3.2764 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{1844 \sqrt{1-2 x} (5 x+3)^{5/2}}{567 (3 x+2)^{3/2}}+\frac{74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{63 (3 x+2)^{5/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}-\frac{62596 \sqrt{1-2 x} (5 x+3)^{3/2}}{3969 \sqrt{3 x+2}}+\frac{1353340 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{35721}+\frac{270668 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{35721}-\frac{904798 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{35721} \]

[Out]

(1353340*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/35721 - (62596*Sqrt[1 - 2*x]
*(3 + 5*x)^(3/2))/(3969*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(21
*(2 + 3*x)^(7/2)) + (74*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(63*(2 + 3*x)^(5/2)) -
(1844*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(567*(2 + 3*x)^(3/2)) - (904798*Sqrt[11/3]*
EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/35721 + (270668*Sqrt[11/3]*El
lipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/35721

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Rubi [A]  time = 0.490711, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{1844 \sqrt{1-2 x} (5 x+3)^{5/2}}{567 (3 x+2)^{3/2}}+\frac{74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{63 (3 x+2)^{5/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}-\frac{62596 \sqrt{1-2 x} (5 x+3)^{3/2}}{3969 \sqrt{3 x+2}}+\frac{1353340 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{35721}+\frac{270668 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{35721}-\frac{904798 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{35721} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(9/2),x]

[Out]

(1353340*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/35721 - (62596*Sqrt[1 - 2*x]
*(3 + 5*x)^(3/2))/(3969*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(21
*(2 + 3*x)^(7/2)) + (74*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(63*(2 + 3*x)^(5/2)) -
(1844*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(567*(2 + 3*x)^(3/2)) - (904798*Sqrt[11/3]*
EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/35721 + (270668*Sqrt[11/3]*El
lipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/35721

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Rubi in Sympy [A]  time = 48.8236, size = 201, normalized size = 0.91 \[ - \frac{1310 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{3969 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{74 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{441 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{21 \left (3 x + 2\right )^{\frac{7}{2}}} + \frac{1250 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{189 \sqrt{3 x + 2}} + \frac{181180 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{35721} - \frac{904798 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{107163} + \frac{2977348 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1250235} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**(9/2),x)

[Out]

-1310*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(3969*(3*x + 2)**(3/2)) - 74*(-2*x + 1)**(
5/2)*(5*x + 3)**(3/2)/(441*(3*x + 2)**(5/2)) - 2*(-2*x + 1)**(5/2)*(5*x + 3)**(5
/2)/(21*(3*x + 2)**(7/2)) + 1250*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(189*sqrt(3*x +
2)) + 181180*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/35721 - 904798*sqrt(33)
*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/107163 + 2977348*sqrt(35)*el
liptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/1250235

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Mathematica [A]  time = 0.292841, size = 109, normalized size = 0.49 \[ \frac{2 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (396900 x^4+9846603 x^3+17788023 x^2+11107911 x+2337569\right )}{(3 x+2)^{7/2}}+\sqrt{2} \left (452399 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2685410 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{107163} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(9/2),x]

[Out]

(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2337569 + 11107911*x + 17788023*x^2 + 984660
3*x^3 + 396900*x^4))/(2 + 3*x)^(7/2) + Sqrt[2]*(452399*EllipticE[ArcSin[Sqrt[2/1
1]*Sqrt[3 + 5*x]], -33/2] - 2685410*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]],
-33/2])))/107163

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Maple [C]  time = 0.03, size = 510, normalized size = 2.3 \[{\frac{2}{1071630\,{x}^{2}+107163\,x-321489} \left ( 72506070\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-12214773\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-24429546\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-16286364\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -3619192\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-56641404\,{x}^{2}-92958492\,x-21038121 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(9/2),x)

[Out]

2/107163*(72506070*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*1
1^(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)-12214773*
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)-24429546*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)-162863
64*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]*x^6+21483280*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))-3619192*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+559608399*x^4+297981972*x^3-56641404*x^2-929584
92*x-21038121)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(9/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(9/2),x, algorithm="fricas")

[Out]

integral((100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((81
*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*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)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(9/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(9/2), x)