3.2866 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=218 \[ \frac{352875016 \sqrt{1-2 x} \sqrt{3 x+2}}{124509 \sqrt{5 x+3}}-\frac{5307272 \sqrt{1-2 x} \sqrt{3 x+2}}{11319 (5 x+3)^{3/2}}+\frac{120324 \sqrt{1-2 x}}{1715 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{576 \sqrt{1-2 x}}{245 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{6 \sqrt{1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{10614544 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18865 \sqrt{33}}-\frac{352875016 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18865 \sqrt{33}} \]

[Out]

(6*Sqrt[1 - 2*x])/(35*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (576*Sqrt[1 - 2*x])/(24
5*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (120324*Sqrt[1 - 2*x])/(1715*Sqrt[2 + 3*x]*
(3 + 5*x)^(3/2)) - (5307272*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11319*(3 + 5*x)^(3/2))
+ (352875016*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(124509*Sqrt[3 + 5*x]) - (352875016*E
llipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(18865*Sqrt[33]) - (10614544*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(18865*Sqrt[33])

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Rubi [A]  time = 0.526557, 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 \[ \frac{352875016 \sqrt{1-2 x} \sqrt{3 x+2}}{124509 \sqrt{5 x+3}}-\frac{5307272 \sqrt{1-2 x} \sqrt{3 x+2}}{11319 (5 x+3)^{3/2}}+\frac{120324 \sqrt{1-2 x}}{1715 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{576 \sqrt{1-2 x}}{245 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{6 \sqrt{1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{10614544 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18865 \sqrt{33}}-\frac{352875016 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18865 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(6*Sqrt[1 - 2*x])/(35*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (576*Sqrt[1 - 2*x])/(24
5*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (120324*Sqrt[1 - 2*x])/(1715*Sqrt[2 + 3*x]*
(3 + 5*x)^(3/2)) - (5307272*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11319*(3 + 5*x)^(3/2))
+ (352875016*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(124509*Sqrt[3 + 5*x]) - (352875016*E
llipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(18865*Sqrt[33]) - (10614544*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(18865*Sqrt[33])

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Rubi in Sympy [A]  time = 46.6249, size = 201, normalized size = 0.92 \[ \frac{352875016 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{124509 \sqrt{5 x + 3}} - \frac{5307272 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{11319 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{120324 \sqrt{- 2 x + 1}}{1715 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{576 \sqrt{- 2 x + 1}}{245 \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{6 \sqrt{- 2 x + 1}}{35 \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{352875016 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{622545} - \frac{10614544 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{622545} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

352875016*sqrt(-2*x + 1)*sqrt(3*x + 2)/(124509*sqrt(5*x + 3)) - 5307272*sqrt(-2*
x + 1)*sqrt(3*x + 2)/(11319*(5*x + 3)**(3/2)) + 120324*sqrt(-2*x + 1)/(1715*sqrt
(3*x + 2)*(5*x + 3)**(3/2)) + 576*sqrt(-2*x + 1)/(245*(3*x + 2)**(3/2)*(5*x + 3)
**(3/2)) + 6*sqrt(-2*x + 1)/(35*(3*x + 2)**(5/2)*(5*x + 3)**(3/2)) - 352875016*s
qrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/622545 - 10614544*sqr
t(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/622545

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Mathematica [A]  time = 0.349505, size = 109, normalized size = 0.5 \[ \frac{2 \left (\frac{\sqrt{1-2 x} \left (119095317900 x^4+305707177080 x^3+294023389014 x^2+125573817736 x+20093773321\right )}{(3 x+2)^{5/2} (5 x+3)^{3/2}}+4 \sqrt{2} \left (44109377 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-22216880 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{622545} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[1 - 2*x]*(20093773321 + 125573817736*x + 294023389014*x^2 + 3057071770
80*x^3 + 119095317900*x^4))/((2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + 4*Sqrt[2]*(44109
377*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 22216880*EllipticF[ArcS
in[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/622545

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Maple [C]  time = 0.037, size = 502, normalized size = 2.3 \[{\frac{2}{-622545+1245090\,x}\sqrt{1-2\,x} \left ( 3999038400\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-7939687860\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-15350063196\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-9880500448\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2117250096\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-42875753542\,{x}^{2}-85386271094\,x-20093773321 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/622545*(1-2*x)^(1/2)*(3999038400*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)-7939687860*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)-15350063196*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^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)-9880500448*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)+1
066410240*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))-2117250096*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+492319036260*x^4+282339600
948*x^3-42875753542*x^2-85386271094*x-20093773321)/(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. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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