3.2711 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx\)

Optimal. Leaf size=280 \[ \frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{1287 (3 x+2)^{11/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}-\frac{20992 \sqrt{1-2 x} (5 x+3)^{3/2}}{81081 (3 x+2)^{9/2}}+\frac{245282464136 \sqrt{1-2 x} \sqrt{5 x+3}}{20440925505 \sqrt{3 x+2}}+\frac{3523482724 \sqrt{1-2 x} \sqrt{5 x+3}}{2920132215 (3 x+2)^{3/2}}+\frac{73596464 \sqrt{1-2 x} \sqrt{5 x+3}}{417161745 (3 x+2)^{5/2}}-\frac{2174468 \sqrt{1-2 x} \sqrt{5 x+3}}{11918907 (3 x+2)^{7/2}}-\frac{7391549624 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1858265955 \sqrt{33}}-\frac{245282464136 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1858265955 \sqrt{33}} \]

[Out]

(-2174468*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(11918907*(2 + 3*x)^(7/2)) + (73596464*Sq
rt[1 - 2*x]*Sqrt[3 + 5*x])/(417161745*(2 + 3*x)^(5/2)) + (3523482724*Sqrt[1 - 2*
x]*Sqrt[3 + 5*x])/(2920132215*(2 + 3*x)^(3/2)) + (245282464136*Sqrt[1 - 2*x]*Sqr
t[3 + 5*x])/(20440925505*Sqrt[2 + 3*x]) - (20992*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/
(81081*(2 + 3*x)^(9/2)) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(39*(2 + 3*x)^(13/
2)) + (362*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(1287*(2 + 3*x)^(11/2)) - (24528246413
6*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1858265955*Sqrt[33]) - (73
91549624*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1858265955*Sqrt[33]
)

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Rubi [A]  time = 0.673338, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{1287 (3 x+2)^{11/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}-\frac{20992 \sqrt{1-2 x} (5 x+3)^{3/2}}{81081 (3 x+2)^{9/2}}+\frac{245282464136 \sqrt{1-2 x} \sqrt{5 x+3}}{20440925505 \sqrt{3 x+2}}+\frac{3523482724 \sqrt{1-2 x} \sqrt{5 x+3}}{2920132215 (3 x+2)^{3/2}}+\frac{73596464 \sqrt{1-2 x} \sqrt{5 x+3}}{417161745 (3 x+2)^{5/2}}-\frac{2174468 \sqrt{1-2 x} \sqrt{5 x+3}}{11918907 (3 x+2)^{7/2}}-\frac{7391549624 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1858265955 \sqrt{33}}-\frac{245282464136 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1858265955 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(15/2),x]

[Out]

(-2174468*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(11918907*(2 + 3*x)^(7/2)) + (73596464*Sq
rt[1 - 2*x]*Sqrt[3 + 5*x])/(417161745*(2 + 3*x)^(5/2)) + (3523482724*Sqrt[1 - 2*
x]*Sqrt[3 + 5*x])/(2920132215*(2 + 3*x)^(3/2)) + (245282464136*Sqrt[1 - 2*x]*Sqr
t[3 + 5*x])/(20440925505*Sqrt[2 + 3*x]) - (20992*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/
(81081*(2 + 3*x)^(9/2)) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(39*(2 + 3*x)^(13/
2)) + (362*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(1287*(2 + 3*x)^(11/2)) - (24528246413
6*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1858265955*Sqrt[33]) - (73
91549624*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1858265955*Sqrt[33]
)

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Rubi in Sympy [A]  time = 66.908, size = 258, normalized size = 0.92 \[ - \frac{1142 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{43659 \left (3 x + 2\right )^{\frac{9}{2}}} - \frac{362 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{9009 \left (3 x + 2\right )^{\frac{11}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{39 \left (3 x + 2\right )^{\frac{13}{2}}} + \frac{245282464136 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{20440925505 \sqrt{3 x + 2}} + \frac{3523482724 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2920132215 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{73596464 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{417161745 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{1254958 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{11918907 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{245282464136 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{61322776515} - \frac{7391549624 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{65039308425} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**(15/2),x)

[Out]

-1142*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(43659*(3*x + 2)**(9/2)) - 362*(-2*x + 1)*
*(3/2)*(5*x + 3)**(3/2)/(9009*(3*x + 2)**(11/2)) - 2*(-2*x + 1)**(3/2)*(5*x + 3)
**(5/2)/(39*(3*x + 2)**(13/2)) + 245282464136*sqrt(-2*x + 1)*sqrt(5*x + 3)/(2044
0925505*sqrt(3*x + 2)) + 3523482724*sqrt(-2*x + 1)*sqrt(5*x + 3)/(2920132215*(3*
x + 2)**(3/2)) + 73596464*sqrt(-2*x + 1)*sqrt(5*x + 3)/(417161745*(3*x + 2)**(5/
2)) + 1254958*sqrt(-2*x + 1)*sqrt(5*x + 3)/(11918907*(3*x + 2)**(7/2)) - 2452824
64136*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/61322776515 -
7391549624*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/65039308
425

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Mathematica [A]  time = 0.478676, size = 117, normalized size = 0.42 \[ \frac{\frac{48 \sqrt{2-4 x} \sqrt{5 x+3} \left (89405458177572 x^6+360618554767050 x^5+606171513555828 x^4+543590753927373 x^3+274263621177573 x^2+73802680969881 x+8272877174903\right )}{(3 x+2)^{13/2}}-1973150325440 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{490582212120 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(15/2),x]

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(8272877174903 + 73802680969881*x + 27426362117
7573*x^2 + 543590753927373*x^3 + 606171513555828*x^4 + 360618554767050*x^5 + 894
05458177572*x^6))/(2 + 3*x)^(13/2) + 3924519426176*EllipticE[ArcSin[Sqrt[2/11]*S
qrt[3 + 5*x]], -33/2] - 1973150325440*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]]
, -33/2])/(490582212120*Sqrt[2])

_______________________________________________________________________________________

Maple [C]  time = 0.06, size = 862, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(15/2),x)

[Out]

2/61322776515*(44950830851430*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^6*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2
)-89405458177572*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^6*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]
5720*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^5*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-357621832710288*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
^5*(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^4*(3+5*x)^(
1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-596036387850480*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)-529810122533760*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)+268
2163745327160*x^8+133187646967200*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)-264905061266880*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)+110867
73017544216*x^7+35516705857920*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)
-70641349671168*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]
842*x^6+3946300650880*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Elliptic
F(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))-7849038852
352*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))+14880670165585224*x^5+440313
7275106857*x^4-1855445492717208*x^3-1998778232441424*x^2-639405497204220*x-74455
894574127)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(13/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{15}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(15/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(15/2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(15/2),x, algorithm="fricas")

[Out]

integral(-(50*x^3 + 35*x^2 - 12*x - 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((2187*x^7 +
10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*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)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**(15/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{15}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(15/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(15/2), x)