3.2857 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{1344984 \sqrt{1-2 x} \sqrt{3 x+2}}{3773 \sqrt{5 x+3}}+\frac{60684 \sqrt{1-2 x}}{1715 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{436 \sqrt{1-2 x}}{245 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{6 \sqrt{1-2 x}}{35 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{40456 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715}+\frac{1344984 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715} \]

[Out]

(6*Sqrt[1 - 2*x])/(35*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (436*Sqrt[1 - 2*x])/(245*
(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (60684*Sqrt[1 - 2*x])/(1715*Sqrt[2 + 3*x]*Sqrt[
3 + 5*x]) - (1344984*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3773*Sqrt[3 + 5*x]) + (134498
4*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1715 + (40456*Sq
rt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1715

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Rubi [A]  time = 0.44351, antiderivative size = 191, 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 \[ -\frac{1344984 \sqrt{1-2 x} \sqrt{3 x+2}}{3773 \sqrt{5 x+3}}+\frac{60684 \sqrt{1-2 x}}{1715 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{436 \sqrt{1-2 x}}{245 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{6 \sqrt{1-2 x}}{35 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{40456 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715}+\frac{1344984 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715} \]

Antiderivative was successfully verified.

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

[Out]

(6*Sqrt[1 - 2*x])/(35*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (436*Sqrt[1 - 2*x])/(245*
(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (60684*Sqrt[1 - 2*x])/(1715*Sqrt[2 + 3*x]*Sqrt[
3 + 5*x]) - (1344984*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3773*Sqrt[3 + 5*x]) + (134498
4*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1715 + (40456*Sq
rt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1715

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Rubi in Sympy [A]  time = 40.1686, size = 172, normalized size = 0.9 \[ - \frac{1344984 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{3773 \sqrt{5 x + 3}} + \frac{60684 \sqrt{- 2 x + 1}}{1715 \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{436 \sqrt{- 2 x + 1}}{245 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}} + \frac{6 \sqrt{- 2 x + 1}}{35 \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}} + \frac{1344984 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{18865} + \frac{40456 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{18865} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1344984*sqrt(-2*x + 1)*sqrt(3*x + 2)/(3773*sqrt(5*x + 3)) + 60684*sqrt(-2*x + 1
)/(1715*sqrt(3*x + 2)*sqrt(5*x + 3)) + 436*sqrt(-2*x + 1)/(245*(3*x + 2)**(3/2)*
sqrt(5*x + 3)) + 6*sqrt(-2*x + 1)/(35*(3*x + 2)**(5/2)*sqrt(5*x + 3)) + 1344984*
sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/18865 + 40456*sqrt(3
3)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/18865

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Mathematica [A]  time = 0.307103, size = 105, normalized size = 0.55 \[ \frac{2 \left (-\frac{\sqrt{1-2 x} \left (90786420 x^3+178568982 x^2+116993058 x+25529443\right )}{(3 x+2)^{5/2} \sqrt{5 x+3}}-6 \sqrt{2} \left (112082 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-56455 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{18865} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-((Sqrt[1 - 2*x]*(25529443 + 116993058*x + 178568982*x^2 + 90786420*x^3))/((
2 + 3*x)^(5/2)*Sqrt[3 + 5*x])) - 6*Sqrt[2]*(112082*EllipticE[ArcSin[Sqrt[2/11]*S
qrt[3 + 5*x]], -33/2] - 56455*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]
)))/18865

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Maple [C]  time = 0.036, size = 386, normalized size = 2. \[ -{\frac{2}{188650\,{x}^{2}+18865\,x-56595}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 3048570\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-6052428\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+4064760\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-8069904\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1354920\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2689968\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-65934172\,x-25529443 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/18865*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(3048570*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)-6052428*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)+40
64760*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)-8069904*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)+1354920*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))-2689968*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+
266351544*x^3+55417134*x^2-65934172*x-25529443)/(2+3*x)^(5/2)/(10*x^2+x-3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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