3.2763 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{7/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{6464 \sqrt{1-2 x} (5 x+3)^{5/2}}{81 \sqrt{3 x+2}}+\frac{74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 (3 x+2)^{3/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}+\frac{11036}{81} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{48478}{729} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{48478 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3645}+\frac{136028 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3645} \]

[Out]

(-48478*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/729 + (11036*Sqrt[1 - 2*x]*Sq
rt[2 + 3*x]*(3 + 5*x)^(3/2))/81 - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(15*(2 + 3
*x)^(5/2)) + (74*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(27*(2 + 3*x)^(3/2)) - (6464*S
qrt[1 - 2*x]*(3 + 5*x)^(5/2))/(81*Sqrt[2 + 3*x]) + (136028*Sqrt[11/3]*EllipticE[
ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3645 - (48478*Sqrt[11/3]*EllipticF[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3645

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Rubi [A]  time = 0.485549, 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 \[ -\frac{6464 \sqrt{1-2 x} (5 x+3)^{5/2}}{81 \sqrt{3 x+2}}+\frac{74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 (3 x+2)^{3/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}+\frac{11036}{81} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{48478}{729} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{48478 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3645}+\frac{136028 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3645} \]

Antiderivative was successfully verified.

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

[Out]

(-48478*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/729 + (11036*Sqrt[1 - 2*x]*Sq
rt[2 + 3*x]*(3 + 5*x)^(3/2))/81 - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(15*(2 + 3
*x)^(5/2)) + (74*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(27*(2 + 3*x)^(3/2)) - (6464*S
qrt[1 - 2*x]*(3 + 5*x)^(5/2))/(81*Sqrt[2 + 3*x]) + (136028*Sqrt[11/3]*EllipticE[
ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3645 - (48478*Sqrt[11/3]*EllipticF[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3645

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Rubi in Sympy [A]  time = 49.5497, size = 201, normalized size = 0.91 \[ - \frac{8906 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{3969 \sqrt{3 x + 2}} - \frac{74 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{189 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{15 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{3208 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{1323} - \frac{35020 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{5103} + \frac{136028 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{10935} - \frac{533258 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{127575} \]

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)**(7/2),x)

[Out]

-8906*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(3969*sqrt(3*x + 2)) - 74*(-2*x + 1)**(5/2
)*(5*x + 3)**(3/2)/(189*(3*x + 2)**(3/2)) - 2*(-2*x + 1)**(5/2)*(5*x + 3)**(5/2)
/(15*(3*x + 2)**(5/2)) - 3208*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*sqrt(5*x + 3)/1323
- 35020*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/5103 + 136028*sqrt(33)*ellip
tic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/10935 - 533258*sqrt(35)*elliptic_f
(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/127575

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Mathematica [A]  time = 0.34477, size = 109, normalized size = 0.49 \[ \frac{\frac{6 \sqrt{1-2 x} \sqrt{5 x+3} \left (24300 x^4-45090 x^3-461043 x^2-517257 x-158237\right )}{(3 x+2)^{5/2}}+\sqrt{2} \left (935915 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-136028 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{10935} \]

Antiderivative was successfully verified.

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

[Out]

((6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-158237 - 517257*x - 461043*x^2 - 45090*x^3 + 2
4300*x^4))/(2 + 3*x)^(5/2) + Sqrt[2]*(-136028*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]], -33/2] + 935915*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/1
0935

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Maple [C]  time = 0.03, size = 396, normalized size = 1.8 \[ -{\frac{1}{109350\,{x}^{2}+10935\,x-32805} \left ( 8423235\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1224252\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1632336\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1458000\,{x}^{6}+3743660\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -544112\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +2559600\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}+4298988\,{x}^{2}-8361204\,x-2848266 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

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)^(7/2),x)

[Out]

-1/10935*(8423235*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)-1224252*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(1/1
1*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)-1632336*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)-1458000*x^6+3743660*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ell
ipticF(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))-54411
2*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))+2559600*x^5+28370520*x^4+32990
058*x^3+4298988*x^2-8361204*x-2848266)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/
(2+3*x)^(5/2)

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

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)^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{3 \, x + 2}}, x\right ) \]

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)^(7/2),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)/((27
*x^3 + 54*x^2 + 36*x + 8)*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)**(7/2),x)

[Out]

Timed out

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

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)^(7/2),x, algorithm="giac")

[Out]

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