x + 7884*x^2))/(2 + 3*x)^(5/2) +
Sqrt[2]*(292*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 105*EllipticF[
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/1715

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Maple [C]  time = 0.031, size = 386, normalized size = 2.4 \[{\frac{2}{17150\,{x}^{2}+1715\,x-5145} \left ( 945\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2628\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1260\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3504\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+420\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1168\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +78840\,{x}^{4}+114894\,{x}^{3}+22859\,{x}^{2}-28522\,x-10743 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1715*(945*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)-2628*2^(1/2)*Ell
ipticE(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)+1260*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)-3504*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)+420*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))-1168*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))+78840*x^4+114894*x^3+22859*x^2-28522*x-10743)*(1-2*x)^(1/2)*(3+5
*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{5 \, x + 3}}{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(5*x + 3)/((27*x^3 + 54*x^2 + 36*x + 8)*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((3+5*x)**(1/2)/(2+3*x)**(7/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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