3.2681 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=218 \[ \frac{10312712 \sqrt{1-2 x} \sqrt{3 x+2}}{1617 \sqrt{5 x+3}}-\frac{155104 \sqrt{1-2 x} \sqrt{3 x+2}}{147 (5 x+3)^{3/2}}+\frac{116044 \sqrt{1-2 x}}{735 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{556 \sqrt{1-2 x}}{105 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{2 \sqrt{1-2 x}}{5 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{310208 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{245 \sqrt{33}}-\frac{10312712 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{245 \sqrt{33}} \]

[Out]

(2*Sqrt[1 - 2*x])/(5*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (556*Sqrt[1 - 2*x])/(105
*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (116044*Sqrt[1 - 2*x])/(735*Sqrt[2 + 3*x]*(3
+ 5*x)^(3/2)) - (155104*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(147*(3 + 5*x)^(3/2)) + (1
0312712*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1617*Sqrt[3 + 5*x]) - (10312712*EllipticE[
ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(245*Sqrt[33]) - (310208*EllipticF[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(245*Sqrt[33])

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Rubi [A]  time = 0.523331, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{10312712 \sqrt{1-2 x} \sqrt{3 x+2}}{1617 \sqrt{5 x+3}}-\frac{155104 \sqrt{1-2 x} \sqrt{3 x+2}}{147 (5 x+3)^{3/2}}+\frac{116044 \sqrt{1-2 x}}{735 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{556 \sqrt{1-2 x}}{105 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{2 \sqrt{1-2 x}}{5 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{310208 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{245 \sqrt{33}}-\frac{10312712 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{245 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x])/(5*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (556*Sqrt[1 - 2*x])/(105
*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (116044*Sqrt[1 - 2*x])/(735*Sqrt[2 + 3*x]*(3
+ 5*x)^(3/2)) - (155104*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(147*(3 + 5*x)^(3/2)) + (1
0312712*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1617*Sqrt[3 + 5*x]) - (10312712*EllipticE[
ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(245*Sqrt[33]) - (310208*EllipticF[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(245*Sqrt[33])

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Rubi in Sympy [A]  time = 45.6274, size = 201, normalized size = 0.92 \[ \frac{10312712 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2695 \sqrt{3 x + 2}} + \frac{148408 \sqrt{- 2 x + 1}}{231 \sqrt{3 x + 2} \sqrt{5 x + 3}} - \frac{372 \sqrt{- 2 x + 1}}{7 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{556 \sqrt{- 2 x + 1}}{105 \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{2 \sqrt{- 2 x + 1}}{5 \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{10312712 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{8085} - \frac{310208 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{8575} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10312712*sqrt(-2*x + 1)*sqrt(5*x + 3)/(2695*sqrt(3*x + 2)) + 148408*sqrt(-2*x +
1)/(231*sqrt(3*x + 2)*sqrt(5*x + 3)) - 372*sqrt(-2*x + 1)/(7*sqrt(3*x + 2)*(5*x
+ 3)**(3/2)) + 556*sqrt(-2*x + 1)/(105*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)) + 2*sq
rt(-2*x + 1)/(5*(3*x + 2)**(5/2)*(5*x + 3)**(3/2)) - 10312712*sqrt(33)*elliptic_
e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/8085 - 310208*sqrt(35)*elliptic_f(asin
(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/8575

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Mathematica [A]  time = 0.361465, size = 109, normalized size = 0.5 \[ \frac{2 \left (\frac{\sqrt{1-2 x} \left (3480540300 x^4+8934240060 x^3+8592783498 x^2+3669873602 x+587237237\right )}{(3 x+2)^{5/2} (5 x+3)^{3/2}}+4 \sqrt{2} \left (1289089 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-649285 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{8085} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[1 - 2*x]*(587237237 + 3669873602*x + 8592783498*x^2 + 8934240060*x^3 +
3480540300*x^4))/((2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + 4*Sqrt[2]*(1289089*Ellipti
cE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 649285*EllipticF[ArcSin[Sqrt[2/11]
*Sqrt[3 + 5*x]], -33/2])))/8085

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Maple [C]  time = 0.036, size = 502, normalized size = 2.3 \[{\frac{2}{-8085+16170\,x}\sqrt{1-2\,x} \left ( 116871300\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-232036020\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-448602972\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-288755936\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -61876272\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-1253036294\,{x}^{2}-2495399128\,x-587237237 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/8085*(1-2*x)^(1/2)*(116871300*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1
/2)-232036020*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)[phone]*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)-448602972*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)[phone]*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)-288755936*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)[phone]*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))-61876272*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^5+14387939820*x^4+8251326936*x^3-1253036294
*x^2-2495399128*x-587237237)/(2+3*x)^(5/2)/(3+5*x)^(3/2)/(-1+2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-2 \, x + 1}}{{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(-2*x + 1)/((675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)
*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)**(7/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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