3.2788 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^{11/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=253 \[ \frac{14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt{5 x+3}}-\frac{3415750480 \sqrt{3 x+2} \sqrt{1-2 x}}{27783 \sqrt{5 x+3}}+\frac{113020952 \sqrt{1-2 x}}{9261 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{813208 \sqrt{1-2 x}}{1323 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{11660 \sqrt{1-2 x}}{189 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{652 \sqrt{1-2 x}}{81 (3 x+2)^{7/2} \sqrt{5 x+3}}+\frac{20549264 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9261}+\frac{683150096 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9261} \]

[Out]

(14*(1 - 2*x)^(3/2))/(27*(2 + 3*x)^(9/2)*Sqrt[3 + 5*x]) + (652*Sqrt[1 - 2*x])/(8
1*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x]) + (11660*Sqrt[1 - 2*x])/(189*(2 + 3*x)^(5/2)*Sq
rt[3 + 5*x]) + (813208*Sqrt[1 - 2*x])/(1323*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (11
3020952*Sqrt[1 - 2*x])/(9261*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (3415750480*Sqrt[1 -
2*x]*Sqrt[2 + 3*x])/(27783*Sqrt[3 + 5*x]) + (683150096*Sqrt[11/3]*EllipticE[Arc
Sin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/9261 + (20549264*Sqrt[11/3]*EllipticF[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/9261

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Rubi [A]  time = 0.613359, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt{5 x+3}}-\frac{3415750480 \sqrt{3 x+2} \sqrt{1-2 x}}{27783 \sqrt{5 x+3}}+\frac{113020952 \sqrt{1-2 x}}{9261 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{813208 \sqrt{1-2 x}}{1323 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{11660 \sqrt{1-2 x}}{189 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{652 \sqrt{1-2 x}}{81 (3 x+2)^{7/2} \sqrt{5 x+3}}+\frac{20549264 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9261}+\frac{683150096 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9261} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(5/2)/((2 + 3*x)^(11/2)*(3 + 5*x)^(3/2)),x]

[Out]

(14*(1 - 2*x)^(3/2))/(27*(2 + 3*x)^(9/2)*Sqrt[3 + 5*x]) + (652*Sqrt[1 - 2*x])/(8
1*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x]) + (11660*Sqrt[1 - 2*x])/(189*(2 + 3*x)^(5/2)*Sq
rt[3 + 5*x]) + (813208*Sqrt[1 - 2*x])/(1323*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (11
3020952*Sqrt[1 - 2*x])/(9261*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (3415750480*Sqrt[1 -
2*x]*Sqrt[2 + 3*x])/(27783*Sqrt[3 + 5*x]) + (683150096*Sqrt[11/3]*EllipticE[Arc
Sin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/9261 + (20549264*Sqrt[11/3]*EllipticF[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/9261

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Rubi in Sympy [A]  time = 56.5091, size = 230, normalized size = 0.91 \[ \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}}}{27 \left (3 x + 2\right )^{\frac{9}{2}} \sqrt{5 x + 3}} - \frac{683150096 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{9261 \sqrt{3 x + 2}} - \frac{49155320 \sqrt{- 2 x + 1}}{3969 \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{813208 \sqrt{- 2 x + 1}}{1323 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}} + \frac{11660 \sqrt{- 2 x + 1}}{189 \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}} + \frac{652 \sqrt{- 2 x + 1}}{81 \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}} + \frac{683150096 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{27783} + \frac{20549264 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{27783} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)/(2+3*x)**(11/2)/(3+5*x)**(3/2),x)

[Out]

14*(-2*x + 1)**(3/2)/(27*(3*x + 2)**(9/2)*sqrt(5*x + 3)) - 683150096*sqrt(-2*x +
1)*sqrt(5*x + 3)/(9261*sqrt(3*x + 2)) - 49155320*sqrt(-2*x + 1)/(3969*sqrt(3*x
+ 2)*sqrt(5*x + 3)) + 813208*sqrt(-2*x + 1)/(1323*(3*x + 2)**(3/2)*sqrt(5*x + 3)
) + 11660*sqrt(-2*x + 1)/(189*(3*x + 2)**(5/2)*sqrt(5*x + 3)) + 652*sqrt(-2*x +
1)/(81*(3*x + 2)**(7/2)*sqrt(5*x + 3)) + 683150096*sqrt(33)*elliptic_e(asin(sqrt
(21)*sqrt(-2*x + 1)/7), 35/33)/27783 + 20549264*sqrt(33)*elliptic_f(asin(sqrt(21
)*sqrt(-2*x + 1)/7), 35/33)/27783

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Mathematica [A]  time = 0.404677, size = 115, normalized size = 0.45 \[ \frac{2 \left (-\frac{3 \sqrt{1-2 x} \left (138337894440 x^5+456548966244 x^4+602551975428 x^3+397527527442 x^2+131099014240 x+17289178827\right )}{(3 x+2)^{9/2} \sqrt{5 x+3}}-4 \sqrt{2} \left (85393762 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-43010905 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{27783} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^(11/2)*(3 + 5*x)^(3/2)),x]

[Out]

(2*((-3*Sqrt[1 - 2*x]*(17289178827 + 131099014240*x + 397527527442*x^2 + 6025519
75428*x^3 + 456548966244*x^4 + 138337894440*x^5))/((2 + 3*x)^(9/2)*Sqrt[3 + 5*x]
) - 4*Sqrt[2]*(85393762*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 430
10905*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/27783

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Maple [C]  time = 0.037, size = 624, normalized size = 2.5 \[ -{\frac{2}{277830\,{x}^{2}+27783\,x-83349}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 13935533220\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-27667578888\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-73780210368\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-73780210368\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-32791204608\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -5465200768\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-405988496886\,{x}^{2}-289561969758\,x-51867536481 \right ) \left ( 2+3\,x \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)/(2+3*x)^(11/2)/(3+5*x)^(3/2),x)

[Out]

-2/27783*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(13935533220*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^4*(3+5*x)^(1/2)*(2+3*x
)^(1/2)*(1-2*x)^(1/2)-27667578888*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^4*(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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)-7378021036
8*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)*Ellipt
icF(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)-73780210368*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)-32791204608*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+2752697920*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/11*1
1^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))-5465200768*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+2245664953836*x^4+577
509238368*x^3-405988496886*x^2-289561969758*x-51867536481)/(2+3*x)^(9/2)/(10*x^2
+x-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(11/2)),x, algorithm="maxima")

[Out]

integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(11/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}{{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(11/2)),x, algorithm="fricas")

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(-2*x + 1)/((1215*x^6 + 4779*x^5 + 7830*x^4 + 684
0*x^3 + 3360*x^2 + 880*x + 96)*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)**(5/2)/(2+3*x)**(11/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(11/2)),x, algorithm="giac")

[Out]

integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(11/2)), x)