3.2921 \(\int \frac{(2+3 x)^{5/2}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=156 \[ \frac{7 (3 x+2)^{3/2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{4157 \sqrt{1-2 x} \sqrt{3 x+2}}{19965 \sqrt{5 x+3}}-\frac{107 \sqrt{1-2 x} \sqrt{3 x+2}}{1815 (5 x+3)^{3/2}}-\frac{412 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3025 \sqrt{33}}+\frac{4157 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3025 \sqrt{33}} \]

[Out]

(-107*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1815*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^(3/2))/
(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (4157*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(19965*S
qrt[3 + 5*x]) + (4157*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3025*S
qrt[33]) - (412*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3025*Sqrt[33
])

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Rubi [A]  time = 0.337073, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{7 (3 x+2)^{3/2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{4157 \sqrt{1-2 x} \sqrt{3 x+2}}{19965 \sqrt{5 x+3}}-\frac{107 \sqrt{1-2 x} \sqrt{3 x+2}}{1815 (5 x+3)^{3/2}}-\frac{412 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3025 \sqrt{33}}+\frac{4157 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3025 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-107*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1815*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^(3/2))/
(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (4157*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(19965*S
qrt[3 + 5*x]) + (4157*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3025*S
qrt[33]) - (412*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3025*Sqrt[33
])

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Rubi in Sympy [A]  time = 31.4805, size = 143, normalized size = 0.92 \[ - \frac{4157 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{19965 \sqrt{5 x + 3}} - \frac{107 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{1815 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{4157 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{99825} - \frac{412 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{105875} + \frac{7 \left (3 x + 2\right )^{\frac{3}{2}}}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4157*sqrt(-2*x + 1)*sqrt(3*x + 2)/(19965*sqrt(5*x + 3)) - 107*sqrt(-2*x + 1)*sq
rt(3*x + 2)/(1815*(5*x + 3)**(3/2)) + 4157*sqrt(33)*elliptic_e(asin(sqrt(21)*sqr
t(-2*x + 1)/7), 35/33)/99825 - 412*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x +
1)/11), 33/35)/105875 + 7*(3*x + 2)**(3/2)/(11*sqrt(-2*x + 1)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.378212, size = 97, normalized size = 0.62 \[ \frac{\sqrt{2} \left (\frac{5 \sqrt{6 x+4} \left (20785 x^2+22313 x+5881\right )}{\sqrt{1-2 x} (5 x+3)^{3/2}}+10955 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-4157 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{99825} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2]*((5*Sqrt[4 + 6*x]*(5881 + 22313*x + 20785*x^2))/(Sqrt[1 - 2*x]*(3 + 5*x
)^(3/2)) - 4157*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 10955*Ellip
ticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/99825

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Maple [C]  time = 0.033, size = 267, normalized size = 1.7 \[ -{\frac{1}{598950\,{x}^{2}+99825\,x-199650}\sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 54775\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-20785\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+32865\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -12471\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +623550\,{x}^{3}+1085090\,{x}^{2}+622690\,x+117620 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/99825*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(54775*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)-20785*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)+32865*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))-12471*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))+623550*x^3+1085090*x^2+622690*x+117620)/(3+5*x)^(3/2)/(6*x^2+x
-2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{3 \, x + 2}}{{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(9*x^2 + 12*x + 4)*sqrt(3*x + 2)/((50*x^3 + 35*x^2 - 12*x - 9)*sqrt(5*
x + 3)*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((2+3*x)**(5/2)/(1-2*x)**(3/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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