3.2898 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{81164 \sqrt{1-2 x} \sqrt{5 x+3}}{108045 \sqrt{3 x+2}}-\frac{15601 \sqrt{1-2 x} \sqrt{5 x+3}}{15435 (3 x+2)^{3/2}}+\frac{163 \sqrt{1-2 x} \sqrt{5 x+3}}{735 (3 x+2)^{5/2}}-\frac{28174 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045}+\frac{81164 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045} \]

[Out]

(163*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(735*(2 + 3*x)^(5/2)) - (15601*Sqrt[1 - 2*x]*S
qrt[3 + 5*x])/(15435*(2 + 3*x)^(3/2)) - (81164*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(108
045*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) + (8
1164*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/108045 - (281
74*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/108045

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Rubi [A]  time = 0.423821, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{81164 \sqrt{1-2 x} \sqrt{5 x+3}}{108045 \sqrt{3 x+2}}-\frac{15601 \sqrt{1-2 x} \sqrt{5 x+3}}{15435 (3 x+2)^{3/2}}+\frac{163 \sqrt{1-2 x} \sqrt{5 x+3}}{735 (3 x+2)^{5/2}}-\frac{28174 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045}+\frac{81164 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045} \]

Antiderivative was successfully verified.

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

[Out]

(163*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(735*(2 + 3*x)^(5/2)) - (15601*Sqrt[1 - 2*x]*S
qrt[3 + 5*x])/(15435*(2 + 3*x)^(3/2)) - (81164*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(108
045*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) + (8
1164*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/108045 - (281
74*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/108045

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Rubi in Sympy [A]  time = 38.2695, size = 172, normalized size = 0.9 \[ - \frac{81164 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{108045 \sqrt{3 x + 2}} - \frac{15601 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{15435 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{163 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{735 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{81164 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{324135} - \frac{309914 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{3781575} + \frac{11 \left (5 x + 3\right )^{\frac{3}{2}}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81164*sqrt(-2*x + 1)*sqrt(5*x + 3)/(108045*sqrt(3*x + 2)) - 15601*sqrt(-2*x + 1
)*sqrt(5*x + 3)/(15435*(3*x + 2)**(3/2)) + 163*sqrt(-2*x + 1)*sqrt(5*x + 3)/(735
*(3*x + 2)**(5/2)) + 81164*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7),
35/33)/324135 - 309914*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/
35)/3781575 + 11*(5*x + 3)**(3/2)/(7*sqrt(-2*x + 1)*(3*x + 2)**(5/2))

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Mathematica [A]  time = 0.250031, size = 104, normalized size = 0.54 \[ \frac{\frac{6 \sqrt{5 x+3} \left (730476 x^3+936351 x^2+292777 x-4877\right )}{\sqrt{1-2 x} (3 x+2)^{5/2}}+\sqrt{2} \left (546035 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-81164 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{324135} \]

Antiderivative was successfully verified.

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

[Out]

((6*Sqrt[3 + 5*x]*(-4877 + 292777*x + 936351*x^2 + 730476*x^3))/(Sqrt[1 - 2*x]*(
2 + 3*x)^(5/2)) + Sqrt[2]*(-81164*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -3
3/2] + 546035*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/324135

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Maple [C]  time = 0.036, size = 386, normalized size = 2. \[ -{\frac{1}{3241350\,{x}^{2}+324135\,x-972405}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 4914315\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-730476\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+6552420\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-973968\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+2184140\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -324656\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}+5123676\,x-87786 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/324135*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(4914315*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)-730476*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)+65
52420*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)-973968*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)+2184140*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))-324656*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^4+412
39098*x^3+25637628*x^2+5123676*x-87786)/(2+3*x)^(5/2)/(10*x^2+x-3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^(7/2)*(-2*x + 1)^(3/2)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}{{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^(7/2)*(-2*x + 1)^(3/2)),x, algorithm="fricas")

[Out]

integral(-(25*x^2 + 30*x + 9)*sqrt(5*x + 3)/((54*x^4 + 81*x^3 + 18*x^2 - 20*x -
8)*sqrt(3*x + 2)*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((3+5*x)**(5/2)/(1-2*x)**(3/2)/(2+3*x)**(7/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^(7/2)*(-2*x + 1)^(3/2)),x, algorithm="giac")

[Out]

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