3.2672 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=189 \[ -\frac{116464 \sqrt{1-2 x} \sqrt{3 x+2}}{147 \sqrt{5 x+3}}+\frac{19268 \sqrt{1-2 x}}{245 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{416 \sqrt{1-2 x}}{105 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{5 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{38536 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{245 \sqrt{33}}+\frac{116464}{245} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(2*Sqrt[1 - 2*x])/(5*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (416*Sqrt[1 - 2*x])/(105*(
2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (19268*Sqrt[1 - 2*x])/(245*Sqrt[2 + 3*x]*Sqrt[3
+ 5*x]) - (116464*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(147*Sqrt[3 + 5*x]) + (116464*Sqr
t[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/245 + (38536*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(245*Sqrt[33])

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Rubi [A]  time = 0.432462, 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 \[ -\frac{116464 \sqrt{1-2 x} \sqrt{3 x+2}}{147 \sqrt{5 x+3}}+\frac{19268 \sqrt{1-2 x}}{245 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{416 \sqrt{1-2 x}}{105 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{5 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{38536 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{245 \sqrt{33}}+\frac{116464}{245} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x])/(5*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (416*Sqrt[1 - 2*x])/(105*(
2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (19268*Sqrt[1 - 2*x])/(245*Sqrt[2 + 3*x]*Sqrt[3
+ 5*x]) - (116464*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(147*Sqrt[3 + 5*x]) + (116464*Sqr
t[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/245 + (38536*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(245*Sqrt[33])

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Rubi in Sympy [A]  time = 38.0217, size = 172, normalized size = 0.91 \[ - \frac{116464 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{245 \sqrt{3 x + 2}} - \frac{1676 \sqrt{- 2 x + 1}}{21 \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{416 \sqrt{- 2 x + 1}}{105 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}} + \frac{2 \sqrt{- 2 x + 1}}{5 \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}} + \frac{116464 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{735} + \frac{38536 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{8575} \]

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)**(3/2),x)

[Out]

-116464*sqrt(-2*x + 1)*sqrt(5*x + 3)/(245*sqrt(3*x + 2)) - 1676*sqrt(-2*x + 1)/(
21*sqrt(3*x + 2)*sqrt(5*x + 3)) + 416*sqrt(-2*x + 1)/(105*(3*x + 2)**(3/2)*sqrt(
5*x + 3)) + 2*sqrt(-2*x + 1)/(5*(3*x + 2)**(5/2)*sqrt(5*x + 3)) + 116464*sqrt(33
)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/735 + 38536*sqrt(35)*ellipt
ic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/8575

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Mathematica [A]  time = 0.324374, size = 105, normalized size = 0.56 \[ \frac{2}{735} \left (-\frac{3 \sqrt{1-2 x} \left (2620440 x^3+5154174 x^2+3376856 x+736871\right )}{(3 x+2)^{5/2} \sqrt{5 x+3}}-2 \sqrt{2} \left (29116 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-14665 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((-3*Sqrt[1 - 2*x]*(736871 + 3376856*x + 5154174*x^2 + 2620440*x^3))/((2 + 3*
x)^(5/2)*Sqrt[3 + 5*x]) - 2*Sqrt[2]*(29116*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]], -33/2] - 14665*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/735

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Maple [C]  time = 0.035, size = 386, normalized size = 2. \[ -{\frac{2}{7350\,{x}^{2}+735\,x-2205}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 263970\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-524088\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+351960\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-698784\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+117320\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -232928\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{4}[phone]\,{x}^{3}+4798614\,{x}^{2}-5709342\,x-2210613 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

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)^(3/2),x)

[Out]

-2/735*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(263970*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)-524088*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)+351960
*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)-698784*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)+117320*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
))-232928*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^4+23063724*
x^3+4798614*x^2-5709342*x-2210613)/(2+3*x)^(5/2)/(10*x^2+x-3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-2 \, x + 1}}{{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(-2*x + 1)/((135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*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)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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